
Class 
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CDESIIGHT DEPOSIT. 



WORKS OF 
WALTER LORING WEBB 

PUBLISHED BY 

JOHN WILEY & SONS, Inc. 



Railroad Construction. — Theory and Practice. 

A Text-book for th^ Use of Students in Col- 
leges and Technical Schools. Seventh Edition. 
Rewritten and Enlarged, xvii -f- 847 pages, 
4iby6f; 225 figures and 10 plates. Flexible 
Binding, $5.00 net. 

Technic of Surveying Instruments and 
Methods. 

Including General and Detailed Instructions 
for Field and Office Work of Extended Stu- 
dents' Surveys. In collaboration with Prof. 
J. C. L. Fish, xiv + 319 pages, 4 by 6^^; 59 
figures. Fexible Binding; $2.50 net. 

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{Author of Section on Railroads.) 

4 M by 7. Flexible Binding, $6.00 net. 



RAILROAD CONSTRUCTION 

THEORY AND PRACTICE 

A TEXT-BOOK FOR THE USE OF STUDENTS 
IN COLLEGES AND TECHNICAL SCHOOLS, 

AND 

A HAND-BOOK FOR THE USE OF ENGINEERS 
IN FIELD AND OFFICE, 



BT 



WALTER LORING WEBB, C.E., 

It 

Member Arnerican Society of Civil Engineers; Member American Railway 

Engineering Association; Assistant Professor of Civil- Engineering (RailT 

road Engineering) in the University of Pennsylvania, 1893-1901' 

Major, Engineer Corps, U. S. A., 1917-1920; etc. 



SEVENTH EDITION, REVISED AND ENLARGED 

TOTAL ISSUE, SEVENTEEN THOUSAND 



NEW YORK 

JOHN WILEY & SONS, Ii^c. 

London: CHAPMAN & HALL, Limited 

1922 



^f' 









Y^^V^ 



Copyright, 1899, 1903, 1908, 1913, 1917, 1922 

BY 

WALTER LORING WEBB. 



/ 



I 1/ 



' r- 



PRESS OF 

BRAUNWORTH & CO. 

BOOK MANUFACTURERS 

BROOKLYN, N. Y. 



OCi 16 V;2 

C1A683736 



PREFACE TO SEVENTH EDITION 



The author wishes to reiterate, with ' even greater emphasis, 
the statement made in the second paragraph of the preface to 
the sixth edition. There are few, outside of railroad circles, 
who realize the great work which is being accomplished by the 
American Railway Engineering Association. Much of this work 
has been done during the past five years. One of the notable 
features is the work of the Special Committee on " Stresses in 
Track." A very condensed account of the work of this Com- 
mittee is given in the new added Chapter XXV. Numerous 
corrections and revisions have also been made throughout this 
edition to make it conform to the decisions of the recent con- 
ventions of the Association. 

Some of the more important changes, additions, or develop- 
ments of subjects, which have been made in this edition, are as 
follows : 

(a) The shrinkage of embankments and the subsidence of sub- 
soil under them — Chapter III. 

(6) Laws governing the life of ties; developments in sub- 
stitutes for wooden ties — Chapter VIII. 

(c) Rails; present status of specification^' testing; life of 
rails; failures; intensity of pressure; rail weS|^Chapter IX. 

(d) Rail joints; causes of failures — Chapter feC. ^ 

(e) Water tanks; principles of constructionf^-^aptef; XII. 
(/) Yards and terminals ; hump yards; gwme^-Chapter 

XIII (nearly rewritten). | ^ 

(g) Train resistance; resistance of passenger cars, frei^WQars; 

resistance through switches — Chapter XVI. t "^ 

(h) Stresses in track, in rails, ties and ballast; static and 

dynamic stresses — Chapter XXV (new). 

Walter Loring Webb. 

Philadelphia, Pa., 
Dec., 1921. 









15^ 




PREFACE TO SIXTH EDITION. 



•The revision of the fifth edition has been so extensive that 
it has almost amounted to a rewriting of the book. Compara- 
tively few pages have been left without some revision. 

The last few years have seen a greater advance in the science 
of railroad construction than any similar period in its previous 
history. This has been largely due to the combined work of 
the several Standing Committees of the American Railway 
Engineering Association. The writer has received special per- 
mission to quote from the Association's publications and has 
availed himself of the privilege, because he considers that the 
decisions of such an Association are, in general, the highest 
authority obtainable. 

Considerable new matter has been added on the general sub- 
ject of railroad surveys, and the handling of surveying parties. 
One feature of the additions has been the emergency medical 
and surgical treatment which the engineer-in-charge, as respon- 
sible head of the party, must sometimes supply when regular 
professional advice is absolutely unobtainable and the engineer 
must choose between seeing the victim die (or become perma- 
nently injured), or assuming the unwelcome responsibility of 
applying simple instructions plus common sense. It usually 
means choosing the lesser of two evils. The author wishes to 
acknowledge his indebtedness to his friends, Dr. G. Victor 
Janvier and Dr. Henry P. DeForest, for advice and the revision 
of these sections, which may thus be depended on to be tech- 
nically correct. 

Those familiar with the former editions of this work will note 
that the computations previously given for the unit values of 
saving one foot (or mile) of distance, one degree of curvature, or 
one foot of rise-and-fall, have now been omitted. This is due 
to the belief, as expressed by the Economics Committee of the 

y 



VI PREFACE TO SIXTH EDITION". 

Am. Rwy. Eng. Assoc, that all previously published methods 
of making such calculations are unreliable since they ignore 
certain operating conditions peculiar to each road, and that the 
application of such unit figures may lead to unwarranted con- 
clusions. It may be that a method will be sometime dfevised 
by which some simple and satisfactory form of unit value may 
be used. At present, the most practicable method yet proposed 
is to compute the costs of -operating two suggested routes on 
the basis of an assumed amount and kind of traffic and compare 
the results. 

Walter Loring Webb. 
Philadelphia, Pa., 
Nov., 1916. 



TABLE OF CONTENTS. 



CHAPTER I. 

EAILROAD SURVEYS. 



PAGE 
Reconnoissance . , 1 

1. Cjtiaracter of a reconri,oissance survey. 2. Selection of a gen- 
eral route. 3. Valley route. 4. Cross-country route. 5. Moun- 
tain route. 6. Existing maps. 7. Determination of relative 
elevations. Barometrical method. 8. Horizontal measurements, 
bearings, etc. 9. Importance of a good reconnoissance. 

Preliminary surveys, . , 14 

10. Character of a survey. 11. Cross-section method. 12. 
Cross-sectioning. 13. Stadia method. 14. Form for stadia notes. 
15. The reduction of stadia observations. 16. Stadia method vs. 
cross-section method. 17. "First" an4 "second" preliminary 
surveys. 

Location surveys ...,., 24 

18. "Paper location." 19. Preparation of the notes. 20. 
Surveying methods. 21. Form of notes. 22. Number of men 
required in surveying parties. 

Maintenance op surveying parties 36 

23. Economy and efficiency. 24. Country hotels and farm 
houses. 25. Camping outfits. 26. Tent floors. 27. Tent stoves. 
28. Dining tables. . 29, Cooking utensils, table ware, tools, etc. 
30. Drawing tables. 31. Stationery and map chest. 32. Pro- 
visions. 33. Beds. 34." Transportation. 35. Clothing. 

Medical and surgical treatment. ;, 47 

36. Responsibility of engineer-in-charge. 37. Appliances. 38. 
Antiseptics. 3.9. Drinking water. 40. Bleeding. 41. Ailments 
and diseases; medicines. 42. Drowning; electric shock; as- 
phyxiation. 43. Fractures. 44. Snake or insect bites. 45. 
Wounds. 

CHAPTER II. 

alinement. 
Simple curves 55 

46. Designation of curves. 47. Metric curves. 48. Length 
of a subchord. 49. Length of a curve. 50. Curve notation. 51. 
Elements of a curve. 52. Relation between T, E, and A. 53. 
Elements of a 1° curye. 54. Exercises. 55. Curve location by 
deflections. 56. Instrumental work. 57. Curve location by 

vii 



viii TABLE OF CONTENTS. 

PAGB 

two transits. 58. Curve location by tangential offsets. 59. Curve 
location by middle ordinates. 60. Curve location by offsets from 
the long chord. 61, Use and value of the above methods. 62. 
Obstacles to location. 63. Modifications of location. 64. Limita- 
tions in location. 65. Determination of the curvature of existing 
track. 66. Problems. 

Compound curves 77 

67. Nature and use. 68. Mutual relations of the parts of a 
compound curve having two branches. 69. Modifications of loca- 
tion. 70. Problems. 

Transition curves 82 

71. Superelevation of the outer rail on curves. 72. Practical 
rules for superelevation. 73. Transition from level to inclined 
track. 74. Fundamental principle of transition curves. 75. Va- 
rieties of transition curves. 76. Proper length of spiral. 77. 
Symbols. 78. Deflections. 79. Location of spirals and circular 
curve with respect to tangents. 80. Field-work. 81. To replace 
a simple curve by a curve with spirals. 82. Application of tran- 
sition curves to compound curves. 83. To replace a compound 
curve by a curve with spirals. 

Vertical, curves 100 

84. Necessity for their use. 85. Required length. 86. Form 
of curve. 87. Numerical example. 

CHAPTER III. 

earthwork. 

Form of excavations and embankments 104 

88. Usual form of cross-section in cut and fill. 89. Terminal 
pyramids and wedges. 90. Slopes. 91. Compound sections. 
92. Width of roadbed. 93. Form of subgrade. 94. Ditches. 
95. Effect of sodding the slopes, etc. 

Earthwork surveys 112 

96. Relation of actual volume to the numerical results. 97. 
Prismoids. 98. Cross-sectioning. 99. Position of slope-stakes. 
100. Setting slope-stakes by means of "automatic" slope-stake 
rods. 

Computation of volume 118 

101. Simple approximations. 102. Approximate volume, level 
sections. 103. Numerical example, level sections. 104. Equiva- 
lent sections. 105. Three-level sections. 106. Computation of 
products. 107. Irregular sections. 108. Volume of an irregular 
prismoid. 109. Numerical example; approximate volume, irreg- 
ular sections. 110. Prismoidal correction. 111. Correction for 
triangular prismoid. 112. Correction for level sections. 113. 
Prismoidal correction for "equivalent" sections. 114. Prismoidal 
correction for three-level sections. 115. Prismoidal correction; 
irregular sections. 116. Magnitude of the probable error of this 
method. 117. Numerical illustration of the accuracy of the ap- 
proximate rule. 118. Cross-sectioning irregular sections. 119. 



TABLE OF CONTENTS. IX 

PAGE 

Side-hill work. 120. Borrow-pits. 121. Correction for curvature. 
122. Eccentricity of the center of gravity. 123. Center of gravity 
of side-hill sections. 124. Example of curvature correction. 125. 
Accuracy of earthwork computations. 126. Approximate com- 
putations from profiles. 

Formation of embankments 149 

127. Shrinkage of earthwork. 128. Proper allowance for shrink- 
age or subsidence. 129. Methods of forming embankments. 

Computation of haul 155 

130. Nature of subject. 131. Mass diagram. 132. Properties 
of the mass curve. 133. Area of the mass curve. 134. Value of 
the mass diagram. 135. Changing the grade line. 136. Limit of 
free haul. 

Elements of the cost of earthwork 163 

137. Analysis of the total cost into items. 138. Loosening. 
139. Loading. 140. Hauling. 141. Choice of method of haul 
dependent on distance. " 142. Spreading. 143. Keeping roadways 
in order. 144. Trimming cuts to their proper cross-section. 

145. Repairs, wear, depreciation, and interest on cost of plant. 

146. Superintendence and incidentals. 147. Contractor's profit 
and contingencies. 148. Limit of profitable haul. 

Blasting 184 

149. Explosives, 150. Drilling. 151. Position and direction 
of drill-holes, 152. Amount of explosive. 153. Tamping. 154. 
Exploding the charge. 155. Cost. 156. Classification of ex- 
cavated material. 157. Specifications for earthwork. 

CHAPTER IV. 

trestles. 

158. Extent of use. 159. Trestles vs. embankments. 160. Two 

principal types 194 

Pile trestles 196 

161. Pile bents. 162. Methods of driving piles. 163. Pile- 
driving formulae. 164. Pile-points and pile-shoes, 165. Details 
of design. 166. Specifications for timber piles. 167. Pile-driving 
— principles of practice. 168. Cost of pile trestles. 

Framed trestles 205 

169. Typical design. 170. Joints. 17L Multiple-story con- 
struction. 172. Span. 173. Foundations. 174. Longitudinal 
bracing. 175. Lateral bracing. 176. Abutments. 

Floor systems 211 

177. Stringers. 178. Corbels. 179. Guard-rails. 180. Ties on 
trestles. 181. Superelevation of the outer rail on curves. 182. 
Protection from fire. 183. Timber. 184. Cost of framed timber 
trestles. 

Design op wooden trestles 217 

185. Common practice. 186. Required elements of strength, 
187. Strength of timber. 188. Loading. 189. Factors of safety. 
190. Design of stringers. 191. Design of posts. IQ2, Desigp 
Qf cap^ £^nd sills, 1Q3, Bracing, 



^ TABLE OF CONTENTS. 

CHAPTER V. 

TUNNELS. 

PAGE 

Surveying. 227 

194. Surface surveys. 105. Surveying down a shaft. l96. 
Underground surveys. 197. Accuracy of tunnel surveying. 

Design • 25f2 

198. Cross-sections. l99. Gfrade. 200. Lining. 201. Shafts. 
202. Drains. 

CoNS'TRucf ION ....;. 237 

203. Headings. 204. Enlargement. 205. Distinctive features 
of various methods of construction. 206. Ventilation during con- 
struction. 207. Excavation for the portals. 208. Tunnels vs. 
open cuts. 209. Cost of tunneling. 

CHAPTER Vl 

CULVERTS AND MINOR BRIDGES. 

^10. Definitiofi and object. 211. Elements of the design. .... . 245 

Area of the waterway- , 246 

212. Elements involved. 213. Methods of computation of area. 
214. Empirical formulae. 2l5. Value of empirical formulse. 216. 
Results based on observation. 217. Degree of accuracy required. 

Pipe culverts. 250 

218. Advantages. 219. Construction. 220. Iron-pipe culverts. 
221. Tile-pipe culverts. 

Box CULVERTS 254 

222. Wooden box culverts. 223. Stone box culverts. 224. Old . 
rail culverts. 225. Reinforced concrete culverts. 

Arch culverts 258 

226. Influence of design on flow. 227. Examples of arch-cul- 
vert design. 

Minor openings , 260 

228. Cattle-guards. 229. Cattle-passes. 230. Standard stringer 
and I-beam bridges. 

CHAPTER VII. 

ballast. 

231. Purpose and requirements. 232. Materials. 233, Cross- 
sections. 234. Classification of railroads. 235. Recommended 
sections for the several classifications. 236. Proper depth of ballast. 
237. Methods of laying ballast. 238. Cost. 238a. Specifications. 265 

CHAPTER VIII. 

TIES AND OTHER FORMS OF RAIL SUPPORT. , . :.' 

239. Various methods of supporting rails. 240. Economics 8f 

ties. . . ........ o ..... 276 

WoODEN TIES. ......;... 277 

241. Choice bi wood. 242. Durability. 2^3. Dimensions. 
244. Spacing. 245. Specifications. 246. Regulatioiis for lajdng 
and renewing ties. 247. Dating nails. 248. Cost of ties. 



TABLE OF CONTENTS. XI 



PAGE 

Preservative processes for wooden ties 282 

249. General principle. 250. Creosoting. 251. Burnfettizihg. 
252. Kyanizing. 253. Zinc-tannin t)rocess. 254. Zinc-creosote 
emulsion process. 255. Two-injection zinc creosote process. 256. 
Cost of treating. 257. Economics of treated ties. 

Metal TIES. 290 

258. Extent of use. 259. FOrms and dimensions of some metal 
ties. 260. Durability. 261. Econoinics of steel ties. 263. Bowls 
or plates. 264. Longitudinals. 265. Reinforced concrete ties. 

CHAPTER IX. 

*■ rails. 

266. Early forms. 267. Present standard forms. 268. Weight 
for various kinds of traffic. 269. Effect of stiffness on traction. 
270. Length of rails. 271. Expafision of rails. 272. Rules for 
allowing for temperature. 273. Standard specifications. 273a. 
Chemical composition. 2735. Physical requirements. 273c. Clas- 
sifica;tion. 273d. Branding. 273e. Dimensions and drilling. 273/. 
Finishing. 274. Life of rails. 275. Intensity of pressure on rails. 
275o. Flow of metal. 276. Rail wear on tangents. 276a. Rail 
wear on curves. 277. Experimental determination of rail wear. 
278. Cost of rails 296 

CHAPTER X. 

haii^fastenings. 

Rail-joints 314 

- 279. Theoretical requirements for a perfect joint. 280. Effi- 
ciency of any type of rail-joint. 281. Effect of rail-gap at joints. 
282. Supported, suspeMed, and bridge joints. 283. Failures of 
rail-joints. 284. Standard angle-bars. 285. Specifications iov 
steel splice-bars. 

Tie-plAtes 320 

286. Advantages. 287. Eleinehts of the design. 288. Methods 
of setting. 

Spikes. . . .; .'.■•■■* ■' • 324 

289. Requirements. 290. tJriviiig. 291. Screw spikes. 292. 
Wooden spikes. 

Track-bolts and nut-locks 330 

293. Essential requirements. 294. Design of track-bolts. 295. 
Design of nut-locks. 

CHAPTER XI. 

switches and crossings. 

Switch construction 335 

296. Essential elements of a switch. 297. Frogs. 298. To find 
the frog number. 299. Stub switches. 300. Point switches. 301. 
Switch-stands. 302. Tie-rods. 303. Gua,rd-rails. 



XU TABLE OF CONTENTS. 



PAGE 

Mathematicai. design of switches > 342 

304. Design with circular lead rails. 305. Standard design, 
using straight frog-i-ails and straight point-rails. 306. Design for 
a turnout from the outer side of a curved track. 307. Design 
for a turnout from the inner side of a curved track. 308. Con- 
necting curve from a straight track. 309. Connecting curve from 
a curved track to the outside. 310. Connecting curve from a 
curved track to the inside. 3 11., Crossover between two parallel 
straight tracks. 312. Crossover between two parallel curved 
tracks. 313. Practical rules for switch-laying. 314. Slips. 

Crossings 361 

315. Two straight tracks. 316. One straight and one curved 
track. 317. Two curved tracks. 

CHAPTER XII. 
miscellaneous structures and buildings. 

Water stations and water supply 367 

318. Location. 319. Required qualities of water. 320. Mechani- 
ical cleaning. 321. Chemical purification. 322. Foaming and 
priming. 323. Boiler compounds. 324. Tanks. 325. Pumping. 
326. Track tanks. 327. Stand pipes. 

Buildings 377 

328. Station platforms. 329. Minor stations. Freight 
houses. 330. Two tpyes. 331. Fire risk. 332. Dimensions. 
333. Platforms. 334. Floors. 335. Doors. 336. Roofs project- 
ing over platforms. 337. Lighting. 338. Scales. 339. Ramps. 
340. Section houses. Engine houses. 341. Form. 342. Doors. 
343. Length. 344. Materials of construction. 345. Engine pits. 
346. Smoke jacks. 347. Floors. 348. Drop pits. 349. Heating. 
350. Window lighting. 351. Electric lighting. 352. Piping. 
353. Tools. 354. Hoists. 355. Turntables. Locomotive coal- 
ing stations. 356. Hand shoveling. 357. Locomotive crane. 
358. Coaling trestle. 359. Coal conveyors. 360. Oil houses. 
361. Section tool houses. 362. Sand houses. 363. Ash pits. 

Snow structures 391 

364. Snow fences. 365. Snow sheds. 

Fences 393 

366. Wire fences. 367. Types. 368. Posts. 369. Braces. 370. 
Concrete posts. 371. Construction details. 

Signs 396 

372. Highway signs. 373. Trespass signs. 374. Marker posts, 
375. Bridge warning. 

CHAPTER XIII. 

YARDS AND TERMINALS. 

376. Value of proper design. 377. Definitions. 378. General 
principles. 379, Minor freight yards, 3§0, Hump yards. 381, 



TABLE OF CONTENTS. XIU 



PAGE 

Ladder tracks. 382. Track scales. 383. Transfier cranes. 384. 
Engine yards or terminals. 384a. Passenger terminals 400 



CHAPTER XIV. 

block signaling. 

General principles 412 

385. Two fundamental systems. 386. Manual systems. 387. 
Development of the manual system. 388. Permissive blocking. 
389. Automatic systems. 390. Distant signals. 391. Advance 
signals. 

Mechanical details 418 

392. Signals. 393. Wires and pipes. 394. Track circuit for 
automatic signaling. 

CHAPTER XV. 

rolling stock. 

Wheels and rails 425 

395. Effect of rigidly attaching wheels to their axles. 396. 
Effect of parallel axles. 397. Effect of coning wheels. 398. 
Effect of flanging locomotive driving wheels. 399. Action of a 
locomotive pilot-truck, 400. Types of locomotive wheel bases. 

locomotives. 

General structure .-..,, 433 

401. Frame. 402. Boiler. 403. Fire box. 404. Area of grate. 
405. Superheaters. 406. Reheaters. 407. Coal consumption. 
408. Oil-burning locomotives. 409. Heating surface. 410. Loss 
of efficiency of steam pressure. 411. Tractive power. 

Running gear 444 

412. Equalizing levers. 413. Counterbalancing. 414. Mutual 
relations of the boiler power, tractive power and cylinder power 
for various types. 415. Life of locomotives. 

CARS. 

416. Capacity and size of cars. 417. Stresses to which car- 
frames are subjected. 418. The use of metal. 419. Draft gear. 
420. Gauge of wheels and form of wheel tread 455 



TRAIN-BRAKES. 

421. Introduction. 422. Laws of friction as applied to this 

problem 461 

Mechanism of brakes 465 

423. Hand-brakes. 424. "Straight" air brakes. 425. Auto- 
matic air brakes. 426. Tests to measure the efficiency of brakes. 
427. Brake shoes. 



XIV . TABLE OF CONTENTS. 

CHAPTER XVI. 

TBAIN RESISTANCE. 

PAGE 

428. Classification of the various forms. 429. Resistances inter- 
nal to the locomotive. 430. Velocity resistances. 431. Wheel 
resistances. 432. Grade resistance. 433. Curve resistance. 434. 
Brake resistance. 435. Inertia resistance. 436. Dynamometer 
tests. 437. Gravity or "drop" tests. 438. Resistance of cars 
through switches. 439. American Railway Engineering Associa- 
tion Formula. 439a. Passenger-car resistance 471 

CHAPTER XVII. 

COST OF RAILROADS. 

440. General considerations. 441. Preliminary financiering. 
442. Surveys and engineering expenses. 443. Land and land 
damages. 444. Clearing and grubbing. 445. Earthwork. 446. 
Bridges, trestles and culverts. 447. Trackwork. 448. Buildings 
and miscellaneous structures. 449. Interest on construction. 
450. Rolling stock. 451. Detailed pstimatp of the cost of a line 
j)J road 490 

CHAPTER XVIII. 

THE POWER OP A LOCOMOTIVE. 

452. Pounds of steam produced. 453. Numerical example. 
454. Weight of steam per stroke at full cut-off. 455. Pounds of 
steam and per cent of cut-oflf for multiples of M velocity. 45^. 
Draw-bar pull. 457. EfTect of increasing the rate of coal qon- 
sumption. 458. Effect of using a better quality of coal. 459. 
Check with approximate rule. 460. Tractive force at higher 
velocities. 461. Effect of grade on tractive power. 462. Accel- 
eration-speed curves. 463. Retardation-speed curves. 464. 
Drifting. 465. Review of computed po\ver of on^ locomotive. 
466. Selection of route. 467. Rating of locomotives 500 

CHAPTER XIX. 

THE PROMOTION OF RAILBOAD PROJECTS. 

468. Method of formation of railroad corporations. 469. The 
two classes of financial interests, the security and profits of each. 

470. The small margin between profit and loss to the projectors. 

471. Extent to which a railro£(,d is a mojiopply. 472. Profit 
resulting from an. increase in busiAess done; loss resulting from a 
decrease. 473. Estimation of probable volume of traflSc, and of 
probable growth. 474. Probable number of trains per day. In- 
crease with growth of traffic. 475. Effect on traflSc of an increase 
in facilities. 476. Loss caused by inconveniei^t terminals and 



TABIiE OF CONTENTS. XV 

PAGE 

by stations far removed from business centers. 477. General 
principles which should govern the expenditure of money for rail- 
road purposes. 478. Study of railroad economics — its nature and 
•limitations. 479. Outline of the engineer's duties 522 

CHAPTER XX. 

OPERATING EXPENSES. 

480. E>istribution of gross revenue. 481. Operating "^expenses 
ppr train mile. 482. Reasons for uniformity in expenses per train 
mile. 483. Detailed classification of expenses with ratios to the 
total expense. 484. Amounts and percentages of the various items. 536 

Maintenance op way and structures 539 

485. Track materials.' 486. Roadway and track. 487. Main- 
tenance of track structures. 
Maintenance of equipment 543 

488. Repairs, renewals, and depreciation of steam and electric 
locomotives. 

Transportation 544 

489. Yard-engine expenses. 490. Road enginemen. 491. Fuel 
for road locomotives. 492. Road trainmen. 493. Train supplies 
and expenses. 494. Clearing wrecks, loss, damage, and injuries 
to persons and property. 495. Operating joint tracks and facil- 
ities, switching charges, etc. 

CHAPTER XXI. 

DISTANCE. 

496. Relation of distance to rates and expenses. 497. The con- 
ditions other than distance that affect the cost; reasons why rates 

are usually based on distance 550 

Effect of distance on receipts. 551 

498. Classification of traffic. 499. Method of division of through 
rates between the roads run over. 500. Effect of a change in the 
length of the home road on its receipts from through competitive 
trafl&c. 501. The most advantageous conditions for roads forming 
part of a through competitive route. 502. Effect of the variations 
in the length of haul and the classes of the business actually done. 
503. General conclusions regarding a change in distance. 504. 
Justification of decreasing distance to save time. 505. Effect of 
change of distance on the business done. 

CHAPTER XXII. 

curvatxjre. 

506. General objections to curvature. 507. Financial value of 
the danger of accident due to curvature. 508. Effect of curvature 
on travel. 509. Effect on operation of trains 557 



3iVl TABLE OF CONTENTS. 



PAGE 

Compensation for curvature , , , 561 

510. Reasons for compensation. 511. The proper rate of com- 
pensation. 512. The limitations of maximum curvature. 

CHAPTER XXIII. 

grade. 

513. Two distinct effects of grade. 514. Application to the 
movement of trains of the laws of accelerated motion. 515. Con- 
struction of a virtual profile. 516. Variation in draw-bar pull. 
517. Use, value and possible misuse. 518. Undulatory grades; 

advantages, disadvantages, and safe limits 566 

Ruling grades 575 

519. Definition. 520. Choice of ruling grades. 521. Maximum 
train load on any grade. 522. Proportion of traflSc affected by 
the ruling grade. 

Pusher grades 578 

523. General principles underlying the use of pusher engines. 
524. Balance of grades for pusher service. 525. Two-pusher 
grades. 526. Operation of pusher engines. 527. Length of a 
pusher grade. 528. Cost of pusher-engine service. 

Balance of grades for unequal traffic 584 

529. Nature of the subject. 530. Computation of the theoreti- 
cal balance. 531. Computation of relative traffic. 



CHAPTER XXIV. 
the improvement of old lines. 

532. Classification of improvements. 533. Advantages of re- 
locations. 534. Disadvantages of re-locations 588 

Reduction op virtual grade 591 

535. Obtaining data for computations. 536. Use of the data 
obtained. 537. Reducing the starting grade at stations. 

CHAPTER XXV. 

STRESSES IN TRACK. 

538. Nature of the subject. 539. Action of track as an elastic 
structure. 540. Typical track depression profile for static load for 
one or two axles. 541. Bending moment and depression in a rail due 
to a group of loads. 542. Special instruments and devices for 
makin'g tests. 543. Pressure transmitted from tie to ballast. 544. 
Transverse stresses in the tie. 545. Effect of counterbalancing. . 596 

Appendix. The adjustments of instruments 612 

Azimuth 620 

Index 825 



TABLE OF CONTENTS. XVll 



Tables. page 

I. Radii of curves 628 

II. Tangents, external distances, and long chords for a 1° curve 632 
Ila. Excess length of sub-chords 635 

III. Switch leads and distances 635 

IV. Transition curves. Functions of the ten-chord spiral 637 

V. Logarithms of nunabers 640 

VI. Logarithmic sines and tangents of small angles 660 

VII. Logarithmic sines, cosines, tangents, and cotangents 663 

VIII. Logarithmic versed sines, and external secants 708 

IX. Natural sines, cosines, tangents; and cotangents 753 

t X. Natural versed sines and external secants 776 

P XI. Reduction of barometer reading to 32° F 799 

XII. Barometric elevatipns 800 

XIII. Coefficients for corrections for temperature and humidity . . 800 

XIV. Useful trigonometrical formulae 801 

XV. Useful formulae and constants 803 

XVI. Squares, cubes, square roots, cube roots and reciprocals. . . 804 

XVII. Cubic yards per 100 feet of level sections 821 

t^ XVIII. Annual charge against a tie, based on the original cost and 

1^ assumed life of the tie 824 

XIX. Superelevation of the outer rail (in feet) for various veloc- 
ities and degrees of curvature 83 

XX. Moduli of rupture for various timbers 220 

XXI. Working unit stresses for structural timber 221 

XXII. Number and kinds of cross ties, used in U. S., 1915 277 

XXIII. Angles and dimensions of standard designs for rails 299 

XXIV. Angles and dimensions of standard designs for splice bars. . 319 
XXV. Rectangular coordinates of curved rail of switches 358 

XXVI. Quantity of reagents required to remove incrusting or 

corrosive matter from water 370 

XXVIII. Cost of fuel for various types of pumps and engines. 374 

XXIX. Locomotive resistances 473 

XXX. Number of cross ties per mile 494 

XXXI. Tons per mile of rails of various weights . 495 

XXXII. Splice-bars and bolts for various weights of rail 496 

XXXIII. Railroad spikes 497 

XXXIV. Track bolts 497 

XXXV. Number of rail-joints and track-bolts per mile of track. . . . 497 

XXXVI. Average evaporation in locomotive boilers 501 

XXXVII. Weight of steam used in one foot of stroke in locomotives.. 503 

XXXVIII. Maximum cut-off and pounds of steam per I. H. P. hour. . 504 

XXXIX. Per cent cylinder tractive power for various multiples of M 505 

XL. Locomotive rating discounts 520 

XLI. Analysis of operating expenses of railroads in the United 

States in 1912 540, 541 

XLII. Velocity head of trains 570 

XLIII. Tractive power of various types of locomotives 577 

XLIV. Cost for each mile of pusher-engine service ,,..,.,..,,.,. 583 



EAILROAD CONSTEUCTION. 



CHAPTER I. 
Railroad surveys. 

The proper conduct of railroad surveys presupposes an 
adequate knowledge of almost the whole subject of railroad 
engineering, and particularly of some of the complicated ques- 
tions of Railroad Economics, which are not generally studied 
except at the latter part of a course in railroad engineering, if 
at all. This chapter will therefore be chiefly devoted to methods 
of instrumental work, and the problem of choosing a general 
route will be considered only as it is influenced by the topog- 
raphy or by the application of those elementary principles of 
Railroad Economics which are self-evident or which may be 
accepted by the student until he has had an opportunity of 
studying those principles in detail 

The student-engineer should be warned against the hasty and 
inadequate surveying which has resulted in so much miscon- 
struction in this country. This kind of surveying was especially 
common forty or fifty years ago, and the methods have more or 
less continued. The demand for railroad facilities was then so 
urgent that lax methods were tolerated. A general route would 
be selected which, at first sight, seemed most obvious and it 
would be immediately staked out in a manner suitable to a 
location survey. After correcting some of the most glaring 
faults, the survey was considered complete and the road was 
constructed accordingly. The cost of such a survey is compara- 
tively small, but it is almost inevitable that the line is not as 
good as could have been obtained with a greater amount of 



2 RAILROAD CONSTRUCTION. § 1. 

examination and study. The cost of construction and the 
future cost of operating such a line is always unnecessarily high. 
The money wasted in construction, plus the capitalized value of 
the annual waste in future operating expenses, is frequently a 
hundred times the cost of the extra study and surveying which 
would have avoided these faults. This has been unquestionably 
proved by the innumerable cases of reconstruction of portions 
of old lines which could have been constructed originally on the 
lines as revised at even less cost. The engineer is not always 
responsible for ill-advised hasty work. An impatient Board 
of Directors often insists on commencing to " throw dirt " 
before a proper survey has been made. The engineer should 
make, if necessary, the most earnest representations and even 
strenuous demands, that he be given the requisite time, oppor- 
tunity and money to conduct his survey in such a manner as 
to investigate thoroughly every possibility for improving the 
alinement. 

A railroad survey ordinarily consists of three parts: (a) 
the reconnoissance; (6) the preliminary survey, and (c) the 
definite location. As explained later, circumstances may modify 
the relative importance of these divisions, but under ordinary 
circumstances all three are necessary. 



RECONNOISSANCE SURVEYS. 

I. Character of a reconnoissance survey. A reconnoissance 
survey is a very hasty examination of a belt of country to deter- 
mine which of all possible or suggested routes is the most prom- 
ising and best worthy of a more detailed survey. It is essentially 
very rough and rapid. It aims to discover those salient features 
which instantly stamp one route as distinctly superior to another 
and so narrow the choice to routes which are so nearly equal 
in value that a more detailed survey is necessary to decide 
between them. 

A map should be prepared, at a scale not smaller than one 
mile to the inch, which should show all general routes which are 
conceivably possible. It is particularly important that the 
mere lack of data should not exclude consideration of some 
general route which might be superior to the one or ^lore pbvioug 
routes which have already been picked out, 



§ 2. RAILROAD SURVEY^. 3 

2. Selection of a general route. The general question of 
running a railroad between two towns is frequently a financial 
rather than an engineering question. Financial considerations 
usually determine that a road must pass through certain more 
or less important towns between its termini. It is also pos- 
sible that there may be certain topographical features in any 
route between two determined towns on the line, such as a low 
saddle in crossing a ridge or a difficult crossing of a large river, 
which, with the towns, may be considered as control points, and 
the problem may be narrowed down to the determination of 
the best route between these consecutive control points. But 
care should be taken that control points are not too hastily 
considered as fixed and unalterable, especially if it results in 
very unfavorable grades and alinement between consecutive 
points. 

The reconnoissance survey should include the determination 
of the location and relative elevations of all these control points. 
These data should be obtained with sufficient accuracy to 
compute the necessary ruling grade and the general character 
of the alinement, and the map as thus amplified should be 
studied by comparing the several possible routes and elimin- 
ating all those which are unquestionably less favorable than 
others. 

The engineer should avoid, especially in a rough and wooded 
country, the influence that an existing highway, or even a path 
through the woods or of a clearing of the trees, may have in 
determining the choice of routes. Mere ease of travel, as long 
as it is not glaringly wrong, has caused many prepossessions in 
favor of ar certain route, when a much better line could be obtained 
by plunging through the woods or over swampy or rocky ground. 
As a first trial in selecting the route, the bearing of a line joining 
two consecutive control points should be determined and then 
an effort should be made to find a general route which will have 
the least possible variation from that straight line, without sac- 
rificing the limits of ruling grade, curvature and general type 
or cost of construction which may have been fixed for the 
road. 

A difficult line between two control points should be studied 
by beginning at either end for two independent studies. The 
very obvious route, starting from A toward B, may lead into 
very difficult construction, which may be avoided by com- 



^ RAILROAD CONSTRUCTION. §3. 

mencing at P and finally reaching A on a route which, while 
practicable, would not be considered attractive when starting 
from A. 

When a railroad runs through a thickly settled and very flat 
country, where, from a topographical standpoint, the road may 
be run by any desired route, the '*' right-of-way agent " some- 
times has a greater influence in locating the road than the 
engineer. But such modifications of alinement, on account of 
business considerations, are foreign to the engineer's side of the 
pvibject, and it will be hereafter assumed that topography alone 
determines the location of the line. The consideration of those 
larger questions combining finance and engineering (such as 
passing by a town on account of the necessary introduction of 
heavy grades in order to reach it), will be considered in later 
chapters. \.i 

3. Valley route. This is perhaps the simplest problem. If 
two control points to be connected lie in the same valley, it is fre- 
quently onl}^ necessary to run a line which shall have a nearly 
uniform grade. The reconnoissance problem consists largely in 
determining the difference of elevation of the two termini of 
this division and the approximate horizontal distance so that the 
proper grade may be chosen. If there is a large river running 
through the valley, the road will probably remain on one side 
qr the other throughout the whole distance, and both banks 
should be examined by the reconnoissance party to determine 
which is preferable. If the river may be easily bridged, both 
thanks may be alternately used, especially when better alinement 
is thereby secured. A river valley has usually a steeper slope 
in the upper part than in the lower part. A uniform grade 
throughout the valley will therefore require that the road climbs 
up the side slopes in the lower part of the valley. In case the 
"ruling grade"* for the whole road is as great as or greater 
than the steepest natural valley slope, more freedom may be 
used in adopting that alinement which has the least cost— ^ 
regardless of grade. The natural slope of large rivers is almost 
invariably so low that grade has no influence in determining the 
choice of location. When bridging is necessary, the river 



* The ruling grade may here be loosely defined as the maximum grade 
which is permissible. This definition is not strictly true, as may be seen later 
when studying Railroad Economics, but it may here serve the purpose. 



§4. RAILROAD SURVEYS. S 

banks should be examined for suitable locations for • abutments 
and piers. If the soil is soft and treacherous, much difficulty 
may be experienced and the choice of route may be largely 
determined by the difficulty of bridging the river except at 
certain favorable places. 

4. Cross-country route. A cross-country route always has one 
Or more summits to be crossed. The problem becomes more 
complex oil account of the greater number of possible solutions 
and the difficulty of properly weighing the advantages and dis- 
advantages of each. The general aim should be to choose the 
lowest summits and the highest stream crossings, provided that 
by so doing the grades between these determining points shall 
be as low as possible ^-nd shall not be greater than the ruling 
grade of the road. Nearly all railroads combine cross-country 
and valley routes to some extent. Usually the steepest natural 
slopes are to be found on the cross-country routes, and also the 
greatest difficulty in securing a low through grade. An approx- 
imate determination of the ruling grade is usually made during 
the reconnoissance. If the ruling grade has been previously 
decided on by other considerations, the leading feature of the 
reconnoissance survey will be the determination of a general 
route along which it will be possible to survey a line whose 
maximum grade shall not exceed the ruling grade. 

5. Mountain route. The streams of a mountainous region 
frequently have a slope exceeding the desired ruling grade. In 
such cases there is no possibility of securing the desired grade 
by following the streams. The penetration of such a region 
may only be accomplished by "development" — accompanied 
perhaps by tunneling. "Development" consists in deliber- 
ately increasing the length of the road between two extremes 
of elevation so that the rate of grade shall be as low as desired. 
The usual method of accomplishing this is to take advantage of 
some convenient formation of the ground to introduce some 
lateral deviation. The methods may be somewhat classified as 
follows: 

(a) Running the line up a convenient lateral valley, turning 
a sharp curve and working back up the opposite slope. As 
shown in Fig. 1, the considerable rise between A and B was 
sunnounted by starting off in a very different direction from 
the general direction of the road; then, when about one-half of 
the desired rise had been obtained, the line crossed the valley 



6 



RAILROAD CONSTRUCTION. 



§5. 



and continued the climb along the opposite slope, (b) Switch" ' 
back. On the steep side-hill BCD (Fig. 1) a very considerable: 
gain in elevation vas accomplished by the switchback CD. 
The gain in elevation from B to D is very great. On the other 
hand, the speed must always be slow; there are two complete 
stoppages of the train for each run; all trains must run back- 
ward from C to D. (c) Bridge spiral. When a valley is so 
narrow at some point that a bridge or viaduct of icisonable 
length can span the valley at a considerable elevation above the 




Fig. 1. 

bottom of the valley, a bridge spiral may be desirable. In Fig. 2 
the line ascends the stream valley past A, crosses the stream At] 
B, works back to the narrow place at C, and there crosses itself, 
having gained perhaps 100 feet in elevation, (d) Turing 
spiral (Fig. 3). This is the reverse of the previous plan. 11? 
implies a thin steep ridge, so thin at some place that a tunnel 
through it will not be excessively long. Switchbacks and 
spirals are sometimes necessary in mountainous countries, but 
they should not be considered as normal types of construction. 
A region must be very difficult if these devices c&nnot be 
avoided, 



PLATE t 





[To face jinge C) 



§5. 



RAILROAD SURVEYS. 



On Plate I are shown three separate ways (as actually con- 
structed) of running a railroad between two points a little over 
three miles apart and having a difference of elevation of nearly 
1100 feet. At A the Central R. R. of New Jersey runs under 
the Lehigh Valley R. R. and soon turns off to the northeast for 
about six miles, then doubles back, reaching D, a fall of about 
1050 feet Avith a track distance of about 12.7 miles. The 
L. V. R. R. at A runs to the westward for six to seven miles, 





"^^i*^ 

^^""\\^\^\^ 



Fig. 2. 



Fig. 3. 



then turns back until the roads are again close together at D. 
The track distance is about 14 miles and the drop a little greater, 
since at A the L. V. R. R. crosses over the other, while at D they 
are at practically the same level. From B to C the distance is 
over eleven miles. From A directly down to D the C. R. R. of 
N. J. runs a "gravity" road, used exclusively for freight, on 
which cars alone are hauled by cable. The main-line routes 
are remarkable examples of sheer "development." Even as 
constructed the L. V. R. R. has a grade of about 95 feet per 
mile, and this grade has proved so excessive for freight work 
that the company has constructed a cut-off (not shown on the 
map) which leaves the main line at A, nearly parallels the 



3 RAILROAD CONSTRtJGTION. § 6. 

C. R. R. to C, and then running in a northeasterly direction 
again joins the main line beyond Wilkesbarre. The grade is 
thereby cut down to 65 feet per mile. 

Rack railways and cable roads, although types of mountain 
j*ailroad construction, will not be here considered. 

,6. Existing maps. The maps of the U. S. Geological Survey 
are exceedingly valuable as far as they have been completed. 
So far as topographical considerations are concerned, they 
almost dispense with the necessity for the reconnoissance and 
"first preliminary" surveys. Some of the State Survey maps 
will give practically the same information. County and town- 
ship maps can often be used for considerable information as tathe 
relative horizontal position of governing points, and even some 
approximate data regarding elevations may be obtained by a 
study of the streams. Gf course such information will not dis- 
pense with surveys, but will assist in so planning them as to 
obtain the best information with the least work. When the 
relative horizontal positions of points are reliably indicated on a 
map, the' reconnoissance may be reduced to the determina- 
tion of the relative elevations of the governing points of the 
route. 

7. Determination of relative elevations. A recent description 
,Df European methods includes spirit-leveling in the reconnois- 
sance work. This may be due to the fact that, as indicated 
above, previous topographical surveys have rendered unnecessary 
Uie "exploratory" survey which is required in a new country, 
imd that their reconnoissance really corresponds more nearly to 
our preliminary. 

The perfection to which barometrical methods have been 
^Drought has rendered it possible to determine differences of 
elevation with sufficient accuracy for reconnoissance purposes 
by tjie combined use of a mercurial and an aneroid barometer. 
The mercurial bjarometer should be kept at " headquarters," an4 
readings should be taken on it at such frequent intervals that 
any fluctuation is noted, and throughout the period that observa- 
tions with the aneroid are taken in the field. At each observa- 
tion there should also be recorded the time, the reading of the 
^attached thermometer, and the temperature of the external 
air. For uniformity, the mercurial readings should then b§ 
"reduced to 32° F." The form of notes for the niercurial 
barometer readings should be as follows : 



J 



§v. 



RAILROAD SURVEYS 



9 



Time. 


Merc. 
Barom. 


Attached 
Therm. 


Reduction 
to 32° F. 


External 
Therm. 


Corrected 
reading. 


7:00 A.M. 
:15 
:30 
:45 


29.872 
.866 
.858 
.850 


72° 
73.5 
75 
76 


— .117 
.121 
.125 
.127 


73° 
75 

76 

77 


29.755 
.745 
.733 
.723 



The corrections in column 4 are derived from Table XI by 
interpdlatioQ. 

Before starting out, a reading of the aneroid should be taken 

! at headquarters coincident with a reading of the mercurial. 

i The difference is one value of the correction to the aneroid. 
As soon" as the aneroid is brought back another comparison of 
readings should be made. Even though there has been con- 

I siderable rise or fall of pressure in the interval, the difference 
in readings (the correction) should be substantially the same 
provided the aneroid is a good instrument. If the difference 
of elevation is excessive (as when climbing a high mountain) 
even the best aneroid will "lag" and not recover its normal 
reading for several hours, but this does not apply to such dif- 
ferences of elevation as are met with in railroad work. The 
best aneroids read directly to -^-^ of an inch of mercury and 
may be estimated to yww^ of an inch- — which corresponds 
to about 0.9 foot difference of elevation. In the field there 
should be read, at each point whose elevation is desired, the 
aneroid, the time, and the temperature. These readings, cor- 
rected by the mean value of the correction between the aneroid 
and the mercurial, should then be combined with the reading 
of the mercurial (interpolated if necessary) for the times of 
the aneroid observations and the difference of elevation ob- 
tained. The field notes for the aneroid should be taken as 
shown in the first four columns of the tabular form. The " cor- 
rected aneroid" readings of column 5 are found by correcting 
the readings of column 3 by the mean difference between the 
mercurial and aneroid when compared at morning and night 
Column 6 is a copy of the "corrected readings" from the office 
notes, interpolated when necessary for the proper time. Column 
7 is similarly obtained. Col. 8 is obtained from cols. 4 and 5, 
and col. 9 from cols. 6 and 7, with the aid of Table XII. The 
correction for temperature (col. 11), which is generally small 
unless the difference of elevation is large, is obtained with the 



10 



RAILROAD CONSTRUCTION. 



§7. 



(Left-hand page of Notes.) 



Time. 


Place. 


Aneroid. 


Therm. 


Corr. 
Aner. 


Corr. 
Merc. 


7:00 


Office 
Z/0 
saddle-back 
river cross. 


29.628 
29.662 
29.374 
29.548 


73° 

72° 
63° 
70° 




29 . 756 


7:10 
7:30 
7:50 


29.789 
29.501 
29.675 


29.748 
29.733 
29.720 



aid of Table XIII. The elevations in Table XII are elevations 
above an assumed datum plane, where under the given atmos- 
pheric conditions the mercurial reading would be 30". Of 
course the position of this assumed plane changes with varying 
atmospheric conditions and so the elevations are to be con- 
sidered as relative and their difference taken. See '^ Technic 
of Surveying Instruments and Methods," Prob. 28, by Webb 
and Fish; John Wiley & Sons. Important points should be 
observed more than once if possible. Such duplicate obser- 
vations will be found to give surprisingly concordant results even 
when a general fluctuation of atmospheric pressure so modifies 
the tabulated readings that an agreement is not at first apparent. 
Variations of pressure produced by high winds, thunder-storms, 
etc., will generally vitiate possible accuracy by this method. 
By "headquarters" is meant any place whose elevation above 
any given datum is known and where the mercurial may be 
placed and observed while observations within a range of several 
miles are made with the aneroid. If necessary, the elevation of 
a new headquarters may be determined by the above method, 
but there should be if possible several independent observations 
whose accordance will give a fair idea of their accuracy. 

The above method should be neither slighted nor used for 
more than it is worth. When properly used, the errors are 
compensating rather than cumulative. When used, for example, 
to determine that a pass B is 260 feet higher than a determined 
bridge crossing at A which is six miles distant, and that another 
pass C is 310 feet higher than A and is ten miles distant, the 
figures, even with all necessary allowances for inaccuracy, will 
give an engineer a good idea as to the choice of route especially 
as affected by ruling grade. There is no comparison between 
the time and labor involved in obtaining the above information 
by barometric and by spirit-leveling methods, and for recon* 



§8. 



RAILROAD SURVEYS. 



11 



. 






(Right-hand page of Notes.) 


Temp, at 
headqu. 


Approx. 
field read. 


Approx. 
headq. read. 


Diff. 


Corr. for 
temp. 


Diff. 
elev. 


75* 

76 

77 


192 
457 
297 


230 
244 
256 


- 38 
+ 213 
+ 41 


-(+ 2) 
+ ( + 10) 
+ (+ 2) 


— 40 
+ 223 
+ 43 



noissance purposes the added accuracy of the spirit-leveUng 
method is hardly worth its cost. 

8. Horizontal measurements, bearings, etc. When reliable 
maps are unobtainable, rapid exploratory surveys become essen- 
tial. Since accuracy is sacrificed for rapidity in such surveys, 
more or less approximate methods are used. *' An experienced 
saddle-horse, whose speeds at his various gaits have been learned 
accurately by previous timing," is quoted from Beahan * as 
one means of rapidly measuring distances. The percentage of 
probafele error is evidently large. A pedometer (or pace- 
measurer) is probably more accurate, but its accuracy depends 
on a knowledge of the average length of the observer's pace.. 
Due allowance must be made for the fact that the length of pace 
will vary very greatly depending on whether the surface is 
smooth and level, or is plowed ground, or marshy, or slippery, or 
consists of rough boulders covered with moss, or is a wilderness 
of brambles, fallen trees, bogs, etc. It will also depend on 
whether the observer is fatigued or is in fresh physical condition. 
Under such a variety of conditions the counting of steps for long 
distances is sometimes a farce. Even when the surface is fairly 
smooth and easy, precautions must be taken that paces are not 
counted during the pauses at important points while bearings 
are being taken and other data recorded. An odometer which 
records the revolutions of a wheel of known circumference is 
far more accurate. Such a machine has been made so that it 
may be trundled like a wheelbarrow and thus go through the 
woods and over ground that would be impassable to any horse- 
drawn vehicle. The attachment of an odometer to the wheels 
of a wagon is very tempting, since it permits the engineer to 
ride, but it is probably an unreliable method for the reason men- 



=1= "The Field Practice of Railway Location," p. 34. 



12 RAILROAD CONSTRUCTION. f ^. 

tioned in Art. 2 — permitting the ease of travel over a road 
practicable for a horse and vehicle to deflect the engineer from; 
his true course, which is perhaps over rough ground which is 
impassable for a vehicle. 

When the country is quite open and clear of underbrush, 
very tapid \V^ork may be done by the stadia method, which is 
many times more accurate ihan any of the methods previ-' 
ously mentioned. Some of the accuracy possible with stadia 
may be sacrificed for extreme rapidity and sights may be 
made 1200 and even 2000 feet long. By taking very few, 
if any, " side-shots," the progress is very rapid ' and many 
miles per day may be covered, with the advantage that the 
three elements of distance, azimuth and relative elevation 
may be obtained with as great accuracy as is necessary for 
an exploratory survey. The method of using the stadia will be 
described later. 

The bearings of the various lines forming the skeleton of the 
survey, and also the bearings of the courses of streams and of side 
lines from the stations on the skeleton line, may be taken most 
easily with a prismatic compass. This instrument has a cil*- 
"cular card, or sometimes a metal ring, attached to the needlife. 
'JThe edge of the card is graduated into degrees and is usually 
immbered consecutively (instead of by quadrants), from 0° 
tip io 360°. This is advantageous since the one number, with- 
out any qualifying letters, NE or NW, determines the quadrant 
definitely without danger of confusion or error. The observer 
sights through a narrow slit in the desired direction and, by 
means of the prismatic reflector, can read directly the number 
of degrees, measured to the right, and usually from the magnetic 
South. The makers of prismatic compasses do not always 
number the graduations in the same manner, and, therefore, the 
engineer, who is accustomed to one particular instrument, should 
carefully study the markings of any new instrument. In any 
case it should be remembered that the prism reflects the numbers 
on that side of the movable card or ring which is toward ike db- 
server rather than on the side toward the object sighted at. The 
prismatic compass has the special advantage that, like a sextant, 
it can be used when supported only by hand, while aii ordinary 
sight compass of equal accuracy would require a tripod, or, at 
least, a Jacob's staff. The declination of the needle in that 
section of the country can be readily determined with sufficient 



§9. RAILROAD SURVEYS." 13 

g-ccuracyfor the purposes of such a survey. Usually the decli- 
nation may be ignored. Any errors due to local attraction are 
never cumulative, but apply only to the point where those indi- 
vidual observations are taken. The angle between two Unea 
radiating from any station may be obtained by subtracting one 
bearing from the other. 

Relative elevations may be obtained systematically, using a 
barometer, as already explained, but much filling in may be 
done with the use of a hand-level. Experience soon teaches an 
engineer that there are many optical illusions about the slopes 
of ground which have the practical effect of making the apparent 
slope different from the actual, and, in the case of low grade, 
may make an actual down grade appear as an up grade. For 
example, when looking along an actual but slight down grade, 
especially if there are no obstructions or natural objects which 
the eye can use as a comparative scale, the eye is apt to fore- 
shorten the distance, which has the effect of lessening the appa- 
rent down grade and perhaps of making it appear as a slight up 
grade. The hand level will immediately detect such errors and 
its frequent use by a reconnoissance engineer will not only 
enable him to avoid many errors he might otherwise make, but 
will also be an effective means of training him to guard against 
such optical illusions. Such a simple and effective instrument 
should always be at hand and it should be tested with sufficient 
frequency to know that it is always as accurate as such an 
instrument can be. The bubble should be as sensitive as is 
practicable for an instrument which is held in the hand. A 
well-made hand level has a bubble of the right sensitiveness, but 
even a super-sensitive level may be utilized and still better 
work done by supporting it steadily on the top of a light wooden 
stick about five feet long. 

9. Importance of a good reconnoissance. The foregoing instru- 
ments and methods should be considered only as aids in exer- 
cising an educated common sense, without which a proper 
location cannot be made. The reconnoissance survey should 
command the best talent and the greatest experience available. 
If the general route is properly chosen, a comparatively low 
order of engineering skill can fill in a location which will prove 
a paying railroad property; but if the general route is so chosen 
that the ruling grades are high and the business obtained is small 
and subject to excessive competition, no amount of perfection in 



14 RAILROAD CONSTRUCTION. § 10. 

detailed alinement or roadbed construction can make the road a 
profitable investment. 

PRELIMINARY SURVEYS. 

10. Character of survey. A. preliminary railroad survey is 
properly a topographical survey of a belt of country which has 
been selected during the reconnoissance and within which it is 
estimated that the located line will lie. The width of this belt 
will depend on the character of the country. When a railroad 
is to follow a river having very steep banks the choice of 
location is sometimes limited at places to a very few feet of 
width and the belt to be surveyed may be correspondingly 
narrowed. 

But even in such a case, the width surveyed should be suffi- 
cient to include not only every possible location of " slope- 
stakes " but also should indicate the contours and nature of 
any soil which might give trouble by sliding, after an excavation 
has been made at the base. It is justifiable and proper to survey 
a belt considerably wider than it is expected to use, for experi- 
ence shows that, while there is generally but little or no direct 
utilization of the extra area surveyed, it frequently becomes 
essential to know something of the character of the ground 
considerably to one side of where it was expected to run 
the line and the inclusion of this area in the original survey 
has saved an expensive trip to obtain a very small amount of 
data. 

In very flat country the desired width may be only limited by 
the ability to survey points with sufficient accuracy at a consider- 
able distance from what may be called the " backbone line " of 
the survey. 

11. Cross-section method. This is the only feasible method 
in a wooded country, and is employed by many for all kinds 
of country. The backbone line is surveyed either by observ- 
ing magnetic bearings with a compass or by carrying forward 
absolute azimuths with a transit. The compass method has 
the disadvantages of limited accuracy and the possibih'ty of 
considerable local error owing to local attraction. On the other 
hand there are the advantages of greater simplicity, no necessity 
for a back rodman, and the fact that the errors are purely 
local and not cumulative, and may be so limited, with care, that 



§11. 



KAILEOAD SURVEYS. 



15 



they will cause no vital error in the subsequent location survey. 
The transit method is essentially more accurate, but is liable 
to be more laborious and troublesome. If a large tree is en- 
countered, either it must be cut down or a troublesome opera- 
tion of offsetting must be used. If the compass is employed 




Fig. 4. 



under these circumstances, it noed only be set up on the far side 
of the tree and the former bearing produced. An error in 
reading a transit azimuth will be carried on throughout the 
survey. An error of enly five minutes of arc will cause an oft- 
set of nearly eight feet in a mile. Large azimuth errors may, 
however, be avoided by immediately checking each new azimuth 



id EAlLROAB CONSTHtJCTlON. | 12. 

with a needle reading. It is advisable to obtain true azimuth 
at the beginning of the survey by an observation on the sun* or 
Polaris, and to check the azimuths every few miles by azimuth 
observations. Distances along the backbone line should be 
measured with a chain or steel tape and stakes set every 100 
feet. When a course ends at a substation, as is usually the case, 
the remaining portion of the 100 feet should be measured along 
the next course. The level party should immediately obtain the 
elevations (to the nearest tenth of a foot) of all stations, and also 
of the lowest points of all streams crossed and even of dry gullies 
which would require culverts. 

12. Cross-^sectioning. It is usually desirable to obtain con- 
tours at five-foot intervals This may readily be done by the 
use of a Locke level (which should be held on top of a simple 
five-foot stick), a tape, and a rod ten feet in length graduated 
to feet and tenths. The method of use may perhaps he best 
explained by an example. Let Fig. 5 represent a section per- 
pendicular to the survey line — such a section as would be made 
by the dotted lines in Fig. 4. C represents the station point. 
Its elevation as determined by the level is, say, 158.3 above 
datum. When the Locke level on its five-foot rod is placed at 
C, the level has an elevation of 163.3. Therefore when a point 
is found (as at a) where the level will read 3.3 on the rod, that 
point has an elevation of 160.0 and its distance from the center 
gives the position of the 160-foot contour. Leaving the long 
rod at that point (a), carry the level to some point (6) such that 
the level will sight at the top of the rod; b is then on the 165- 
foot contour, and the horizontal distance ab added to the hori- 
zontal distance ac gives the position of that contour from the 
center. The contours on the lower side are found similarly. 
The first rod reading will be 8.3, giving the 155-foot contour. 
Plot the results in a note-book which is ruled in quarter-inch 
squares, Using a scale of 100 feet per inch in both directions. 
Plot the work up the page; then when looking ahead along the 
line, the work is properly oriented. When a contour crosses 
the survey line, the place of crossing may be similarly deter- 
mined. If the ground flattens out so that five-foot contours are 
very far apart, the absolute elevations of points at even fifty- 
foot distances from the center should be determined. The 

* The method of making such observations is given in the Appendix. 



§13. 



RAILROAD SURVEYS. 



17 



method is exceedingly rapid. Whatever error or inaccuracy 
occurs is confined in its effect to the one station where it 
occurs. The work being thus plotted in the field, unusually 
irregular topography may be plotted with greater certainty and 
no great error can occur without detection. It would even be 
possible by this method to detect a gross error that might have 
been made bv the level party, 




Fig. 5. 




Fi«. 6. 



13. Stadia method. This method is best adapted to fairiy 
open country where a "shot" t^ any desired point may be 
taken without clearing. The backbone survey line i^s (tbe same 



18 EAILROAD CONSTRUCTION. § 13. 

as in the previous method except that each course is hmited to 
the practicable length of a stadia sight. The distance between 
stations should be checked by foresight and backsight— also the 
vertical angle. Azimuths should be checked by the needle. 
Considering the vital importance of levehng on a railroad survey 
it might be considered desirable to run a line of levels over the 
stadia stations in order that the leveling may be as precise as 
possible; but when it is considered that a preliminary survey is 
a somewhat hasty survey of a route that maij be abandoned, and 
that the errors of leveling by the stadia method (which are con- 
pensating) may be so minimized that no proposed route would 
be abandoned on account of such small error, and that the effect 
of such an error may be usually neutralized by a slight change in 
the location, it may be seen that excessive care in the leveling 
of the preliminary survey is hardly justifiable. 

A stadia party should include a locating engineer (or chief of 
party), and perhaps an assistant, a transitman, a recorder and 
four rodmen, beside axemen. The transitman should have noth- 
ing to do but attend to his instrument. After setting up the 
transit at an advanced station, a backsight should be taken 
to the previous station. If the vertical circle is full 360°, the 
telescope should be plunged and sighted on the backsight with 
vernier A reading the same as the foresight to the station occu- 
pied. If the vertical arc is semi-circular (or less), no vertical 
angle can be taken with the telescope plunged and, therefore, 
vernier A should read 180° more (or less) than the foresight. 
The lower plate should be very firmly clamped, and then, after 
loosening the upper plate, a reference sight and reading on some 
well-defined natural object should be taken. If there is any 
reason to suspect that the instrument has been disturbed while 
occupying that station, the reference point can be sighted at 
and the instrument can be re-alined, and re-leveled, if necessary, 
without sending a rodman back to the previous station. 
When taking a backsight the rod reading for distance should 
be taken first and immediately compared with the previously 
recorded foresight. Since the distance between stations will 
always be taken with especial care so as to avoid " blunders " 
of an even 10, 20 or perhaps 100 feet, the foresights and back- 
sights should agree to within the proper limits of the stadia 
method. Similarly the vertical angle should agree with the 
previous reading, hut with opposite sign. If especial care is 



§ 13. KAILROAD SURVEYS. 19 

taken in leveling the instrument immediately before taking both 
foresights and backsights, these readings should agree to within 
one minute, or even 30 seconds, with a good transit. The 
height of the telescope above the ground at the new station must 
be measured, and the middle wire sighted at that reading on the 
rod (called the H. I.)] when taking any vertical angle. Theo- 
retically the rod reading for distance should be taken when the 
telescope is pointing at the proper vertical angle for that shot, 
but this will mean, in general, that both the upper and lower 
cross wires will read odd amounts and that an inconvenient 
subtraction must be made to get the difference, which is the 
" rod reading." But it may be demonstrated that no error of 
distance, amounting to the lowest practicable unit of measure- 
ment, can result if the telescope is raised or lowered just enough 
to set it on the nearest even foot mark. The routine of observ- 
ing a shot is therefore as follows: (a) swing the instrument (the 
upper plate) horizontally until the telescope sights at the rod 
and clamp the horizo^ital motion — but very lightly and perhaps 
not at all; (6) raise or lower the telescope until the middle cross 
wire is sighting at the H. I., reading on the rod; a target on the 
rod may be set at the H. I. reading for each set-up and it will 
facilitate the work; (c) read the vertical angle and report it 
to the recorder, standing at hand; (d) raise or lower the tele- 
scope just enough so that the lower wire is on the nearest even 
foot mark and read (calhng it out to the recorder) the number of 
even feet of interval from the lower to the upper wire and the odd 
amount at the top at the reading of the upper wire; (e) dismiss 
the rodman, who is then directed to another point by the chief 
of party; (/) read the azimuth on the horizontal plate. By 
that time another rodman has been located at a point where an 
observation is required, and the routine is repeated. The work 
of the transitman is thus very strenuous, without any recording 
work, and the progress of the party depends on him. He, there- 
fore, should not be required to direct the party or even to record 
his notes, since every moment spent in that way delays the entire 
party by that amount. The recorder also has all that he can 
do to record the notes (with perhaps some sketches), as fast as 
the transitman calls them off. Usually four rodmen can be 
kept very busy, and they must be on the run between the suc- 
cessive points at which they hold their rods. One of the rod- 
men or one of the axemen, if axernen are employed, carries and 



20 



RAILROAD CONSTRUCTION. 



§ 14. 



diives the stalies; whlcK are only required at tHe instrument 
points. One or more axemen are generally useful in lopping off 
branches or cutting down saplings which interfere with desirable 
sights. The chief of party has plenty to do in directing the 
rodmen and axemen so that shots may be taken at points which 
will give the most significant information, and also in picking 
out the proper location for the advance station at some place 
from which a maximum of information may be observed with 
one set-up of the transit. A well-drilled organization and 
"team work" are necessary. The best work is done when every 
man^ is kept busy. Several hundred shots per day can be obr 
served when it is considered advisable to obtain much detailed 
information and the average number of shots per set-up is large. 
On the other hand, when the stadia method is used for a rapid 
exploratory survey, only a few side shots (at some stations per?- 
haps none at all) will be taken at each station. In such a case, 
the tt)tal' number of shots taken during a day will be compara^ 
tively small, but the progress will be very rapid, and the salient 
features of several miles of a proposed route can be obtained in a 
day. 

14. Form for stadia notes. 



^Left-hand page.] 










Inst, at 


Azim. 


Rod 


Vert, angle 


r; Elev. 


Sighting at 


A24. 

HI =4^.9 .... 
M = 629.2... 


264° 27' , 
83° 10' 
184° 23' 
5° 47' 


.622 
528 
264 
218(175) 


-0° 18' 
+1° 16' 
-2° 18' 
+26° 20' 


i 

: 
: 


A23 

A25 
bend in creek 
top of bluff 







The usual six-column note-book can be utilized by ruling an 
extra line (shown dotted in the Form of Stadia Notes), in the 
fifth column, since the column is wide enough for both the " dif- 
ference of elevation " and the " elevation." The " rod reading " 
(3d column) as recorded should include the (f+c), which in 
almost all. American transits equals 1.0 to 1.3 feet. Sijice the 
wire-interval ratio is almost invariably 1 : 100, the rod interval 
in hundredths of a foot is considered as the number of feet of dis- 
tance", except that one even foot is added for the (/+c). The 
sample figures given above are typical of all that needs to be 
taken in the field. The " difference of elevation " and the " ete- 
vation " are computed and entered later. 



m 



§ 14. RAILROAt) StJRVEYS. 21 

The " difference of elevation " may be mathematicaiUy com- 
puted from the formula 

D = k r I sin 2a + (f+c) sin a, 

in which D is the difference of elevation, /c is a constant, usually 
100, r is the rod intercept and. a is the angle of elevation — or 
depression. The mathematical solution of such an equation 
for every shot that is taken (except the very few shots which are 
level) is very laborious and impracticable. But the work of 
reduction Can be shortened by a justifiable approximation. By 
changing the factor of (/+c) from sin a to | sin 2a, the formula 
may be written 

Z)' = [/cr+(/+c)]isin2a. 

The first term (that within the bracket) is the number recorded 
under "Rod" in the Form of Notes (622, 528, etc.). The 
second term (| sin 2(x) may be taken from " Stadia Tables," of 
which many are published, although the tables usually give 
these numbers merely as the factors by which the dista,nce is to 
be multiplied in order to obtain the " Difference of Elevatioti," 
and do not mention that the factor is really | sin 2(x. The error 
of the approximation (when (/+c) =1 foot) is less than 0,01 foot 
for a vertical angle of 15° and less than 0.1 foot for the unusual 
angle of 30°. Since 0.1 foot is the usual lowest unit of measure- 
ment for stadia elevations, probably 99% of all stadia work cati 
use such an. approximation without appreciable error. The 
special cases with high angles can be computed separately if it 
is considered necessary. The algebraic sign of the vertical angle 
Should always be recorded, even if it is plus, or upward; the sign 
H- is a positive statement that it is plus and that the sign was 
tibt forgotten. The difference of elevation likewise should always 
iiave a + or — sign. Adding the difference of elevation to the 
elevation of the station (or subtracting it), gives the elevation of 
each point. 

Theoretically the true horizontal distance for all inclined sights 
fe always less than the nominal distance, as given by the rod 
f<e^ading, Tlie formula for true distance is 

» 
L=kr cos^ a -\- (J -{-c) cos a. 



22 KAILROAD CONSTRUCTION. §15. 

As before, we may use the approximation of combining the 
(/+c) with the kr and say that 

^' = [A;r+(/+c)] cos^a, 

and that the correction, which is subtracted from [A;r+(/+c)], 
and not from kr, is 

Corr. = [fcr + (/+c)] sin2 a. 

The error of this approximation is usually insignificant, as illus- 
trated below. Since sin^ « is very much less than cos^ a. for the 
usual small values of «, it is easier and more accurate to compute 
the smaller quantity and mentally subtract it from the nominal 
reading. When the vertical angle and the distance are both 
small, the horizontal correction is within the lowest unit of 
measurement (one foot), and should, therefore, be ignored. The 
engineer soon learns the approximate limits at which the com- 
bination of vertical angle and distance will make a correction 
necessary. In the above notes no correction is necessary except 
in the last case, the angle being 26° 20'. The exact mathe- 
matical computation is as follows, the rod interval being 2.17 
and (/+c) = l, 

L = 217 cos2 26° 20' + 1 cos 26° 20' = 175.20. 

Using the approximate rule, the correction = 218 sin^ 26° 20' 
= 42.90. 

218-42.90 = 175.10. 

The above calculations have been carried to hundredths of a foot 
for the sole purpose of illustrating that the discrepancy between 
the approximate and the theoretical value is only 0.10 foot, even 
for this unusually large angle, and considering that the rod 
interval is read only to the nearest 0.01 foot, which corresponds 
to one foot of distance, this discrepancy is utterly inappreciable. 
15. The reduction of stadia observations is most easily 
accomplished by using a stadia slide rule, which has one loga- 
rithmic scale for distances and for the computed differences 
of elevation or corrections to distance, and also two other scales 
one of which gives values for | sin 2a., and the other givesvalucs 



§ 16. RAILROAD SURVEYS. 23 

for sin^ a. Some scales give values of cos^ a. To illustrate the 
difference, in the above case, it is evidently easier to read 43 
(two significant figures) than to read 218, which has three fig- 
ures. When the distance is over 1000 (four figures), the diffi- 
culty is even greater. The necessity for subtracting the cor- 
rection is of no appreciable importance. In this case, the cor- 
rection would be read from the slide rule as 43, and mentally 
subtracting 43 from 218, we write at once 175, which is recorded 
in parenthesis in the Rod column. The draftsman, when plot- 
ting the notes, uses this distance (175) instead of 218. Using a 
slide rule, two men can very quickly compute the differences of 
elevation for the entire day's work in a very short time. A very 
little practice will enable them to run down the Hst, picking out 
the observations, usually less than 10% of the total number, 
where the combination of distance and vertical angle is suffi- 
ciently great to make it necessary to compute a horizontal cor- 
rection. The stadia slide rule is so small that it may readily be 
carried into the field and used there if desired, in which respect it 
has a great advantage over diagrams, which are sometimes used 
for the same purpose. 

i6. Stadia method vs. cross-section method. There is still a 
difference of opinion among engineers as to the choice of these 
two methods. When a large part of the route is thickly wooded, 
the cross-section method is preferable. In open country the 
stadia method is more rapid and more economical. Although it 
would be inadvisable to change from one method to the other 
every mile or so, a very considerable economy is possible by 
alternating the two methods according to the character of the 
country. The locating engineer can plan such change of method 
during his reconnoissance. The real efficiency of the stadia 
method is due to the fact that the preHminary survey should be 
considered as the topographical smrvey of an area or belt, and 
not the survey of a line, and that in open country the stadia 
method is the most efficient method of obtaining such topo- 
graphical data. But the efficiency depends on the handfing of 
the party. When a valley widens out with easy slopes and the 
possible area in which the location may He is correspondingly 
widened, it is far easier and more accurate to widen the belt 
surveyed by stadia shots of 1000 feet if necessary. 

17. " First " and " Second " preliminary surveys. Some 
engineers advocate two preliminary surveys. When this is done^ 



f 



24 RAILEOAD CONSTRUCTION. § 18. 

! _ • I 

the first IS a very rapid survey, made perhaps with a compass, 
and is only a better grade of reconnoissance. Its aim is to 
rapidly develop the facts which will decide for or against any 
proposed route, so that if a route is found to be unfavorable 
another more or less modified route may be adopted without 
having wasted considerable time in the survey of useless details. 
By. this time the student should have grasped the fundamental 
idea that both the reconnoissance and preliminary surveys are , 
not surveys of lines but of areas, that their aim is to survey 
only those topographical features which would have a deter- 
mining influence on any railroad line which might be constructed 
through that particular territory, and that the value of a locating 
engineer is largely measured by his ability to recognize those 
determining influences with the least amount of work from his 
surveying corps. Frequently too little time is spent on the 
comparative study of preliminary lines. A line will be hastily 
decided on after very little study; it will then be surveyed with 
minute detail and estimates carefully worked up, and the claims 
of any other suggested route will then be handicapped, if not 
disregarded, owing to an unwillingness to discredit and throw 
away a large amount of detailed surveying. The cost of two or 
three extra preliminary surveys {at critical sections and not over 
the whole line) is utterly insignificant compared with the prob- 
able improvement in the " operating value " of a line located 
after such a comparative study of preliminary lines. 

! LOCATION SURVEYS. 

18. " Paper location." When the preliminary survey has 
been plotted to a proper scale (usually 200 feet per inch), and the 
contours drawn in, a study may be made for the location survey. 
Disregarding for the present the effect on location of transition 
curves, the alinement may be said to consist of straight lines (or 
" tangents ") and circular curves. The '' paper location " there- 
fore, consists in plotting on the preliminary map a succession of 
straight lines which are tangent to the circular curves connecting 
them. It may be assumed that the general route of the prelim- 
inary survey has been so well selected, as the result of the recon- 
noissance survey, that it is possible to construct a line without 
excessive earthwork between consecutive control points, and 
that the grades are within the ruUng grade. If the preliminary 



f 



§18. 



RAILROAD SURVEYS. 



25 



surfey has been run by locating stations every 100 or 200 feet 
(see § 11 and Fig. 4), the profile of this line gives the first approx- 
imation toward the rate of grade, and from this may be deter* 
mined whether one uniform grade between the cotitrol points iS 































^ 


— 


















1 


■ ^ 


brp 


































— .,i£^ 


■^ 




































«-' 












~'" V 


















— 


L^ 


— 


^^^^ 


— 






J 


?^ 


b^ 


F^ 




:z^ 


-^- 


1 








^^^^ 


^-^ 


— 


[ jB^ ■! 




-A= 


=^ 




^ 


P^- 


i"*'~^ 


^^v- 


¥ 
















1 — - 
1 — 




^ 


C^ 


— /' 














\ 


t*^ 






















^ 






= 


^ 


==^ 






— - 


s — ' 


^^ 


-sr- 










i 


'^^ 






^ 


















!v^ 






^^^^ 


^q 




:v^ 


^ 






^ 








1 














^ 








x^'^ — 

M 






fr^ ■ 






















- S 
















p" 



























[=^ 


^ 


=:^ 


• - - 


_ „ 


hH 


-■rtU— . 











Fig, 7, Single Grade Between Contbol Points. 

practicable, or whether two or more different grades must be 
Used. If the stadia method was used, the profile of a line run- 
ning through the station points will serve the same purpose. Iri 
Fig. 7 let AMZ represent, on a very small scale, the surface 
profile between two control points, A and Z, which are, perhapSj 




Fig. 8. Two Grades Between Control Points, 



two miles apart. The upper dotted hne shows the elevations of 
the highest points in the surveyed belt at each of the several 
stations, and the lower hne the corresponding lowest points. If 
the straight hne AZ does not go outside of these dotted lines, it 
indicates that the uniform grade AZ will have " supporting 
ground " for the entire distance and that such a grade is prac- 
ticable and should be tentatively selected (or at least invest^- 



26 RAILROAD CONSTRUCTION. § 18. 

gated) for that stretch. If the straight Hne AZ passes outside 
the belt of the dotted lines, as in Fig. 8, it implies that there was 
some definite reason why no higher supporting ground could be 
found near M' , or the preliminary survey, if properly made,, 
would have covered that ground. It then becomes necessary 
to adopt two grades, such as AM' and M'Z. Three or more 
grades might prove necessary or desirable in some cases. 

Having determined, at least tentatively and approximately, 
the rate of grade, set a pair of dividers at such a distance (to 
scale) that the distance times the rate of grade equals the con- 
tour interval. For example, with a contour interval of 5 feet 
and a 2% grade, 

distance X .02 = 5, 
or 

distance = 5 -^ .02 = 250. 

Then, with dividers set at 250 feet, put one leg where the line 
previously located crosses a contour and put the other leg where 
it reaches the contour next above — or below, if a down grade. 
Then step to the next contour and so on. If the desired starting 
point is not on a contour, the distance for the first step should be 
proportionately shortened. A strict application of this method 
would probably make a sidehill line run around short gullies 
where the curvature would need to be excessively sharp. To 
avoid such sharp curvature, these narrow gullies must be crossed 
by bridges, trestles or high embankments. To carry a grade 
across such a place, the length of step of the dividers should be 
doubled or trebled and the step should be to the second or third 
contour above or below. The line running through these suc- 
cessive points located on the contours will be practically a surface 
line which has nearly the desired grade. The cut and fill would 
be almost nothing — except " side-hill work," and the crossing of 
gullies. No accuracy need be expected on this preliminary trial 
since the distance is somewhat greater than the air-line distance 
AZ. It would, in general, be impossible to run a practicable 
combination of tangents and proper curves through these points, 
but such a line is very suggestive of a proper alinement which 
will fulfill the grade and curvature conditions and along which 
the cut and fill will be reasonably small. 

If there are long stretches where, in each case, the line joining 
a group of consecutive points is nearly straight, the tangents will 



§ 18. RAILROAD SURVEYS. 27 

predominate and should be located first and then connected by 
curves. If the line has numerous and long bends, it may be 
preferable to select the curves first and then connect them with 
tangents. For such work a series of curves, drawn to proper 
scale, varying by even degrees from 1° up to 15° or 20°, or what- 
ever is the maximum allowable curvature, and drawn on any 
transparent material such as tracing cloth, celluloid or glass, is 
very useful, since different curves may be tried in turn until the 
curve which best fits the ground is discovered. The contours 
and other fixed features should have been inked in and then 
the trial lines and curves may be marked in lightly with a soft 
pencil, so that trial Hnes may be easily erased until a satisfactory 
line is obtained. The number of possible combinations is infinite, 
but certain conditions must be fulfilled which narrows the choice. 
(1) The connecting tangents must not be too short; 100, 200 
and even 300 feet are used as limits. (2) The curvature must be 
within the adopted limit. If two consecutive curves, which are 
connected by a very short tangent, bend in the same direction, 
it is preferable that they should be combined into one simple 
curve, or into two branches of a compound curve, rather than to 
make a " broken-backed " curve. If they bend in opposite 
directions (making a reverse), even 300 feet is none too long for 
the transition curves which should be used, especially if the 
curves are sharp. Actual reverse curves (changing the direction 
of curvature without any separating tangent) should 7ievej be 
used, except on switch work and track where the speed is always 
slow. It would be far preferable to sharpen the curvature 
enough to introduce a tangent at least 100 feet long. The fol- 
lowing considerations should be kept in mind.* 

" (1) If the location could follow the grade Hne [or surface 
line] precisely, there would be no cuts or fills (practically speak- 
ing) on the center line. 

" (2) Whenever the location lies on the \ i •„ i side of the 

[ up-hiU J 

f fill 

grade contour [or surface line] there will be 



cut 

" (3) The further the location departs from the grade contour 
the greater will be the cut or fill, as the case may be." 

* Art. 50, Part IV, "Technic of Surveying Instruments and Methods," 
by Webb and Fish. John Wiley & Sons. 



28 RAILEOAD CONSTRUCTION. ' § 19. 

After a location line has been selected which seems satisfactory 
from the standpoints of easy curvature, not too short tangents, 
a proper balance of cut and fill, and not too great cuts and fills, 
as will be approximately indicated by its distance from the 
surface line, the volume of earthwork may be estimated with 
sufficient accuracy for comparative purposes by drawing a 
profile of the surface location line and its roadbed line. Con- 
sidering the ease with which such lines may be drawn on the 
preliminary map, it is frequently advisable, after making such 
a paper location, to begin all over, draw a new line over some 
specially difficult section and compare results. Profiles of 
such lines may be readily drawn by noting their intersection 
with each contour crossed. Drawing on each profile the re- 
quired grade line will furnish an approximate idea of the com- 
parative amount of earthwork required. A comparison of 
the areas of cut and fill on the profile will show the approxi- 
mate balance in volume of cut and fill. If it is considered nec- 
essary to compute the volume with greater accuracy, it may be 
done by the use of Table XVII (see also § 126), applying the 
latter part of the table correctively to allow for side slope. 
After deciding on the paper location, the length of each 
tangent, the central angle (see § 51), and the radius of each 
curve should be measured as accurately as possible. Frequent 
tie lines and angles should be determined between the plotted 
location line and the preliminary line. When the preliminary 
line has been properly run, its " backbone " line will lie very 
near the location line and will probably cross it at frequent 
intervals, thus rendering it easy to obtain short and numerous 
tie lines. 

19. Preparation of the notes. This and the actual transfer 
of the paper location to the ground is a problem in surveying 
which is so varied in its character that the ingenuity of the engi- 
neer is required to use the best method adapted to each partic- 
ular case, but a few principles may profitably be kept in mind. 

(1) The scale of the paper location drawing is probably 200 feet 
per inch, unless the difficulties of the problem demand a larger 
scale for a particular stretch of the road, so that the paper loca- 
tion may be more accurate. Since a variation of 1/200 inch in 
the drawing means a variation of one foot on the ground, no 
close checking of the line on any tie-point need be expected. 

(2) Since a very small variation in alinement would, if persisted 



I 19. RAILROAD SURVEYS. 29 

ih, throAV the alinement very far from its desired location, it must 
be expected that there will be more or less adjustment of the 
paper location alinement (numerically) on nearly every tangent 
and curve. (3) The intersection of the preliminary line by a 
paper-location tangent (or the tangent produced) gives a pos- 
sible tie-point. The position of this tie-point on the preliminary 
line must be scaled and the angle between the lines determined 
by measuring the chord of a long arc with its center at the point 
of intersection or by scaling the sine (or tangent) produced by a 
perpendicular from one line to the other from a point whose dis- 
tance from the intersection is a convenient unit length. (4) 
When there is no intersection at some place where a tie is desired, 
j k perpendicular offset from the preliminary line may bfe necessary. 
(5) When the paper location crosses the preliminary line at fre- 
quent int'^rvals (say 500 to 1000 feet), it may be more simple td 
locate the tie-point intersections on the preliminary line and work 
from one to the other, taking up the inevitable inaccuracies by 
slight variations in the length of tangents or curves or by some 
one of the various methods detailed in § 63. When no prac* 
ticable tie can be obtained for a considerable distance (say one- 
haK mile), it may be desirable to determine the ordinates (lat-^ 
itudes and departures) of all the points on the preliminary and 
on the paper location between two consecutive intersections^ 
In such a case the precision would depend entirely on the accuracy 
of scaling the positions of the two intersections and on the accu- 
racy of the preliminary survey. While such a method requires 
considerable office computation, even that is cheaper than an 
extensive revision of a located line in the field. For a further 
development of this method, the student is referred to a 
course of instruction originally written by Prof. J. C. L. Fish, 
of Stanford University, and included in " Technic of Surveying 
Instruments and Methods," by Webb and Fish, published by 
Wiley & Sons. 

As previously stated, the above method has been developed 
as if the final located line were to be made up only of tangents 
and circular curves. But transition curves between the tan- 
gents and circular curves are essential for the easy operation of 
trains. Anticipating the more complete demonstration of the 
subject, ^ 71 j et seq., it may be stated that the effect of the transi- 
tion curve, or " spiral," is to move the curve inward, or toward its 
center, or to move the tangent outward. The effect of this is 



30 EAILKOAD CONSTRUCTION. § 20. 

equivalent to offsetting the tangent outward, or offsetting the 
curve inward, and then connecting the tangent and circular 
curve by a transition curve which gradually crosses the offsetted 
distance. The amount of the offset varies with the degree of 
the central curve and the desired length of the transition curve,, 
but it is seldom more than three or four feet, and is usually much! 
less. No consideration need be given to these offsets when; 
comparing several trial locations. It is only after the paper-j 
location has been settled and it is time to transfer this to the] 
ground that it is necessary to compute these offsets and adjust | 
the lines accordingly. Even then the offsets will seldom be so | 
large that they would appreciably affect the paper location, but 
when the alinement is actually located on the ground, the proper 
offsets should be used and the aHnement laid out as described in 
detail in § 80. 

20. Surveying methods. A transit should be used for aline- 
ment, and only precise work is allowable. The transit stations 
should be centered with tacks and should be tied to witness- 
stakes, which should be located outside of the range of the earth- 
work, so that they will neither be dug up nor covered up. All 
original property lines lying within the limits of the right of way 
should be surveyed with reference to the location line, so that 
the right-of-way agent may have a proper basis for settlement. 
When the property lines do not extend far outside of the re- 
quired right of way they are frequently surveyed completely. 

The leveler usually reads the target to the nearest thousandth 
of a. foot on turning-points and bench-marks, but reads to the 
nearest tenth of a foot for the elevation of the ground at stations. 
Considering that -^-^^-jf of a foot has an angular value of about 
otie second at a distance of 200 feet, and that one division of a level- 
bubble is usually about 30 seconds, it may be seen that it is a 
useless refinement to read to thousandths unless corresponding 
care is taken in the use of the level. The leveler should also 
locate his bench-marks outside of the range of earthwork. A 
knob of rock protruding from the ground affords an excellent 
mark. A large nail, driven in the roots of a tree, which is not 
to be disturbed, is also a good mark. These marks should be 
clearly described in the note-book. The leveler should obtain 
the elevation of the ground at all station-points; also at all 
sudden breaks in the profile line, determining also the distance 
of these breaks from the previous even station. This will in- 



§ 21. RAILROAD SURVEYS. 31 

elude the position and elevation of all streams, and even drj' 
<5ullies, which are crossed 

Measurements should preferably be made with a steel tape, 
care being taken on steep ground to insure horizontal measure- 
ments. Stakes are set each 100 feet, and also at the beginning 
and end of all curves. Transit-points (sometimes called " plugs" 
or "hubs") should be driven flush with the ground, and a 
"witness-stake," having the "number " of the station, should 
be set three feet to the right. . For example, the witness-stake 
I might have on one side "137 + 69.92," and on the other side 
"PC4°R," which would signify that the transit hub is 69.92 
feet beyond station 137, or 13769.92 feet from the beginning of 
the line, and also that it is the "point of curve" of a "4° curve" 
which turns to the right. 

Alinement. The alinement is evidently a part of the loca- 
tion survey, but, on account of the magnitude and importance 
of the subject, it will be treated in a separate chapter. 

21. Form of Notes. Although the Form of Notes cannot be 
thoroughly understood until after curves are studied, it is here 
introduced as being the most convenient place. The right-hand 
page should have a sketch showing all roads, streams, and 
property lines crossed with the bearings of those lines. This 
should be drawn to a scale of 100 feet per inch — the quarter- 
inch squares which are usually ruled in note-books giving con- 
venient 25-foot spaces. This sketch will always be more or less 
distorted on curves, since the center line is always shown as 
straight regardless of curves. The station points ("Sta." in 
first column, left-hand page) should be placed opposite to their 
sketched positions, which means that even stations will be 
recorded on every fourth line. This allows three intermediate 
lines for substations, which is ordinarily more than sufficient. 
The notes should read up the page, so that the sketch Avill be 
properly oriented when looking ahead along the line The 
other columns on the left-hand page will be self-explanatory 
when the subject of curves is understood. If the "calculated 
bearings" are based on azimuthal observations, their agreement 
(or constant difference) with the needle readings will form a 
valuable check on the curve calculations and the instrumental 
work. 

22. Number of men required in surveying parties. No fixed 
rules can be given. The general rule of economy and efficiency 



32 



RAILROAD CONSTRUCTION. 



§22. 



FORM OF NOTES. 



[Left-hand page.] 



Sta. 


Aline- 
ment. 


Vernier. 


Tangential 
Deflection. 


Calculated 
Bearing. 


Needle. 


54 












53 

© +72.2 


P.T. 


9° 11' 


18° 22' 


N 54° 48' E 


N 62° 15' E 


52 




7 57 








51 


f 1 

be • 

(_ 03 


6 15 








50 


6 ti 


4 33 








49 




2 51 








48 


o 

CO GO 
1—1 

1 J 


1 09 








3+32 

47 


P.O. 


0° 








46 








N 36° 26' E 


N 44° 0' E 



should govern, and that is, that the organization should be such 
that all desired data can be obtained at a minimum of cost. 
This general r'ule may be Subject to the modification that the 
early completion of the Survey is sometimes financially so impor- 
tant as to justify the maximum speed, almost regardless of 
expense. A common violation of the general rule of economy 
is the Use of too few men, with the mistaken idea that it is eco- 
nomical. This requires the high-priced efficient men to waste 
their time on work which men at one-half (or even one-third) 
their salary could do sufficiently well, thus delaying the com- 
pletion of the work or depfeciating its quality by undue haste 



§22. 



EAILROAD SURVEYS. 



33 



[Right-hand page.] 




53t60 
d) JAS. WILSON 

52+18 




Wm. brown 




or by neglect to obtain complete data, ^he work should be so 
organized that each man is constantly busy at the kind of work 
for which he is especially qualified, and that n© men shall have 
to wait for others to complete their co-ordinate work. Even if 
100% efficiency is unobtainable, it is very uneconomical to have 
nearly the whole party idle while one or two high-priced men 
do some work which must be done before the party can proceed 
but which could have been done by some extra lower-grade men 
without delaying the party. Reconnoissance. When the ter- 
ritory of the general route has been mapped by the U. S. Geol. 
Survej; there jaay be fto »eed of instrumental work on th€» 



34 RAILROAD CONSTRUCTION. § 22, 

reconnoissance, since the approximate ruling grades and general 
route may perhaps be determined directly from the map, and 
the purpose of the reconnoissance is the examination of physical 
features which would affect or modify the general route. In 
such a case the engineer does his technical work alone and only 
needs a guide and cook in case camping is necessary. When the 
reconnoissance partakes more of the nature of a hasty prelim- 
inary, distances, elevations and the necessary side topography 
being determined by rapid approximate methods, more men 
should be added, keeping in mind that the work should be so 
organized that each member of the party is kept busy at his own 
co-ordinate work, and that the chief engineer is not delayed in 
his own special work by spending his valuable time on a cheaper 
grade of work which an assistant could do sufficiently well. In 
other words, it is economical to add to the party an extra assist- 
ant whenever the work that he can do will so facilitate the 
work of the party as a whole that the value of the salaries 
and expenses saved will more than offset the assistant's 
salary and expenses. Preliminary surveys. No fixed list of 
members of a party is applicable to all conditions. The fol- 
lowing list, with monthly salaries, is given by Mr. Fred Lavis* 
as having been used on each of five parties in surveying the 
Choctaw, Oklahoma & Gulf R. R. The list is very full but 
justifiably so. 

Locating engineer $150 to $175 

Assistant locating engineer 115 125 

Transitman 90 100 

Levelman 80 90 

Draftsman 80 90 

Topographers, two 80 90 

Level rodman 50 

Head chainman 50 

Rear chainman 40 

Tapemen, two 30 

Back flagman , 30 

Stake marker 30 

Axemen, three to five 25 to 30 

Cook 50 

Cook's helper 20 

Double teams and driver, furnish their own feed, 

driver boarded in camp 65 to 90 

* Methods of Railroad Location on the Choctaw, Oklahoma & Gulf 
R.R. Trftus. Am, 3oc. C. E., Vol. LIV, page 104, 



§ 22. KAILROAD SURVEYS. 35 

Other organizations sometimes combine the first two positions 
on this hst and possibly call him " chief of party." For the 
above work, the locating engineer was relieved altogether from 
the detailed direction of the party, which was handled by the 
assistant, and spent nearly all his time in studying the country 
so as to determine how the line should advance. In nearly 
all cases, such expense is justified, perhaps many times over, (1) 
by the saving of uselessly surveying an improper route, (2) by 
an improvement in the operating value of the route selected, or 
(3) by an improvement in route which makes a decrease in 
construction cost. Sometimes those controlling the financial 
side of the project insist that the chief of party shall also run 
the transit, as a measure of " economy." Such a policy cannot 
be too strongly condemned. The work of a transitman requires 
every instant of his time and every minute that he turns from 
his transit to direct the party or study the proper route is a min- 
ute delay for the entire party. It generally means also a deteri- 
oration in the quality of his work as a leader and as a transitman, 
in his effort to hastily do at one time work which requires the 
concentrated efforts of two men. In this survey (described by 
Mr. Lavis) , the skeleton or backbone line was a broken line with 
angles every few hundred feet, and the topography was taken 
by right-angled offsets every hundred feet or oftener, substan- 
tially as described in § 11 and Fig. 4. These offsets were deter- 
mined by a hand level and pacing by one of the two topographers. 
The other topographer, using a transit, with the other two tape- 
men " determined drainage areas, located property lines and 
section corners, got names of property owners, etc." When, as 
is usually the case, such essential work cannot be done by the 
main party without delaying their progress, there is a real econ- 
omy in adding to the party these comparatively low-priced 
assistants. It may be noted that the above party includes two 
chainmen, back flagman and stake-marker, beside three to five 
axemen. The proper number of axemen manifestly depends 
on the amount of necessary cutting, but the chainmen or the 
stake-marker should not be depended on for such work. The 
steady march of the party should not be halted while a stake- 
marker or chainman stops his regular work to cut down a tree. 
One of the duties of the chief of party is to foresee the necessities 
of tree-cutting and clearing, so far in advance that, by the time 
the surveying members of the party have reached the spot, the 



36 RAILROAD CONSTRUCTION. § 23. 

area is clviafed. It is likewise false economy to dispense with 
the stake-marker and require the head chainman to do such work. 
A full corps of such men, properly drilled, can add 20 to 50% 
to the daily progress of the party and much more than save 
their cost, 

MAINTENANCE OF StJRVEY PARTIES. 

23. Economy and efficiency. When considering the treat- 
ment and maintenance of surveying parties, it should be remem- 
bered that a false idea of economy is frequently responsible for 
making the parties too small, overworking the men, depriving 
them of physical comforts and even necessities, and that the 
result is a greater net cost and a great deterioration in the quality 
of the results. A party may cost $40 to $65 per day in salaries 
and expenses. Any policy which depreciates the net output 
of their work 20 to 50% (which is easily possible) in order to 
save a few dollars per day is manifestly poor policy. The men, 
especially those who must use their brains and who presumably 
have a finer nervous organism, have only a quite definite sum 
total of nervous energy. If a considerable part of that energy 
is spent in needlessly long tramps both morning and evening to 
and from work, or if that nervous energy is not maintained by 
plentiful and appetizing food and by sufficient and comfortable 
rest, there is a reduction in efficiency which is often far greater 
than any possible saving in expenses. This idea of developing 
the maximum efficiency of the party is the justification of the 
recommendations made below regarding outfit, equipment, and 
other details about managing a party. 

24. Country hotels and farm houses. In settled sections 
of the country, country hotels and even farm houses are some- 
times available where men can be provided with living facilities 
which are unobtainable in camp life and at less total expense. 
Such accommodations have the. advantage that they obviate a 
considerable capital expenditure to purchase sufficient campi 
outfit. But if suitable accommodations are unobtainable over 
a considerable portion of the route and such accommodations 
as there are on the remaining distance are inconvenient and . 
inadequate, it may be preferable to provide a camping outfit at 
once. Considering the fact that there is a real economy in 
making a survey with a large party and that such a party can 



1 25^ RAILROAD SURVEYS. 37 

seldom if ever be accommodated in a single farmhouse, and that 
there is a lack of efficiency if the party is separated, the farm- 
house plan is frequently impractical. But when villages are so 
located that there is always one within five miles of any point of 
Ijhe line, the house plan may be preferable, since the party 
may b© taken to and from work in conveyances. The economy 
of employing conveyances may be judged by comparing the cost 
of the vehicles and the value of the time and energy saved. A 
five-mile tramp, carrying an instrument, following a full day's 
work surveying, will frequently incapacitate a man from doing 
effective work in the night-work which the higher grade men of 
the party must generally do. The day's work in the field must 
be begun later and ended earlier or else the time and strength 
spent in the morning and evening tramps are uneconomical 
drains on their total nervous energy. 

25. Camping Outfits: Tents. The Choctaw, Oklahoma & 
GuM R.R. survey, previously referred to, provided for each 
party one office tent, with fly, 14X16 feet, three tents, evidently 
without flies, 14X16 feet, and one cook tent 16X20 feet. The 
office tent had 5-foot walls; the others 4-foot. H. M. Wilson 
("Topographical Surveying," p. 817) recommends 9X9 foot 
tents, with 4-foot walls. These are easier to erect but have only 
36% of the floor area of the 14 X 16-foot tents and it would 
require 15 such tents to equal the floor area of the 5 tents 
described above. For a small party the snialler tents would be 
preferable. The canvas should be mildew-proof and free from 
sizing. A " sod-flap " about 8 inches wide, should be attached 
to the bottom of the wall. When this flap is weighted down 
with stones or heavy sticks the wind and weather is kept out. 
Dirt or sod should not be used for weights, since they rot the 
canvas. It pays to use tents which conform to the U. S. Army 
specifications. Some of the specifications as to material and 
workmanship are here quoted: 

" Materials. — Body of tent to be made of Army standard 12m 
ounce cotton duck, 29| inches wide and the sod cloth of Army 
standard 8-ounce cotton duck, 28| inches wide. 

" Workmanship. — To be made by machine in a workmanlike 
manner, all seams to be stitched with two rows of stitching, not 
less than six stitches to the inch, with three-cord twelve-thread 
Sea Island cotton, white. 

" In making tents by hand, to, have not less than two and one-! 



38 RAILROAD CONSTRUCTION. § 26. 

half stitches of equal length to the inch, made with a double 
thread of five-fold cotton twine, drab, well waxed. 

" The seams should be not less than 1 inch in width, flatj 
stitched, and no slack taken in them. 

"Grommet holes. — Made with malleable iron rings, galvanized, 
to be worked with four-thread five-fold cotton twine, well waxed. • 

" Sod cloth. — To be 8 inches in width in the clear from the; 
tabhng, into which it is inserted 1 inch and extending from door';; 
seam to door seam around the tent. 

" Tabling. — On foot of tent when finished to be 2^ inches in 
width." (Adopted July 14, 1911.) 

A ditch should be dug outside the tent, at least on the up-hil 
side, if the ground is at all inclined. This will prevent rain- 
water from draining through the tent. Of course, the bottom of 
the ditch should have a uniform slope draining to an outfall 
amply clear of the tent. 

26. Tent floors. Dry floors are almost essential to health. 
Sectional floors, about 3X9 feet per section, made by fastening 
boards to cross cleats, provide a perfectly dry floor and often 
repay their transportation. A mere layer of canvas, cut to 
proper shape and bound on the edges, is worth providing if the 
ground is dry when the tent is erected and can be kept from 
getting rainsoaked by proper outside drainage. 

27. Tent stoves. For winter work, tents may be made quite 
comfortable with stoves. Oil stoves are convenient when the 
oil can be purchased without excessive cost for transportation. 
" Sibley " stoves, burning wood, are commonly used but they 
require smoke pipes which must pass through the canvas and 
this means that the holes must be properly protected with metal 
or asbestos. If a pipe elbow is provided, the pipe may be taken 
out through one end of the tent. This obviates a hole in the 
roof of the tent (and also the fly) ; it avoids a direct pour of rain 
on the fire or leakage into the tent around the pipe, and also 
the danger of sparks dropping on the canvas. A " Sibley " 
stove for mere heating is a sheet-iron frustum of a cone, about 
3 feet high; diameter at bottom 18 to 30 inches; diameter at 
top 4^ to 6 inches, or so as to fit the stovepipe which is to be 
used. It has no bottom, or in other words, the bare earth forms 
the base. A door, large enough for the insertion of such fuel 
as it is designed to use, is placed in the side. Three or four 
lengths of pipe, one of which should have a damper, and an elbow, 



§ 28. RAILEOAD SURVEYS. 39 

should be provided. Draft at the bottom is obtained, and may 
be easily controlled, by packing earth around the base, leaving 
a small opening which may be easily enlarged or diminished to 
control the draft. Cook stove. A regular 6-hole cooking range, 
perhaps made of wrought-iron or sheet-steel, is essential to cook 
meals for twenty or more hearty men. Sporting outfitters 
supply all sizes of stoves, which must always be selected with 
due regard for the facilities for transportation. Oil stoves are 
commonly used. For still smaller parties, or when no cook 
stove can be permitted in the baggage, a primitive grid may be 
made from four sticks of green timber about 6 inches in diameter 
and 2 to 4 feet long. Notch two of them, each with a pair of 
notches about 10 inches apart. Place the other two sticks 
across the notches and they will steadily support a kettle or a 
frying pan. If the sticks are sufficiently green and the fuel quite 
dry the grid will last some time. A folding grid of iron bars may 
be obtained, which is but a small addition to the weight of the 
baggage. Another method is to suspend a kettle by a chain 
or long hook either from a tripod of sticks or from a horizontal 
stick lying in two forked sticks on each side of the fire. 

28. Dining tables. These are justifiable for a large party 
when the baggage is necessarily great and camp wagons are a 
part of the equipment. Mr. Lavis, in the article previously 
referred to, describes a very good table from the standpoint of 
transportation. The table top consists of three loose planks 
If" X 12" X 18' 0". Two similar boards are used for seats. 
During transportation these boards are placed on the bottom 
of the wagon and, of course, project from the back where they 
form a support for stoves, etc., which can be roped on. These 
boards are supported on three trestles or horses, made as shown. 
For a much smaller party, a table may be improvised by utilizing 
two " mess-boxes," which carry the cooking utensils and table- 
ware. These mess-boxes are about 20 inches wide and high 
and from 24 to 30 inches long. The covers are made to open 
180° and may be fastened horizontally. An " inside cover," 
which can be utilized as a bread board, covers the entire inside 
area of the box. Two such boxes, set together and with the tops 
opened out, provide a fairly even surface four times the area 
of one box. 

29. Cooking utensils, table-ware, tools, etc. The size of the 
party, the individual preferences of the person designing the 



40 



EAILROAD CONSTRUCTION. 



§29. 



outfit and the facilitiee for transportation, vary such hsts almost 
indefinitely. Agate ware has replaced china for plates and cups. 
Aluminum ware, although expensive, is preferable from a cooking 
standpoint and has the advantage of a very material reduction 
in weight. Out of the very great number of lists which have 
been published, the following list of articles is quoted as sug-| 
gestive: Plates, cups, saucers, steel knives and forks, German-!-! 
silver spoons, large and small, carving knives and forks, large', 
cooking forks and spoonSj pepper and salt boxes, tin pans about 



FIT MORTISE AND TENONS 
AT BOTH ENDS OF POSTS 
TIGHT BUT DO NOT FASTEN, 
SO HORSE WILL KNOCK DOWN 









2 SEAT BOARDS 
l?i*x lOx 

.18'0'd.o. 




Fig. 9. — Camp Dining Table. 

6 inches diameter by 1| inches deep, utilized for serving soup, 
<36real, etc., pans and kettles of varying sizes which will " nest " 
and thus facilitate packing, tea kettle, coffee pot, frying pan, 
griddte, cake turner, pie pktes, dripping pan, chopping bowl 
and chopper, colander, flour sieve, coffee mill, broiler, corkscrew 
and can opener, rolling pin, folding table (similar to the drawing 
table described below), wash basins, kerosene oil can, alarm 
clock, spring balance. The last two articles are important. 
The cook is the first man up in the morning — usually before 
daylight — and may need the alarm clock. A single delay, of , 
ieyen ten minutes of such a party, would cost more than a very : 
valuable clock. A spring balance is very essential to the proper 



29. 



BAILROAD BtJRVEYB. 



41 



£tnd economical use of provisions without waste. It pays to 
have a cook who is able to compute, weigh out and use an amount 



CLEATS FASTENED 
TO TOP WITH EIGHJ 




HOLES FOR TWO SCREWS 
WILL FASTEN LEGS SECUR 
FOLOEP 




Fig. 10. — Folding Drafting Table. 



d each kind of provisions so that there will be sufficient, but 
po waste. Besides the above, dish towels are practically essen- 



42 EAILROAD CONSTRUCTION. § 30. 

tial and tablecloths and napkins are easily carried. A table 
oilcloth may replace the ordinary tablecloth. Wash tubs and 
wash board facilitate the washing of table linen and also under- 
wear, so essential to clean, healthy living. Illumination for 
night work must be provided. Reflecting lanterns will answer 
for all tents except the office tent, where good lamps, with 
cylindrical wick and center draft, or similar, should be provided. 
The farther the party travels from " civilization " the greater 
the necessity for providing for emergencies, breakages, etc. 
Axes are essential, apart from their use in the surveying work. 
Extra handles should be provided. A saw, brace and several 
sizes of bits, screw drivers, monkey wrench, files, pliers, hatchet, 
assorted screws and nails, pick, shovel, crowbar, whetstone, 
rope in various sizes, sailor's needles, palm and sewing twine, 
will all be useful and even invaluable in times of emergency. 
Canvas-covered canteens, for each member of the party, when 
passing through arid regions, may be essential. 

30. Drawing tables. Complete topographic drawings, made 
in the field, are absolutely essential. Suitable drawing boards 
are, therefore, required. The design shown in Fig. 10 fulfills all 
the working requirements; it also is easily handled when, packed 
up and is not readily broken. By packing them together in 
pairs, face to face, the surfaces are protected during transpor- 
tation. The table consists essentially of a drawing board with 
stiffening cleats. The legs are hinged to the cleats, the braces 
for each pair of legs being of just such a length that when opened 
the legs stand at the desired angle. The braces are hinged and 
fold up, jackkiiife fashion, so that they nowhere project beyond 
the legs. 

31. Stationery and map chest. Considering that the maps, 
drawings and notebooks may represent thousands of dollars, 
and that they are likely to be injured, if not irreparably ruined, 
by rain, when moving camp or during a cyclonic storm, a strong, 
water-tight chest, of ample capacity for all drawings and note- 
books, should be provided. It should be required that all 
drawings and notebooks should be kept in the chest over night 
and at all other times, except such drawings and notebooks as are 
in actual use. The net inside length should be a little in excess 
of the longest roll or drawing, which is perhaps 36 inches. There 
' should be a tray in the top with numerous compartments or 
boxes for the multitudinous small articles required by a drafts- 



32. 



RAILROAD SURVEYS. 



43 



man. Handles should be provided for convenience and it 
should have a lock. A good " steamer " trunk of requisite size 
will answer the purpose, provided it is waterproof, and it would 
perhaps be cheaper than a chest of similar size, made to order. 
32. Provisions. A "ration" is the estimated amount of food 
required per man per day. For men engaged in strenuous out- 
door work, the food required is far more than that eaten ordina- 
rily. Ration lists should average about 5 to 6 pounds of food 
per day per man. The amount that must be transported may 
be considerably less than this, in view of the fact that e.g., dried 
vegetables may be substituted for fresh vegetables in the ratio 
bf 1 lb. of dried for 3 lbs. of fresh, the water used in cooking 
providing the other two pounds. For explorers, who carry their 
own provisions, and who must cut down every possible ounce of 
baggage, still further concentrations are possible. 



Article 

Fresh meat, including fish and poultry, (a) 

Cured meat, canned meat-, or cheese (6) 

Lard 

Flour, bread or crackers 

Corn meal, cereals, macaroni, sago, or cornstarch 

Baking powder or yeast cakes 

Sugar 

Molasses 

Coffee 

Tea, chocolate or cocoa 

Milk, condensed (c) 

Butter 

Dried fruits (<f) 

Rice or beans 

Potatoes, or other fresh vegetables (e) 

Canned vegetables or fruit 

Spices 

Flavoring extracts 

Pepper or mustard 

Salt 

Pickles 

Vinegar' 



100 rations 


100 lbs. 


50 " 


15 " 


80 " 


15 " 


5 " 


40 " 


^ 1 gal. 


12 lbs. 


2 " 


10 cam 


10 lbs. 


20 " 


20 " 


100 " 


30 " 


1 • « 


s 


i " 


1 < < 


5 


4 " 


3 qts. 


1 " 



" (a) Eggs may be substituted for fresh meat in the ratio of 8 eggs for 1 lb. 
of meat. 

" (b) Fresh meat and cured meat may be interchanged on the basis of 5 lbs. j 
of fresh for 2 lbs. of cured. [This ratio 5:2 is far higher than is usually 
allowed, 5:3 or even less is usually stated as the equivalent ratio.] 

" (c) Fresh milk may be substituted for condensed milk in the ratio of 5 
quarts of fresh for 1 can of condensed. 

" (d) Fresh fruit. may be substituted for dried fruit in the ratio of 5 lbs. 
of fresh for 1 of dried. j \ 

" (e) Dried vegetables may be substituted for fresh vegetables in the ratio 
of 3 lbs. of fresh for 1 lb. of dried." 



44 



RAILBOAD CONSTRUCTION. 



§321 



The list at bottom of p. 43 Is given by H. M. Wilson ("Topo. 
graphic Surveying ") as the ration list of the U. S. Geol. Survey 
The quantities are those required to niake up 100 rations, or the 
food for 5 njen for 20 days, or for 100 men for one day. Thej 
are considered maximum. The sum total is about 525 lbs. oi 
51 lbs. per day per man. 

Wilson states that the cost of the above list of rations should 
not average more than 45 to 55 cents per day for average con-' 
ditions and with a maximum of 75 cents, but considering that 
this statement was written in 1900, some allowance may neecl 
to be made for higher prices since then. 

The list given below represents the provisions actually supplied 
to a mining camp in British Columbia. The list has been reduced 
to the average quantity actually consumed per man per day. 
The food supply averaged nearly 6 lbs. per day per man. 



Meat, etc.: 

Fresh beef 1 . 89 lbs. 

Bacon 076 " 

Ham 060 " 

Codfish 007 •• 

Canpjed salmon 014 can 

Breads, etc.: 

Pilot bread 007 lb. 

Flour .894 " 

Bakihg powder 016 " 

Corn meal 037 " 

Vegetables: 

Potatoes 1 . 421 lbs. 

Turnips 010 " 

Carrots 047 " 

Beets 016 " 

Parsnips .023 " 

Rice .043 " 

Cabbage 101 " 

Dehydrated onions. . . . 0014 ' ' 
rhubarb. .0029 " 

White beans 0014 *' 

Bayo *• ........ .027 " 

Lima '♦ 013 " 

Split peas 006 " 

Rowan «■• '.. .0014 *' 

Canned tomatoes 016 can 

• ' beans 0043 ' * 

•• peas... .0014 •• 

Pearl barley . 0004 lb. 

Rolled oats. ........ .117 !' 

Beverages: 

Tea 021 lb. 

Coffee 036 " 

Milk, condensed 137 can 



Fruit: 

Dried apples ,040 

" pears 033 

' ' peaches 029 

" prunes 020 

• ' apricots 007 

" figs . .030 

Dehydrated cramberries . 004 

Currants . 021 

Jam 001 



lb. 



Condiments, etc: 

Mustard 

Salt 

Pepper 

Vinegar, Klondyke. . 
Worcestershire sauce . 
Catsup 



pint 



lb. 



.001 
.036 " 
.001 " 
.0003 pint 
.0043 " 
.0029 gal. 



Miscellaneous: 

Sugar 594 lb. 

Lard 030 " 

Cheese 016 " 

Cornstarch 007 " 

Extract 049 " 

Curry powder 0007 " 

Cinnamon 0009" 

Hops 0001 " 

iSTutmeg .. .0009 " 

Ginger 0014 " 

Mapleine 0011 oz. 

Candied peel 004 lb. 

Butter. .014 " 

Macaroni 003 ' ' 

Sago Oil *' 

Tapioca 003 ' ' 

Baker's chocolate 0014 " 

Cocoanut 0003 " 

Pickles 003 gal. 

Supplies: candles, .03 lb.; gold dust, 
.003 lb.; soap. .024 bar. 



§33. 



RAILROAD SURVEYS, 



45 



The following list of provisions was bought to start a camp of 
20 to 25 men on the Choctaw, Oklahoma & Gulf R. R. Survey. 
(F. Lavis, Trans. Am. Soc. C. E., Vol. LIV, p. 104.) 



6 hams 


100 cakes soap 


6 pieces of bacon 


1 gal. molasses 


,50 lbs 


. fresh beef 


1 case condensed milk 


■ 1 case eggs 


1 doz. tomato catsup 


"'25 lbs 


. butter 


1 ' ' Worcestershire sauce 


25 " 


lard 


1 gal. pickles 


100 " 


flour, hard wheat 


J doz. lemon extract 


100 " 


flour, soft wheat 


I " vanilla extract 


100 " 


sugar 


1 box dried prunes 


5 " 


baking powder 


5 lbs. raisins 


2 " 


tea 


4 doz. assorted canned fruits 


50 " 


coffee 


1 case tomatoes 


50 " 


navy beans 


1 bushel potatoes 


25 " 


lima beans 


1 kit salt mackerel 


12 " 


buckwheat flour 


20 lbs. salt 


5 " 


macaroni 


1 ' ' mustard 


35 " 


cornmeal 


1 ' ' pepper 


1 cheese, about 15 lbs. 


1 qt. vinegar 


12 packages oatmeal 


^ doz. yeast cakes 


10 lbs 


rice 1 





In addition to the above, there must be provided plenty of 
matches, kerosene oil and perhaps candles. As a matter of 
health conservation, and the prevention of piles, it is wise to 
provide toilet paper and to insist, if necessary, on its use. There 
is economy, when it is practicable, in making wholesale con- 
tracts for all provisions, rather than to buy haphazard from small 
local sources. 

33. Beds. When baggage wagons accompany the party, as is 
virtually necessary to transport other essential equipment, it is 
desirable that they also transport army cots. These fold up so 
as to be easily transportable. It is a wise economy to obtain 
the regular army blankets, since they are what long experience 
has approved. Bedding rolls should be provided for the bed- 
ding. This is essential to keep the bedding in even reasonably 
cleanly condition, especially while moving. The policy of 
requiring each member of the party to provide himself with cot, 
bedding and cover, and to care for them, is debatable. As a 
matter of business economy, the company should buy all cots 
and bedding wholesale. Requiring each one to purchase his 
own is virtually a reduction of salary, for, if a man leaves the 
party, he usually does not care to take his bedding with him, 
except in the hope of realizing something on it. But as all this 
is considered when accepting employment, the company vir- 
tually pays for the bedding by an increase of salary over what 



46 RAILROAD CONSTRUCTION. § 34. 

they would have to pay if bedding were provided. There is the 
same reason for owning bedding as for owning dishes, etc. 
SteriHzing bedding by means of a formaldehyde candle, especially 
after a man has left the party, is a wise sanitary precaution and 
nullifies one of the strongest reasons for individual ownership. 
34. Transportation. The route of travel of a mining engineer, 
a topographical engineer or an explorer, may be over country 
with every variety of surface and slope. But, since a practicable 
railroad route is necessarily on a low grade, except as it may pass 
over a ridge or mountain to be pierced by a tunnel, the question 
of grade does not ordinarily influence the method of transporta- 
tion and wagons can ordinarily be used, provided the nature of 
the surface will permit. Strong and heavy wagons can usually 
pick their way between the camping places, even though long 
detours must be made to avoid swamps or other obstructions. 
The parties surveying the Choctaw, Oklahoma & Gulf R. R., pre- 
viously referred to, used two teams regularly, one of which stayed 
with the topographical party. They used a third team for 
hauling supplies. Two teams of horses can help each other over 
a particularly bad place in the trail or. in the case of accident. 
The wagons should have canvas tops, as a protection against 
rain, especially while moving. Transportation by dogs and 
sledges is only applicable under very limited and unusual condi- 
tions. It implies winter work, which is always uneconomical 
and inefficient compared with summer work, but in a very 
swampy country, where the transportation of any considerable 
amount of baggage is very difficult, and where it freezes during 
the winter to a comparatively smooth surface, such a method 
may be preferable in spite of short daylight hours and other 
disadvantages. " The Duluth, South Shore & Atlantic Rail- 
way employed toboggans during the construction of its road 
throughout the season of 1887." The description of this work, 
and much other useful information is given in a paper by Chas. 
H. Snow, Vol. XXIX, p. 164, in the Trans. Am. Inst. Mining 
Eng'rs. A reconnoissance through a comparatively unexplored 
country, made with the object of discovering a practicable low- 
grade route through a mountainous section, might require that 
all baggage shall be reduced to what may be handled in packs 
carried by horses, mules, Indian ponies or even by men. The 
question of the necessary method of transportation must always 
be studied before beginning a survey, since the entire question 



§ 35. RAILROAD SURVEYS. '47 

of subsistence, and even many features of the method of work, 
must depend on what can be included in the baggage. 

35. Clothing. While it may seem an unwarranted inter- 
ference with personal liberty to control the clothing worn by 
members of the party, it becomes justifiable when the efficiency 
and progress of the party is impaired by bad health or disability, 
which is plainly due to neglect of proper precautions in the way 
of clothing or personal sanitation. Sore feet are responsible for 
a large part of the disablement of men. Washing the feet 
every night, especially when they have become wet, will often 
obviate blisters. Stockings should be heavy, made of " natural 
wool " and should fit tightly enough so that wrinkles will not 
form. Shoes should have heavy soles and should be made of 
such tough leather that they will not easily tear. Rubber boots 
should not be worn; they make the feet tender. Although a 
survejrrng trip is usually considered as the opportunity to use up 
discarded clothing, ordinary clothing is usually very unsuitable 
and quickly becomes unwearable. When camping conditions 
are rough and the work must last for several months, and possi- 
bly years, clothing made of specially suitable material is econom- 
ical. The material should be tough, so that it will not easily 
be torn by brambles, etc. It should be waterproof so as to 
shed rain and yet should be porous. It should be so thoroughy 
shrunken that moisture cannot appreciably shrink it further. 
"Mackinaw" is a soft, rough cloth, all wool, thoroughly shrunken, 
hght, warm and waterproof. It is especially suitable for cold 
weather. " Pontiac " is similar. " Khaki " is a twilled cotton 
and is especially suitable for warm weather clothing. " Jungle 
cloth " is somewhat similar, but is particularly noted for its 
toughness and durability. 

Especial care should be taken in the choice of underclothing, 
so as to avoid sudden chills after becoming overheated. Woolen 
underclothing is almost essential. " Cholera bands," made of 
wool, should always be worn about _ the abdomen in tropical 
countries, 

MEDICAL AND STJRGICAL TREATMENT. 

36. Responsibility of engiiieer-in-charge. Throughout any 
surveying trip, where camping is necessary, professional medical 
aid is usually unobtainable. There rests upon the engineer-in- 



48 RAILEOAD CONSTRUCTION. § 37. 

charge, as the head of the expedition, some measure of responsi- 
bility for the health and care of the party. When some member 
of the party is seriously injured by accident, bitten by a poisonous 
snake or insect, or stricken with a sudden and violent attack qfl 
disease, and competent medical assistance is absolutely unob- 
tainable for several days or even weeks, the head of the party 
must choose between seeing the victim die or boldly performing 
some simple surgical operation or giving medical treatment 
which he would not dream of doing otherwise. It is the lesser of 
two evils and the engineer must not shirk his duty. Even though 
a doctor is perhaps obtainable after many days delay by 
despatching a messenger 50 miles for him, common sense first- 
aid work and the intelligent use of a few simple methods and 
remedies may save life or prevent or mitigate permanent dis- 
ablement. The outfit should include a sufficient supply of the 
medicines and medical appliances which would most probably 
be required. All bottles should be carried in cases to prevent 
breakage and the corks or stoppers secured tightly. When prac- 
ticable, the drugs should be in tablet form, rather than liquid, 
and a normal dose should be marked on each bottle or package. 
They should be doubly labeled and the labels varnished to 
prevent their coming off in a damp climate. All adhesive 
plasters, antiseptic gauze, and such appliances, should be kept 
carefully wrapped up and protected from air and moisture. 

37. Appliances. The very simplest medical outfit should 
include a pair of good scissors, which can be made antiseptically 
clean by wiping off with alcohol ; a knife with two razor-sharp ' 
blades; a probe; a small saw; dentist's forceps; a pair of mousey- 
tooth forceps; a hypodermic syringe and two needles, or tliQ 
more modern individual hypodermic syringe packages ; also indir 
vidual needles and cat-gut " No. 2 chromic " in curved vacuum 
tubes; a two-quart fountain syringe; supplies of sterilized gauze, 
adhesive plaster, needles, safety pins. The engineer shoul4 
thoroughly familiarize himself with the working and manner of 
use of these. Any engineer who is preparing to head an expedi" 
tion into a region where medical attention is unobtainable should 
consider that he can very wisely spend time with some doctor 
friend in learning the elements of the use of all these appliances. 

38. Antiseptics. The engineer should warn his men of the 
danger from the infection of even slight wounds and scratchesj 
especially in hot climates. The best emergency treatment for 



§ 39. JlAILJtOAD SUHVEYB. 49 

any scratch, nail gouge, or riail in the foot, is to apply pure tinc- 
ture of iodine at the base of the wound by cotton on the end of 
a small stick or probe, A more modern safe-guard against 
tetanus, or " lock-jaw," is '' tetanus antitoxin," put up in indi- 
vidual syringe packages. A few of the many effective antiseptics 
are here mentioned : Borie ointment ; one part of powdered boric 
acid added to nine parts of vaseline. Carbolic ointment; one 
part of carbolic acid to nineteen parts of vaseline. Iodoform 
powder promotes rapid healing of sores and wounds; one part 
in eight parts of vaseline is a good healing ointment. Perman- 
ganate of potash; one grain gives a purple color to a gallon of 
water; if the water is impure, the purple color changes rapidly 
to brown and this is a rough test of organic impurity; the crystals 
are soluble in 20 parts of water; it is especially useful in the treat- 
ment of snake bites. In a snake-infested country, it is wise for 
each man to carry permanganate of potash crystals with him, 
for use in emergency, See '^ Snake bites," § 44, 

39. Drinking water. Every chief of party should see to it 
that his party has a pure supply of drinking water and especially 
that this supply is not contaminated by excrement from the 
camp draining into it. If there is any doubt about the purity 
of the supply (especially if so indicated by the permanganate- 
of-potash test) it should be part of the duty of the camp cook to 
maintain a liberal supply of boiled and cooled water. A neglect 
of such a precaution might easily result in an epidemic of typhoid. 
In a region where all streams are contaminated, perhaps by 
decaying vegetation or other natural cause, it may be wise to 
provide canteens, which the cook should furnish each morning 
filled with sterilized water. 

40. Bleeding from an artery or vein can sometimes be stopped 
by pressing the vessel with sufficient pressure to stop the flow 
and continuing the pressure until the blood coagulates. If the 
vein or artery is actually severed but is not too large, the bleed- 
ing may be stopped by the use of a pair of forceps; grasp and 
pinch the vessel and twist it around three or four times. In 
about ten minutes the forceps may be removed. If the vein or 
artery is larger, and especially when it is an artery, which may 
be recognized by spurts of bright red blood, it may be necessary 
to tie the vessel. This may be done with catgut ligature, which 
should previously be boiled to prevent infection. While pre- 
paring for this, bleeding should be stopped by temporary pres- 



50 RAILROAD CONSTRUCTION. §41. 

sure. This is most easily done when the bleeding vessel may be 
pressed against a bone. A tourniquet can be improvised for 
pressing a pad (or even a' stone) against the vein or artery of a 
limb by using a stick and a piece of cloth, or, perhaps, a rope and 
a small block of wood. Fasten the cloth or rope into a loose loop 
around the limb and run the stick through the loop; then twist 
it so that the pad is pressed down as desired. The rope can be 
so disposed as to press the block, which in turn presses the pad 
against the vein or artery. 

41. Ailments and diseases; medicines; treatment. 

Colic or cramp. Essence of ginger, 5 to 20 drops, in a small 
amount of very hot water. 

Diarrhoea. Remove the bowel irritant by a castor-oil purge; 
then, if diarrhoea continues, give one-half teaspoonful of bismuth 
sub-carbonate every two or three hours until relieved. 

Purgatives. Epsom salts; dose, two teaspoonsful in a small 
glass of hot water. Calomel; dose, two to five grains; should 
be followed by Epsom salts. Cascara sagrada; dose, two to 
six grains. Castor oil; dose, one to three tablespoonsful, 
which may be made more palatable by mixing with an equal 
amount of glycerine, and then putting the mixture into a glass 
of lemonade. Any tendency to constipation, which leads to 
intestinal poisoning and appendicitis, may be avoided by 
using a laxative, made as effective as necessary, about once a 
week. 

Emetics. Common salt (two tablespoonsful), or mustard (one 
tablespoonful) or Ipecacuanha (30 grains) or Zinc Sulphate (30 
grains), dissolved in a glass of water. Tickling the throat with 
a feather may sometimes be effective. Strong " Ivory " soap 
suds is excellent. 

Malaria. Five grains of Quinine as a preventive; ten grains, 
three times a day, as an ordinary maximum dose. Larger doses 
are often given but it is dangerous imless under the care of a 
physician. 

Cold-in-head. Rhinitis tablets, given as directed on bottle, 
are effective to break up an incipient cold. ** Dover's powder "; 
dose, five to ten grains. Keep patient warm, with hot-water 
bottles and hot drinks. 

42. Drowning; electric shock, asphyxiation. The trouble and 
the remedy is essentially the same in all three cases; respiration 
has been temporarily suspended and must be promptly restored 



§43. RAILROAD SURVEYS. 51 

by artificial means. Loosen the patient's clothjng, especially 
about the neck. In a drowning case, lay the patient on the 
ground, face down, straddle him and raise him at the hips so 
that the water in the air passages will drain out. Remove from 
the mouth any tobacco, false teeth or anything else that might 
obstruct breathing. Draw the tongue forward with forceps or a 
handkerchief. Then lay him face down, but with the face 
turned to one side so as to facilitate breathing, and with the arms 
extended forward. Then the operator, kneeling astride the 
patient, facing his head, and with the hands pressing on the lower 
ribs, gradually presses down so as to expel the air from the 
lungs. Then he suddenly removes the pressure by swinging 
back, and thus allows air to enter the lungs. Repeat the move- 
ments every four or five seconds, until natural breathing com- 
mences. Considering the fact that this method has successfully 
restored breathing after some hours of unsuccessful effort, 
and also that, in those cases, the patient would have died except 
for the persistency of the effort, the operator must not be dis- 
couraged because his efforts are not immediately successful. 
Promptness in beginning such treatment is so important that 
it is better to commence at once (even outdoors) rather than 
allow any material delay in order to get the patient to a house. 
The patient should be allowed plenty of air; crowding around 
him should be avoided. A blanket, extra clothing, hot bricks 
or stones, or hot-water bags, to restore heat to the body, will be 
of assistance, provided they do not interfere with the respiration 
operations. Do not attempt to make the patient swallow any- 
thing (e. g., a stimulant), until he is fully conscious; otherwise 
he will choke. 

43. Fractures. Obtain medical aid if possible, but if this is 
unobtainable, except after a delay of many days or weeks, and 
it is uncertain even then, it may be preferable to take the chances 
of common-sense treatment, even if unskilled, rather than the 
certain permanent injury due to neglect of all treatment. Frac- 
tures are (a) simple, when the skin is not broken; (b) compound, 
when the skin is so broken that the fractured bone is more or less 
exposed to the air; and (c) comminuted, when there are two or 
more breaks of the same bone; a comminuted fracture may be 
simple or compound. Great care should be used in handling 
the patient immediately after the accident so that a simple frac- 
ture does not become compound. A broken limb should fe^ 



52 KAILROAD CONSTRUCTION. § 44. 

carefully straightened out and bound temporarily with the best 
improvised splints which are available until the patient can be 
removed to a bed. Even if amateur bone setting is decided 
to be advisable, setting should not be attempted if there is exces- 
sive swelling or tenderness. Apply ice or evaporating lotions 
to reduce any swelling. Splints should be made which are of 
proper length and are so rounded and padded with cloth that 
they cannot produce any concentrated pressure. Usually the 
dislocated bones are forced past each other, especially if the frac- 
ture is oblique rather than perpendicular, and it is always neces- 
sary to use considerable force, especially if it is a broken leg, to 
pull the bones back into position, The amateur must use his 
best common sense and knowledge of skeleton anatomy to restore 
the fragments to the same relative position they had previously, 
and then to secure them rigidly stiff with splints. Comparison 
with an unbroken arm or leg will be made even by a skilled 
surgeon, and such a comparison should be carefully studied by 
the amateur. While the binding should be as firm as it is safe 
to make it, it may be so tight as to produce swelling and even 
ulceration, and then the binding must be loosened. Compound 
fractures require the care of the flesh and skin wound in addition 
to the bone setting. The wound should be treated as described 
for wounds, but the sphnts and binding should be designed so 
that the wound can be properly dressed without loosening the 
splints. If the broken bone protrudes through the wound, it 
must be drawn back so that the wound can heal externally, 
even though the bone setting is beyond the skill of the amateur 
surgeon. Setting usually requires about six weeks, but, in the 
case of a limb, the joints above and below the break should be 
very carefully moved after about three weeks, so as to avoid 
stiff joints, special care being taken that there is no strain on the 
healing bone. 

44. Snake or insect bites. The majority of snake bites occur 
on the limbs. In such a case (1) tie a cord or bandage about the 
limb just above the wound as promptly as possible, so as to 
prevent the poisoned blood from getting into the system; (2) 
cut into the wound so as to induce free bleeding; (3) suck the 
wound to aid in drawing out the poisoned blood; there is little 
or no danger in this, provided the mouth is free from sores, and 
provided the mouth is immediately rinsed out, preferably with 
an antiseptic solution, such as a light purple solution of per- 



1. -1 

§ 45. KAILROAD SURVEYS. S3 

manganate of potash; (4) inject into the wound a strong solu- 
tion of permanganate of potash, which may be done hypoder- 
mically or, perhaps, even by rubbing into the wound crystals of 
the drug. When the case is very serious, on account of the 
known deadly character of the poison, and when no permanganate 
of potash is obtainable, heroic measures are sometimes necessary. 
Pure carbolic acid, or caustic, may be used, if available. Cauter- 
izing the wound with white-hot iron, exploding a pinch of gun- 
powder over the wound, shooting away the infected part with a 
gun, or even summary amputation with a hatchet, may some- 
times be considered the lesser of two evils. If the limb has been 
tightly tied, it will, of course, produce great pain, discoloration 
and swelling, which must not be continued too long. A second 
ligature should be tied a few inches above the first. When the 
limb becomes very swelled and painful, loosen the first ligature 
for about ten seconds and again tighten, and then loosen the 
second ligature for ten seconds and again tighten. After fif- 
teen minutes, repeat the loosening and tightening. After about 
eight repetitions, the ligatures may be removed altogether. If 
the poison is partly sucked out, the remainder partly neutralized 
with chemicals, and does not get fully into the system for two 
hours, the danger is greatly diminished. Of course bites on the 
face or body cannot be tied up and can only be treated by suck- 
ing out the poison and by chemicals. Stimulation of the heart 
is usually essential, which may be done with one teaspoonful of 
aromatic spirits of ammonia in two tablespoonsful of water, or 
with alcoholic Hquor, preferably whiskey. One l-30th grain 
strychnine tablets, dissolved in two tablespoonsful of water, is 
also a stimulant. If a hypodermic is available, one tablet may 
be dissolved in thirty drops of sterile water and inserted in the 
back or arm, well under the skin. ^ 

45. Wounds. First, last and all the time, prevent infection. 
The marvelous success of modem surgery is due largely to anti- 
septic methods. Neglect of cleanliness almost inevitably 
induces blood poisoning. A perfectly clean cut, after being 
washed and sterilized with iodine, may be closed with adhesive 
plaster, taking stitches, if necessary, with sterilized catgut or silk 
or linen thread. The stitches may be removed in a week. 
But when the flesh is torn and, especially, when dirt and other 
matter, which is possibly poisonous or infectious, has been forced 
into the wound, there is great danger of blood poisoning, and 



54 



RAILROAD CONSTRUCTION. 



§45a. 



the wound must be cleansed. First,- cover the wound itself 
with a pad which has been soaked in an antiseptic solution and 
then wash the skin (shaving off all hair), all around the wound, 
using first soap and then an antiseptic solution. Then cleanse 
out all foreign matter from the wound, using antiseptics and 
pack the woimd with strip gauze, soaked in the antiseptic, so as 
to extend from the deepest part of it to the outside. This will 
drain the discharges. The dressing should be renewed every- 
day, or even three times per day, according to the severity of 
the wound, until the wound shows a tendency to heal. A gaping 
torn woimd should not be sewed up, except to bring the edges 
together temporarily. 

45a. Medical outfit to be carried. The quantity of medicine 
which should be carried is necessarily guess-work. If the party 
has great good luck, it might bring back the entire supply un- 
touched. On the other hand severe sickness might exhaust 
some of the medicines long before the survey is complete. But 
the following list has been estimated as a reasonably proper 
supply for a party of 25 men which may be out of reach of an 
adequate source of medical supplies for a period of six months. 
The list should be varied somewhat according to the climate. 
The probabilities of disease, snake-bites, etc., in a cold climate 
are not the same as those of the tropics. Some of the following 
articles, those commonly required for " first-aid " work, should 
always be provided, even when the party will never be more than 
a few hours distance from medical assistance. , 



Boric acid, powdered, 5 lbs. 

Carbolic acid, pure, 1 oz. 

Iodoform powder, 2 oz. 

Permanganate of potash, 8 oz. 

Essence of ginger, 2 oz. 

Epsom salts, 50 lbs. 

Calomel, 1000 J^-gr. tablets. 

Cascara sagrada, 1000 5-gr. tablets. 

Castor oil, 4 quarts. 

•Glycerine, 4 quarts. 

Ipecacuanha, 6 oz. 

Individual hypoder. syr. packages; 
Tetanus antitoxin, 12 units; Mor- 
phine 04 ST.), Atropine (Hso gr.); 
(for agonizing pain); 24 units; 
Strychnia (Ho gr.); 24 units; Cam- 
phorated oil; 24 units (for pro- 
found shock) ; Digalen (20 drops) ; 
24 units (for acute heart trouble). 



Bismuth sub-carbonate, 2 lbs. 

Zinc sulphate, 4 oz. 

Quinine 1000 5-gr. tablets. 

Rhinitis, 2000 5-gr. tablets. 

Dover's powder, 1000 5-gr. tablets. 

Caustic, AgNOa, 24 sticks. 

Aromatic sp'ts of ammonia, 1 pint. 

Strychnine tablets, 1000 ^so gr. 

Carbolized vaseline, 12 I-02. jars. 

Sterilized gauze, 5 doz. individual 
1-yard rolls. 

Adhesive plaster, 3 5-yard rolls, 12 
inches wide. 

Needles and catgut. No. 2 chromic, 
in curved vacuum tubes, 12 pack- 
ages. 

Needles, safety pins, etc. 

Instruments, etc., as listed in § 37. 



I 



H 



CHAPTER II. 



ALINEMENT 



In this chapter the ahnement of the center line only of a pair 
of rails is considered. When a railroad is crossing a sumnait in 
the grade line, although the horizontal projection of the aline- 
ment may be straight, the vertical projection will consist of 
two sloping lines joined by a curve. When a curve is on a 
grade, the center line is reall}"- a spiral, a curve of double curva- 
ture, although its horizontal projection is a circle. The center 
line therefore consists of straight lines and curves of single 
and double curvature. The simplest method of treating thera 
is to consider their horizontal and vertical projections separately. 
In treating simple, compound, and transition curves, only the 
horizontal projections of those curves will be considered. 



SIMPLE CURVES. 

46. Designation of curves. A curve may be designated either 
by its radius or by the angle subtended by a chord of unit 
length. Such an angle is known 
as the " degree of curve " and is 
indicated by D. Since the curves 
that are practically used have very 
long radii, it is generally impracti- 
cable to make any use of the actual 
center, and the curve is located 
without reference to.it. If AB in 
Fig. 11 represents a unit chord (C) 
of a curve of radius R, then by the 
above definition the angle AOB 
equals D. Then Fig. 11. 




AOdn\D==^AB = \C, 



R = 



sin ^D' 



(1) 



55 



56 RAILROAD CONSTRUCTION. § 47. 

or, by inversion, 

«i»i»=^- • • (2^ 

The unit chord is variously taken throughout the world as 
100 feet, 66 feet, and 20 meters. In the United States 3 00 
feet is invariably used as the unit chord length, and throughout 
this work it will be so considered. Table I has been computed 
on this basis. It gives the radius, with its logarithm, of all 
curves from a O'' 01' curve up to a 10° curve, varying by single 
minutes. The sharper curves, which are seldom used, are given 
with larger intervals. 

An approximate value of R may be readily found from the 
following simple rule, which should })e memorized : 

r, 5730 

Although such values are not mathematically correct, since R 
does hot strictly vary inversely as Z), yet the resulting value i$ 
within a tenth of one per cent for all commonly used values 
of 7^, and is sufficiently close for many purposes, as will be 
shown later. 

47. Metric Ctirves. The unit chord for railroad cUfvfes on 
the metric system is 20 meters. If a curve has a 100-foot chotd 
and a central angle of 5°, the radius would, of course, be 1146.3 
feet. Since 20 ffletefs =65.6174 feet, a 20-meter chord between 
those same radial lines would subtend an arc with a radius of 
.656174X1146.3 fefet, or 752.16 feet. But this radius, measured 
in meters, would be (.656174X1146.3)^3.28087=229.26 meters, 
which k 1146.3 X. 20. In other words, the radius of any metric 
curve, measured in meters, is numerically one-fifth of the radius, 
measured in feet, of the same degree curve, but in actual length 
is a little less than two-thirds. This practically means that a 
10° curve, nietric, is actually very much sharper than a 10® 
curve, using foot-measure, or that the radius is about 66% as 
much. Therefore, in selecting curves for location, an engineer, 
who is accustomed to the foot-measure system, should remember 
that a 10° curve metric, for example, has approximately the 
same radius as a 15° curve, using foot-measure. While it is 
more convenient for an engineer, who is constantly using the 
metric system for curves, to have tables computed directly on 



§48. 



ALINEMENT. 



57 




Fig. 12. 



this basis, an engineer need not be dependent on such tables, 
since it is only necessary to divide the tabular quantities in the 
foot-table by 5 to obtain the correspond- 
ing quantities for the metric system. This 
applies not only to radii, but also to 
tangents, external distances and long 
chords for a 1 ° curve. A desired logarithm 
may be obtained by subtracting 0.6989700 
from the foot- table logarithm. 

For example, anticipating the explana- 
tion in Art. 53, what is the tangent 
distance of a 6° metric curve, when the 
central angle is 32° 40'. From Table II, we 
find that by the foot-system the tangent 
distance for a 1° curve when the central 
angle is 32° 40' is 1679.1 feet; then for 
a 6° curve it is 1679.1^6=279.85 feet; 
for a 6° metric curve it is 279.85 -r 5 = 
65.97 meters. The radius of the 6° metric 

curve — 955.37-5-5 = 191.074 meters, which is in actual length 
about 66% of 955.37 feet. 

As another illustration of the transformation from the foot- 
system to the metric system, or vice versa, the degree of a curve, 
by the foot system, may be multiplied by .66 and obtain approx- 
imately the degree of the equivalent curve by thQ metric system. 
For example, a 6° curve, foot system, has about the same actual 
radius as a 6X. 66 =3.96° metric curve, or about a 4° curve. 

48. Length of a subchord. Since it is impracticable to 
measure along a curved arc, curves are always measured by 
laying off 100-foot chord lengths. This means that the actual 
arc is always a little longer than the chord. It also means that a 
subchord (a chord shorter than the unit length), will be a little 
longer than the ratio of the angles subtended would call for. 
The truth of this may be seen without calculation by noting that 
two equal subchords, each subtending the angle ^D, will evi- 
dently be shghtly longer than 50 feet each. If c be the length of 
a subchord subtending the angle d, then, as in Eq, 2, 



sin |cZ = 



or, by inversion, 



c 
"2R' 

-2R sin |cZ. 



(3) 




58 RAILROAD CONSTRUCTION. §49. 

d 

The nominal length of a subchord = 100 — . For example, 

a nominal subchord of 40 feet will subtend an angle of -j^ of 
J5°; its true length will be slightly more than 40 feet, and may 

be computed by Eq. 3. The difference 
between the nominal and true lengths 
is maximum when the subchord is , 
about 57 feet long, but with the low J 
degrees of curvature ordinarily used .1 
the difference may be neglected. With ; 
a 10° curve and a nominal chord ] 
length of 60 feet, the true length is I 
60.049 feet. Very sharp curves should 
be laid off with 50-foot or even 25-foot 
chords (nominal length). In such 
jTjQ 13 cases especially the true lengths of 

these subchords should be computed 
and used instead of the nominal lengths. 

For example, assume that a 12° curve begins at Sta. 26+30. 
The first subchord will be nominally 70 feet and actually 
70.066 feet. Assume that the central angle between the 
tangents is 39° 36'. Then the nominal length of curve is 
39.6° ^ 12° :*3.30 stations. 3.30 -.70 = 2.60, the nominal length 
of curve beyond the first station point on the curve. The final 
subchord is nominally 60 feet, but its actual length is 60.070 
feet. 

The values of these subchords for even degrees between 
5° and 30°, and for nominal chord lengths of 10, 20, 30, 
40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90 and 95 feet, are given 
in Table Ila. The excess values increase approximately as 
the square of the degree of curvature, but for intervals of 
1° simple interpolation will be sufficiently accurate for inter- 
mediate values. 

49. Length of a curve. The actual mean length of the two 
rails will be more than the nominal length of the curve, as defined 
above, and even more than the sum of the full 100-foot lengths 
and the true lengths of the subchord lengths at the ends. In 
the above numerical case the mean rail length is 

39.6" Xr^o Xi2 =39.6° X 7^ X478.34 =330.604. 
180 180 



§ 50. ALSNEMENT. 59 

The sum of the two full-chord lengths and the two subchords is 
70.066+200+60.070 = 330.136. A large part of the excess 
(330.604-330.136 = .468) is the excess length (.183) of each 
arc of a 12° curve over the 100-foot chord. The remainder is 
the excess of the 70-foot and 60-foot arcs over the true chord 
lengths. But this excess length is of little practical importance. 
In the above case (a 12° curve) it adds about 0.2% to the length 
of rail that must be bought. The excess varies approximately 
as the square of the degree of curvature. The percentage of 
excess for the entire length of a road is utterly insignificant and 
is swallowed up by the 2% excess which is usually allowed for 
wastage in rail cutting. 

50. Curve notation. The notation adopted by the Amer. 
Rwy. Eng. Assoc, indicates any point where there is a change of 
alinement by two letters, the first of whicK denotes the alinement 
on the side toward station zero and the second that away from 
station zero. Thus, the beginning of a curve, or the change from 
a tangent to a simple curve, is noted as TC] the other end of the 
curve, or the change from a simple curve to a tangent is noted as 
CT. But, since the use of two letters to indicate a point, or the 
use of four letters to indicate a line joining the two points, is 
cumbersome in the algebraic solutions and demonstrations which 
follow (demonstrations which the A. R. E. A. do not give), the 
author has decided to retain the old notation, rather than to try 
to conform to the A. R. E. A. notation. The A. R. E. A. sys- 
tem also indicates the central angle of a curve, or the angle 
between the two tangents, by /. In the first edition of this 
work, the author, following Searles, indicated the central angle 
by A. To make even this change, for the sake of conformity, 
would require a change in all the mathematical work and 
figures involving curves throughout the book. In Fig. 14 
both notations are given, the A. R. E. A. notations being 
given in parentheses. Both notations are also shown in 
Fig. 36, which illustrates a transition curve or spiral. It 
should be .noted that some of the notations coincide for some 
of the elements. 

51. Elements of a curve. Considering the line as running 
from A toward B, the beginning of the curve, at A, is called 
the 'point of curve (PC). The other end of the curve, at B, is 
called the point of tangency (PT). The intersection of the 
tangents is called the vertex (V). The angle made by the 



m 



TIAILKOAD CONSTRUCTION. 



52. 



t-angents at V, which equals the angle made by the radii to 
%\iQ extremities of the curve, is called the central angle (A). AV 
and BY , thp two equal tangents from the vertex to the PC 
and . T, are called the tange I distances (T). The chord 
AB is pallefi the long hard (LC). The intercept BG from 
the middle of the long chord to the middle of the aye is called 
the middle ordi.iate (M). That part of the secant GV from 




Fig. 14. 



the middle of the arc to the vertex is called the external distance 
{E). From the figure it is very easy to derive the following 
frequently used relations: 

r^i^tanfA (4), 

LC = 222siniA (5) 

M=i2versiA. . {&), 

• E = R exsec ^A . . (7)^ 



52. Relation between T, E, and A^ Join A and G in Fig. 14. 

The angle V AG = lA, since it is measured by one half of the. .1 
arc AG between the secant and tangent. 

A€O = 90°-iA. 



§ 53. ALINEMENT. 61 

AV :VG::sm AOV : sin VAG; 
sin AGV-Bva. AGO ^ cos lA; 

T : E : : cos jA : sin |A; 

T=E cot lA (8) 

I 

r Ths sapae relation may be obtained hy dividing Eq. 4 by Eq, 

7, since tan a-^exsec a = cot |a. 

ji 53. Elements of a 1° curve. From Eqs. 1 to Sit is seen that 
the elements of a curve vary directly as 72. It is also seen to 
be very nearly true that R varies inversely as D. If the ele- 
ments of a 1 ° curve for various central angles are calculated and 
i tabulated, the elements of a curve of D° curvature may be 
approximately found by dividing by D the corresponding ele- 
ments of a 1° curve having the same central angle. For small 
central angles and low degrees of curvature the errors involved 
by the approximation are insignificant, and even for larger 
angles the errors are so small that for many purposes they may be 
(lisrega^rded 

In Table II is given the value of the tangent distances, 
external distances, and long chords for a 1° curve for various 
central angles The student should familiarize himself with the 
degree of approximation involved by solving a large number of 
cases under various conditions by the exact and by the approxi- 
mate methods, in order that he may know when the approxi^ 
mate method is sufficiently exact for the intended purpose. 
The approximate method also gives a ready check on the 
exact method. 

A closer value may be obtained by using the " Corrective Tablie " 
found at the end of the main table. The correction is aVways 
additive and is usually very small and often even too insignificant 
for, attention. A glance ai; the corrective table will show whether 
a corjection need be made' and an easily computed interpolatibn 
will show its amount. For example, what is the tangent dis- 
tance for a 6° curve having a central angle of 42^ 1-5'? Inteiv 
polating between 2209.0 and 2218.6, we have 2213'.8 as the 
tangent distance for a 1° curve. Dividing by 6, we have 368-. 97 
as the approximate tangent distance. Interpolkting- in the cor- 
rective table* we have .14 as the correction for a 5° curve and a 



62 



RAILROAD CONSTRtJCTION. 



I 54. ; 



central angle of 42° 15', and ,28 as the correction for a 10° curve. 
Interpolating for 6° between these values of .14 and .28, we have 
.17, which added to 368.97 equals 369.14. The precise value, 
computed from Eq. 4, is 369.12. If the approximate value, even 
after correction, is not considered sufficiently accurate, Eq. 4 
should be used. The student should appreciate that the dis- 
crepancy of even .02 in the above calculation is not due to any 
real error in the main table or the corrective table, but is due to 
the fact that the tangent distances are only computed to the 
nearest tenth of a foot for values over 1000 feet, and this will 
produce such discrepancies. The table should not be used 
where precise values are required. 

54. Exercises, (a) What is the tangent distance of a 4° 20' 
curve having a central angle of 18° 24'? 

(b) Given a 3° 30' curve and a central angle of 16° 20', how 
far will the curve pass from the vertex? [Use Eq. 7] 

(c) An 18° curve is to be laid off using 25-foot (nominal) 
chord lengths. What is the true length of the subchords? 

(d) Given two tangents making a central angle of 15° 24'. 
It is desired to connect these tangents by a curve which shall 
pass 16.2 feet from their intersection. How far down the 
tangent will the curve begin and what will be its radius? (Use 
Eq. 8 and then use Eq. 4 inverted.) 

55. Curve location by deflections. The angle between a 
secant and a tangent (or between two secants intersecting on an 
arc) is measured by one half of the intercepted arc. Beginning 
at the PC (A in Fig. 15), if the 
first chord is to be a full chord 
we may deflect an angle VAa 
(=^D), and the point a, which is 
100 feet from A, is a point on the 
curve. For the next station, b, 
deflect an additional angle bAa 
(=:^Z>) and, with one end of the 
tape at a, swing the other end 
until the 100~foot point is on the 
line Ab. The point b is then on 
the curve. If the final chord cB 
is a subchord, its addiiional deflec- ^^' 
tion {i^d) is something loss than ^D. The last deflection (BAV) is 




§ 56. ALINEMENT. 63 

of course iJ. It is particularly important, when a curve begins 
or ends with a subchord and the d(;flections are odd quantities, 
that the last additional deflection should be carefully com- 
puted and added to the previous deflection, to check the mathe- 
matical work by the agreement of this last computed deflec- 
tion with ^J. 

Example. Given a 3° 24' curve having a central angle of 
18° 22' and beginning at sta. 47 + 32, to compute the deflec- 

I tions-- The nominal length of curve is 18° 22' ^ 3° 24' = 18.367 ^ 
3.40=5,402 stations or 540.2 feet. The curve therefore ends 
at sta. 52 + 72.2. The deflection for sta. 48 is iV<tXK3°24') 
=0.68Xl°.7 = l°.156 = l°09' nearly. For each additional 100 

i feet it is 1° 42' additional. The final additional deflection for 
the final subchord of 72.2 feet is 

-^XK3°24')=1°.2274 = 1° 14' nearly. 

The deflections are 

P. C ... Sta. 47 + 32 0° 

48 0° +1°09' = 1°09' 

49 1° 09' + 1° 42' =2° 51' 

50 2° 51' + 1° 42'=4° 33' 

51 4° 33' + 1° 42' = 6° 15' 

52 6° 15' + 1° 42' = 7° 57' 

P. T 52 + 72.2. 7°57' + l°14'=9°ll' 

As a check 9° ll' = K18° 22') =^J, (See the Form of Notes 
in §21.) 

56. Instrumental work. It is generally impracticable to 
locate more than 500 to 600 feet of a curve from one station. 
Obstructions will sometimes require that the. transit be moved up 
every 200 or 300 feet. There are two methods of setting off 
the angles when the transit has been moved up from the PC. 

(a) The transit may be sighted at the previous transit station 
with a reading on the plates equal to the deflection angle from 
that station to the station occupied, but v/itli the angle set off on 
the other side of 0°, so that when the telescope is turned to 0° it 
will sight along the tangent at the station occupied. Plunging 
the telescope, the forward stations may be set off by deflecting 
the proper deflections from the tangent at the station occupied. 



64 



RAILROAD CONSTRUCTION. 



§56. 



Thig is a very common method and, when the degree of curva'* 
ture is an even number of degrees and when the transit is only 
set at even stations, there is but little objection to it. But the 
degree of curvature is sometimes an odd quantity, and the exi- 
gencies of difficult location frequently require that substations 
be occupied as transit stations. Method (a) will then require 
the recalculation of all deflections for each new station occupied. 
The mathematical work is largely increased and the probability 
of error is very greatly increased and not so easily detected. 
Method (6) is just as simple as method (a) even for the most 
simple eases, and for the more difficult cases just referred to the 
superiority is very great. 

(b) Calculate the deflection for each station and substation 
throughout the curve as though the whole curve were to be lo^ 
cated from the PC. The computations 
may thus be completed and checked (as 
above) before beginning the instrumental 
■rt^'ork. If it unexpectedly becomes neces- 
sary to introduce a substation at • any 
point, its deflection from the PC may be 
readily interpolated. The stations actually 
set from the PC are located as usual. 
Rule. When the transit is set on any 
forward station, backsight to any previous 
station with the plates set at the deflection 
angb for the station sighted at. Plunge 
ths tslsscope and sight at any forward 
station with tho deflection angb originally 
computed for that station. When the 
plates read the deflection angle for the 
station occupied, the telescope is sighting 
along the tangent at that station — which 
is the method of getting the forward tan- 
gent when occupying the PT. Even though 
the station occupied is an unexpected sub- 
station, when the instrument is properly 
oriented at that station, the angle reading 
for any station, forward or back, is that originally computed 
for it from the PC. In difficult work, where there are ob- 
structions, a valuable check on the accuracy may be found by 
sighting backward at any visible station and noting whether 







Fig. 16. 



§66. 



ALINBMENf. 



65 



its deflection agrees with that origirially computed. As a 
numerical illustration, assume a 4° eufVe, with 28° curvature, 
I with stations 0, 2, 4, atid 7 occupied. Aftef getting stations 
1 and 2, set tip the transit at sta. 2 and backsight to sta. 
Iwith the deflection for sta. 0, which is 0^. The reading on sta. 
1 is 2°; when the reading is 4^ the telescope is tangent to 
the curve, and when sighting at 3 and 4 the defections will be 
"6^ and 8°. Occupy 4 ; sight to 2 with a reading ot 4^. When 
the reading is 8*^ the telescope is tangent to the cUfve and, by 
plunging the telescope, 5, 6, and 7 may be located with the 
originally computed deflections of 10°, 12°, and 14°. When oc- 
cupying 7 a backsight may be taken to any visible statioti with 
the plates reading the deflection for that station; theU when 




Fig. 17. 




'-= — ---.^^^^^ 



Fig. 18. 



jthe plates read 14° the telescope will point along the forward 
iitangent. 

The location of curves by deflection angles is the normal 
u-nethod. A few other methods, to be described, should be Con- 
jiSidered as exceptional. 



66 



KAILROAD CONSTRUCTION. 



is; 



57. Curve location by two transits. A curve might be locate 
more or less on a swamp where accurate chaining would c 
exceedingly difficult if not impossible. The long chord A\ 
(Fig. 17) may be determined by triangulation or otherwis' 
and the elements of the curve computed, including (possibbjc 
subchords at each end. The deflection from A and B to ead 
point may be computed. A rodman may then be sent (M 
whatever means) to locate long stakes at points determin^ 
by the simultaneous sightings of the two transits. 

58. Curve location by tangential offsets. When a curve 
very flat and no transit is at hand the following method may 
used (see Fig, 18) ; Produce the back tangent as far forward 
necessary. Compute the ordinates Oa% Ob', Oc' , etc., and tl 
abscissae a' a, b'b, c'c, etc. If Oa is a full station (100 feet), the 



Oa'=Oa' =100 cos ^D, also =i2 sin D; 

Ob'=Oa'^a'V =100 cos iz> + 100 cos fZ), 

also = R sin 2D ; 
Oc' =0a' + a'b' + b'c' = 100(cos |D + cos |Z) + cos W), 

also =jB sin 3Z>; 



etc. 



a'a = 



b'h=a'a + b"b 



100 sin ^D, also = R vers D ; 
= 100siniD + 100sin|i), 

also = R vers 2D ; 
= 100(sin \D + sin |Z) + sin f D) , 

also =22 vers 3I>; 



(C 



(10 



etc. 

The functions \D, |D, etc., may be more conveniently usee 
without logarithms, by adding the several natural trigonometrica 
functions and pointing off two decimal places. It may also 1> 
noted that Ob' (for example) is one half of the long chord fo 
four stations; also that b'b is the middle ordinate for fou 
stations. If the engineer is provided with tables giving the lonj 
chords and middle ordinates for various degrees of curvature 
these quantities may be taken (perhaps by interpolation) fron 
such tables. 

If the curve begins or ends at a substation, the angles anc 
terms will be correspondingly altered. The modifications maj 



! § 59. ALINEMENT. 67 



Uj3e readily deduced on. the same principles as above, and should 
|pe worked out as an exercise by the student, 
a ' In Table II are given the long chords for a 1° curve for various 
lvalues of J. Dividing the value as given bv the degree of the 
y3urve, we have an approximate value which is amply close for 
jii:0W degrees of curvature, especially for laying out curves with- 
out a transit. For example, given a 4° 30' curve, required the 
[Ordinate Oc'. This is evidently one half of a chord of six stations, 
^svith J =27°. Dividing 2675.1 (which is the long chord of a 
ii° curve with J =27°) by 4.5 we have 594.47; one half of this is 
jbhe required ordinate, Oc' =297.23. The exact value is 297.31, 
,Un excess of .08, or less than .03 of 1%. The true values 
ijare always slightly in excess of the value as computed from 
I Table 11. 

r .Exercise. A 3° 40' curve begins at sta. 18 + 70 and runs to 
3ta. 23 + 60. Required the tangential offsets and their corre- 
sponding ordinates. The first ordinate =30 cos K A*(rX3° 40') = 
j 30 X. 99995 =29.9985; the offset =30 sin 0° 33' =30 X. 0096 = 
[0.288. For the second full station (sta. 20) the ordinate = 
j Y long chord for J =2(1° 06' + 3° 40') with D ='3° 40'. Divid- 
|ing 476.12, from Table II, by 3|, we have 129.85. Otherwise, 
|by Eq. 9, the ordinate =30 X cos 0° 33' + 100 cos (1° 06' + 1° 50') 
I =30.00 + 99.87 = 129.87. The offset for sta. 20=30 sin 0° 33' + 
jlOO sin (1° 06' + 1° 50') = 0.288 + 5.12 = 5.41. Work out 
I similarly the ordinates and offsets for sta. 21, 22, 23, and 
23 + 60. 

59. Curve location by middle ordinates. Take first the sim- 
pler case when the curve begins at an even station. If we con- 
isider (in Fig. 19) the curve produced back to z, the chord za = 
[2x100 cos ^D, A'a = 100 cos §Z), and A' A =am=ZM = 100 sin ^D. 
:Set off A A' perpendicular to the tangent and A'a parallel to 
!the tangent. AA'=aa'=&5'=cc', etc. = 100 sin |Z). Set off 
iaa' perpendicular to a' A. Produce Aa' until a'b=A'a, thus 
determining b. Succeeding points of the curve may thus be 
: determined indefinitely. 

Suppose the curve begins with a subchord. As before 

:(ra = A m'=c' cos i<i', and rA = am' =c' sin Id'. Also 52=An' = 

f'(c" cos |d", and sA =zn'=c" sin |d", in which (d' + d")=D. 

iThe points z and a being determined on the ground, aa' may 

be computed and set off as before and the curve continued in 



68 



EAILEOAD CONSTKUCTION. 



§6( 



full stations. A subchord atthe end of the curve may be locate 
by a similar process. 

60. Curve location by offsets from the long chord. (Fig. 21 
Consider at once the general case in which the curve commence 
with a subchord (curvature, d')> continues with one or more fu 




Fig. 19. 



Fig. 20. 



Fig. 21. 



chords (curvature of each, D), and ends with a subchord with 
curvature d". The numerical work consists in computing first 
AB, then the various abscissae and ordinates. AB =2R sin \d. 



Aa' = Aa' =p' cos i(J-d'); 

Ah'=Aa' + a'V =c' cos K^-d') + 100 cos K^-2d'-I>)', 

^c'=Aa' + a'y + &'c' = c' cos iU-dO + lOO cos Ki-2d'-D) 

+ 100cosKi-2d"-D): 
also 

'^AB-Be ^2nsva.\A-d' GO^\U-dr\ . 



aa = aa 



= c'sini(J-d')", 



(11) 



dc =6'6-n&'=c'sinK^-^') + 100sinK^-2rf'-Z>) 

-100 sin K^-2d*-Z>); 
also «=c*sin i(J— d"). 



I 



(12) 



The above formulae are considerably simplified when the > 



l5 61. 



ALINEMENT. 



eo 



Isurve begins and ends at even stations. When the curve is 
very long a regular law becomes very apparent in the formation 
Ipf all terms between the first and last. There are too few terma 
iin the above equations to show the law. 

I 6i. Use and value of the above methods. The chief value 
m the above methods lies in the possibility of doing the work 
I without a transit. The same principles are sometimes em- 
■Iployed, even when a transit is used, when obstacles 4)revent the 
rjse of the normal method (see §62, c). If the terminal tan- 
gents have already been accurately determined, these methods 
ire useful to locate points of the curve when rigid accuracy is 
lot essential. Track foremen frequently use such methods to 
ay out unimportant sidings, especially when the engineer and 
lis transit are not at hand. Location by tangential offsets (or 
3y offsets from the long chord) is to be preferred when the 
3urve is flat (i.e., has a small central angle J) and there is no 
obstruction along the tangent, or long chord. Location by 
niddle ordinates may be employed regardless of the length of 
:.he curve, and in cases when both the tangents and the long 
3hord are obstructed. The above methods are but samples 
jf a large number of similar methods which have been devised. 
The choice of the particular method to be adopted must be 
' determined by the local conditions. 

62. Obstacles to location. In this section will be given only 
a few of the principles involved in thii 
class of problems, with illustrations. The 
engineer must decide, in each case, which 
is the best method to use. It is frequently 
advisable to devise a special solution for 
some particular case. 

a. When the vertex is 'inaccessible. As 
shown in § 56, it is not absolutely essential 
that the vertex of a curve should be 
located on the ground. But it is very evi- 
dent that the angle between the terminal 
tangents is determined with far less prob- 
able error if it is measured by a single 
measurement at the vertex: rather than as 
the result of numerous angle measurements 
Fig. 22. along the curve, involving several posi- 

;tions of the transit and comparatively short sights Some- 




70 RAILROAD CONSTRUCTION. § 62! 

times the location of the tangents is already determined oi 
the ground (as by hn and am, Fig. 22), and it is required t( 
join the tangents by a curve of given radius. Method. Measure 
ab and the angles Vba and baV. A is the sum of these anglesi 
The distances bV and aV are computable from the above data; 
Given A and R, the tangent distances are computable, and then 
Bb and a A are found by subtracting bV and aV from the tan- 
gent distances. The curve may then be run from A, and the 
T/ork may be checked by noting whether the curve as run ends 
at B — previously located from b. 

Example. Assume o6=546 82; angle a = 15° 18'; angle 
h = 18° 22' ; D =3° 40' ; required aA and bB. 

J = 15° 18' + 18° 22' =33° 40' 

Eq. (4) R (3° 40') 3.19392 

tan ^A =tan 16° 50' 9.4808 

r=472.85 2.67472 






v= 7. sin 18° 22' ab 2.73784* 

°'^ -sin 33° 40' log sin 18° 22' 9.49844 

co-log sin 33° 40' . 25621 

a7=310.81 2.49250 

AF=472.85 



aA =162.04 



hV= 5!i5:15M8' ab . 2.73784 

J^ ^ sin 33° 40' log sin 15° 18' 9.42139 

co-log sin 33° 40' 0.25621 



6F=260.29 2.41545 

57=472.85 



65=212.56 

b. When the point of curve (or point of tangency) is inacces- 
sible, j^t some distance (.4s, Fig. 23) an unobstructed line pn 
may be run parallel with AV, nv=py=As=R vers a. 

\ vers a=As-^R. 

ns=ps=.B sin ffi. 



63. 



ALINEMENT. 



71 



At y, which is at a distance ps back from the computed posi- 
tion of A, make an offset sA 
to p. Run p'/z. parallel to the 
tangent A tangent to the 
curve at n makes an angle of a 
with wp. From n the curve is 
run in as usual 

If the point of tangency is 
obstructed, a similar process, 
somewhat reversed, may be 
used. /? is that portion of A still 
to be laid off when m is reached. 
tm=tl=R sin ^. mz=tB==lx=R 
vers ^. 

c. When the central part of 

the curve is obstructed, a is the 

central angle between two points 

of the curve between which 

a may equal any angle, but it is prefer- 




FlG 



I chord may be run. 
able that a should be a multiple 
Df D, the degree of curve, and that 
the points m and n should be on 
3ven stations. mn=2Rsm^a. A 
point s may be located by an offset 
ks from the chord mn by a similar 
method to that outlined in § 60. 

The device of introducing the 
dotted curve mn having the same 
radius of curvature as the other, 
although neither necessary nor 
advisable in the case shown in 
Fig. 24, is sometimes the best 
method of surveying around an 
obstacle. The offset from any point on the dotted curve to 
the corresponding point on the true curve is twice the " ordinate 
to the long cord," as computed in § 60. 

63. Modifications of location. The following methods may 
be used in allowing for the discrepancies between the " paper 
location " based on a more or less rough preliminary survey and 
the more accurate instrumental location, (See § 18.) They are 




Fig. 24. 



72 



RAILROAD CONSTRtJCTION. 



m 



also frequently used in locating new parallel tracks and modify| 
ing old tracks. 

a. To move the forward tangent parallel to itself a distance x 
the point of curve (a) remaining fixed. (Fig. 25.) 



y'h^B'r^x\ 



VV' = 



V'h 



X 



The triangle BmB' is isosceles and Bm-=B'm. 

B'r 



U'-R-^^-OV^mB^ 



X' 



vers B'mB vers A' 



(13; 



/. n'=R+ 



X' 



vers A' 



(14) 



The solution is very similar in case the tangent is moved in- 
ward to Y"B". Note that this method necessarily changes the 



^^5HI^^ 






^^^c^^^;- 


-~-^'^" 




/V^^^ 




V' 


/ / / \^ 


"V^^""^ 


y 


III \ 


\\ A 


V" 


// /' 






LLL 




A 


0' 0" ' 











Fia. 25. 




Fig. 26. 



radius. If the radius is not to be changed, the point of curv« 
must be altered as follows; 

b. To move the forward tangent parallel to itself a distance x, 
the radius being unchanged. (Fig. 26.) In this cas/e the whole 



§64. 



i^LINEMENT. 



73 



i curve is moved bodily a distance 00'=^A' = FF'==J5j8', and 
moved parallel to the first tangent AV 

B'n__ 
sin nBB' 



BB' 



X 



sin A 



-AA' 



(15) 



c. To change the direction of the forward tangent at the point 
of tangency. (Fig- 27.) This problem involves a change (a) in 

the central angle and also requires a 
new radius. An error in the deter- 
mination of the central angle fur- 
nishes an occasion for its use. 

R, A, a, A V, and BV are known. 
A'^J. 




Bs==R vers A. 



R'^R- 



■a, 

Bs^R' vers A'. 
vers A 



vers (J— a)' 
As =R sin A. A's =R' sin A'. 



(16^ 



AA'=A's—As=R' sin A'^R sin A, 



(17) 



The above solutions are given to illustrate a large class of 
problems which are constantly arising. All of the ordinary 
problems can be solved by the application of elementary geome- 
try and trigonometry. 

64. Limitations in location. It may be required to run a 
curve that shall join two given tangents and also pass through a 
given point The point (P, Fig. 
28) is assumed to be deter- 
mined by its distance ( VP) 
from the vertex and by the 
angle AVP^/^. 

It is required to determine 
the radius (R) and the tangent 
distance (AV). A is known. 

PF(? = K180°-i)-/? 
=90°-(Ji+^/?). 
PP'=2FP sin PFG 

=2FPcos(ii+/?). 
PSV^^A. 




SP^VP 



sin /? 
sin 4J' 



Fig. 28. 



74 EAILROAD CONSTRUCTION. § 65. 

\ sm ^ J I sin JJ J 



_VP Ijn''^ 2sin^cosQJ+/?) 
\sinH^ sin 4 J 

5F=FP?i54i4M. i 

sin ^ J 

^F=^,S4-5F 

= J^Tsin(ii +i?) + Vsm2/? + 2sm/?sini4cos(i J +^)]. (18) 
sin^J 

R=AV cot iJ. 

In the special case in which P is on the median Hne OV, 
^=90°- JJ, and (iJ+/?)=90°. Eq. 18 then reduces to 

AV=-J^n+cos iJ) =VP cot p, 

sm f J 

as might have been immediately derived from Eq. 8. 

In case the point P is given by the offset PK and by the 
distance FX, the triangle PKV may be readily solved, giving the 
distance VP and the angle /?, and the remainder of the solution 
will be as above, 

65. Determination of the curvature of existing track, (a) Using 
a transit. Set up the transit at any point in the center of the 
track. Measure in each direction 100 feet to points also in the 
center of the track. Sight on one point with the plates at 0°. 
Plunge the telescope and sight at the other point. The angle 
between the chords equals the degree of curvature. 

(b) Using a tape and string. Stretch a string (say 50 feet 
long) between two points on the inside of the head of the outer 
rail. Measure the ordinate (x) between the middle of the string 
and the head of the rail. Then 

-^ chord^ . . . .,„- 

K= — (very nearly) (19) 

oX 

For, in Fig. 29, since the triangles AOE and ADC are similar, 



§66. 



ALINEMENT. 



75 



AO : AE :: AD : DC or R = ^AD'^x. When, as is usual, 
the arc is very short compared with the 
radius, AD^^AB, very nearly. Making 
this substitution we have Eq. 19, With a 
chord of 50 feet and a 10° curve, the result- 
ing difference in x is .0025 of an inch — far 
within the possible accuracy of such a 
method. The above method gives the 
radius of the inner head of the outer rail. 
It should be diminished by ^g for the radius 
of the center of the track. With easy curvature, however, this 
will not affect the result by more than one or two tenths of one 
per cent. 

The inversion of this formula gives the required middle or- 
dinate for a rail on a given curve. For example, the middle 
ordinate of a 30-foot rail, bent for a 6° curve, is 




Fig. 29. 



a: =900 --(8X955) =.118 foot = 1.4 inches. 



Another much used rule is to require the foreman to have a 
string, knotted at the center, of such length that the middle 
ordinate, measured in inches, equals the degree of curve. To 
find that length, substitute (in Eq. 19) 5730 ^D for R and 
D-i-12 for X. Solving for chord, we obtain chord = 61.8 feet. 
The rule is not theoretically exact, but, considering the uncertain 
stretching of the string, the error is insignificant. ' In fact, the 
distance usually given is 62 feet, which is close enough for all 
purposes for which such a method should be used. 

66. Problems. A systematic method of setting down the 
solution of a problem simplifies the work. Logarithms should 
always be used, and all the work should be so set down that a 
revision of the work to find a supposed error may be readily 
done. The value of such systematic work will become more 
apparent as the problems become more complicated. The two 
solutions given below will illustrate such work. 

a. Given a 3° curve beginning at Sta, 27 + 60 and running 
to Sta, 32 + 45, Compute the ordinates and offsets used in 
locating the curve by tangential offsets. 

h. With the same data as above, compute the distances to 
locate the curve by offsets from the long chord. 

c. Assuro*^ that in J^'y 22 ab is measured as 217,6 feetj tb© 



76 



RAILROAD CONSTRUCTION. 



§66. 



angle a6F = 17°42', and the angle &a7=21° 14'. Join ihe 
tangents by a 4° 30' curve. Determine bB and a A . 

d. Assume that in a case similar to Fig. 23 it was noted 
that a distance (As) equal to 12 feet would clear the building. 
Assume that J =38° 20' and that 2) =4° 40'. Required the 
value of a and the position of n. Solution: 



vers a=As-T-R 



ns=R sin a 



As = 12 
7? (for 4° 40' curve) 
a =8° 01' 



ns = 171.27 



log = 1.07918 

log = 3. 0892 3 

log versa = 7 .9899^ 

log sin a =9. 14445 

log 72 = 3^08923 

log =2723369 



e. Assume that the forward tangent of a 3° 20' curve having 
a central angle of 16° 50' must be moved 3.62 feet inumrd, with- 
out altering the P.C. Required the change in radius. 

/. Given two tangents making an angle of 36° 18'. It is 
required to pass a curve through a point 93.2 feet from the 
vertex, the line from the vertex to the point making an angle 
of 4.i.'21' with the tangent. Required the radius and tangent 
distance Solution: Applying Eq. 18, we have 



2 


log = 


0.30103 


/? = 42°21' 


log Rin = 


9.82844 


iJ = 18°09' 


log sin = 


9.49346 


(^i+/?)=60°30' 


log cos = 


9.69234 


.20667 




9.31527 


log sin^ ^ = 9 . 65688 45382 




2 9.81987 66049 






9.90993 ".81271 






nat. sin 60° 30' 8703 


log = 




1.6830.... 


0.22610 


FP = 93.2 


log = 


1.96941 
2.19551 




log sin 1 J = 


9.49316 


Tang. dist. .4 7=503.36.. 


Jog = 

log cot ^i = 


2 . 70205 




10.48137 


72 = 1536.1 


log = 


3.18642 


7) =3° 44' 







§67. 



ALINEMENT, 



77 



COMPOUND CURVES. 

67. Nature and use. Compound curves are formed by a 
succession of two or more simple curves of different curvature. 
Th3 curves mUst have a common tatigent at the point of com- 
pound curvature (P.C.C). In mountainous regions there is 
frequently a necessity for compound curves having several 
changes of curvature. Such curves may be located separately 
as a succession of simple curves, but a combination of two 
sirnple curves has special properties which are worth investigat- 
ing and utilizing. In the following demonstrations R2 always 
represents the longer radius and R^ the shorter, no matter which 
succeeds the other. T^ is the tangent adjacent to the curve of 
shorter radius (Ri), and is invariably the shorter tangent. J^ is 
the central angle of the curve of radius Ri, but it may be greater 
or less than Jj 

68. Mutual relations of the parts of a compound curve having 
two branches. In Fig. 30, AC and CB are the two branches of 




Fig. 30. 



the compoutid curve having radii of R^ and R2 and central angles 
of Ji and Jj- Produce the arc ^C to n so that AO{n^A. The 
chord Cn produced must intersect B. The line nSj parallel to 
CO2, will intersect BO^ so that Bs=sn=^0/)^^ = R2—Ri. Draw 
Am perpendicular to 0{n,. It will be parallel to hk. 



78 RAILROAD CONSTRUCTION. § ^8. ; 

Br =sn vers Bsn =(i?2— ^i) vers J2> 
mn=AOiYeis AO{n ==J?ivers J; 
Ak=AV sin AVk ^TysinJ; 
Ak=hm^wn + nh=mn + Br. 
.-. Tisin J=7?i vers J + (i?2— -^i) vers J2- • • (20) 
Similarly it may be shown that 

Tj sin J=R2 vers J -(7?2 -Ri) vers Jj. , . (21) 

The mutual relations of the elements of compound curves 
may be solved by these two equations. For example, assume 
the tangents as fixed (.4 therefore known) and that a curve of 
given radius Ri shall start from a given point at a distance T^ 
from the vertex, and that the curve shall continue through a 
given angle J^. Required the other parts of the curve. From 
Eq. 20 we have 

Ti sin J — 7?i vers J 



R2 — R1 



vers Jj 



... ;;,=fl. + ^-^i"^-^ g^ (22) 

vers(J — Ji) 

T2 may then be obtained from Eq. 21. 

As another problem, given the location of the two tangents, 
with the two tangent distances (thereby locating the PC and 
PT), and the central angle of eaich curve; required the two 
radii. Solving Eq. 20 for R^, we have 

T^ sin J—R2 vers J2 



R, 



vers J — vers Jj 



Similarly from Eq. 21 we may derive 

T2 sin J — 7?2(vers J —vers J^) 
vers Ay 

Equating these, reducing, and solving for R2, we have 

Tj sin A vers Ay — T2 sin A (vers A — vers A^ 

^~ vers A2 vers J^— ^vers J— vers Ji)(vers J— vers Jg) ' 

Although the various elements may be chosen as above with 

considerable freedom, there are limitations. For example, in 

Eq. 22, since R2 is always greater than 72j, the term to be | 

added to R^ must be essentially positive — i.e., r^ sin A must be 

vers A • 

greater than R^ vers A. This means that Ty>Ry—. — j-, or that 



§69. 



ALiNEMENT. 



?9 



Ti>Ritsin^^J, or that Ti is greater than the corresponding 
tangent on a simple curve. Similarly it may be shown that T2 
is less than R2 tan §i or less than the corresponding tangent 
on a simple curve. Nevertheless Tj is always greater than Tj. 
In the limiting case when R2=Ri, 7^2 = ^i> ^^^ ^2 = ^1- 

69. Modifications of location. Some of these modifications 
may be solved by the methods used for simple curves. For 
example * 

a. It is desired to move the tangent VB, Fig. 31, parallel to 
itself to VB'. Run a new curve from the P.C.C. which shall 
reach the new tangent at B', where the chord of the old curve 





Fig. 31. 



Fig. 32. 



intersects the new tangent. The solution is almost identical 
with that in § 63, a. 

b. Assume that it is desired to change the forward tangent 
(as above) but to retain the same radius. In Fig. 32 

(i?2— -Ki) cos ^2 =02n; 

(R2-Ri) cos J2' =02V. 

x=02n—02'n' ={R2 — Ri)(cos ^2 — ^^^ "^2')' 



cos i/ 



cos Jo 



X 



(24) 



The P.C.C. is moved backward along the sharper curve an 
angular distance of A2' — A2 = d^ — d/. 

In case the tangent is moved inward rather than outward, 
the solution will apply by transposing Jj ^^^ ^Z' Then we 
I shall have 

cos iz'^cos ij+D D (25) 



80 



RAILROAD CONSTRUCTION. 



§69i 



The P.C.C. is then moved forward. 

c. Assume the same case as (b) except that the larger radiu^ 
comes first and that the tangent adjacent to the smaller radiuq 
is moved. In Fig. 33 

(R^—Ri) cos i, =Ojn; 
(E2-7?i)cosii'=OiV. 

x=0in'—0in 
= (7^2— -^i) (cos i/— cos ^i). 



cos i/ = cos J I 



X 



R2-K 



(26) 




The P.C.C, is moved jonvard 
along the easier curve an angular 
distance of J/ — ii = J2~"^2'- Fia. 33. 

In case the tangent is moved inward, transpose as before and 
we have 



cos J/=cos Ji 



X 



.R2—R1 



(27) 



The P.C.C. is moved backward 

d. Assume that the radius of one curve is to be altered with- 
out changing either tangent. Assume conditions as in Fig. 34. 

For the diagrammatic solution 
assume that i?2 is to be increased 
by O2S. Then, since R2' must 
pass through Oj and extend be- 
yond Oi a distance 0,*S, the, 
locus of the new center must lie 
on the arc drawn about 0, as 
center and with OiS as radius. 
The locus of O2 is a] so given 
by a line Oa'/) parallel to BV 
and at a distance of 7?/ (equal 
to S .. . P.C.C.) from it. The 
new center is therefore at the. 
intersection O2'. An arc with ra- 
dius R2' will therefore be tangent' 
at B' and tangent to the olrf' 
curve produced at new P.C.C. Draw Otti' perpendicular to O2B. | 




Fig. 34. 



§ 70. ALINEMENT. 81 

With O2 as center draw the arc 0{m., and with O2' as center draw 
the arc Oitn'.. mB = m'B' =R^. 

.*. mn^m'n' = {R2 —R^ vers J/ = (i?2— i^i) vers ^2- 

.'. versJ/ = ^^^p'ersJ2 (28) 

0{n, = (7?2 — -^1) sin A 2 ', 
Oin' = (i?2'--Ri) sin J2'. 
BB' =^0{a'-0{ii = (R2'-Ri) sin J/ - (^2 - ^1) sin J 2- (291 

This problem may be further modified by assuming that the 
radius of the curve is decreased rather than increased, or that 
the smaller radius follows the larger. The solution is similar 
and is suggested as a profitable exercise. 

It might also be assumed that, instead of making a given 
change in the radius Ro, a given change BB^ is to be made. J 2' 
and 7^2' ^^^ required. Eliminate R2' from Eqs. 28 and 29 
and solve the resulting equation for J2'. Then determine R2' 
by a suitable inversion of either Eq. 28 or 29. 

As in §§ 62 and 63, the above problems are but a few, although 
perhaps the most common, of the problems the engineer may 
meet with in compound curves. All of the ordinary problems 
may be solved by these and similar methods. 

70. Problems, a. Assume that the two tangents of a com- 
pound curve are to be 348 feet and 624 feet, and that Jj=22° 16' 
and ^2=28° 20'. Required the radii. 

[.4ns. 7^1=326.92; 7^2 = 1574.85.] 

b. A line crosses a valley by a compound curve which is first 
a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is 
afterward decided that the last tangent should be 6 feet farther 
up the hill. What are the required changes? [Note. The 
second tangent is evidently moved outward. The solution cor- 
responds to that in the first part of § 69, c. The P.C.C. is 
moved forward 16.39 feet. If it is desired to know how far the 
P.T. is moved in the direction of the tangent (i.e., the projection 
of BB', Fig. 33, on F'5'), it may be found by observing that it 
is equal to nn' = (7?2— 7?i)(sin J^— sin J/). In this case it equals 
0.65 foot, which is very small because J^ is nearly 90°. The 
value of ^2 (46° 30') is not used, since the solution is independent 
of the value of Jg- The student should, learn to recognize 



82 



KAILROAD CONSTRUCTION. 



§71. 



which quantities are mutually related and therefore essential 
to a solution, and which are independent and non-essential.j 




TRANSITION CURVES. 

71. Superelevation of the outer rair on curves. When a mass 
is moved in a circular path it requires a centripetal force to keep 
it moving in that path. By the principles of mechanics we 
know that this force equals Gv^-^gR, in which G is the weight, 
V the velocity in feet per second, g the acceleration of gravity in 
feet per second in a second, and R the radius of curvature. 
If the two rails of a curved track were laid on a level (trans- 
versely), this centripetal force could only be furnished by the 
pressure of the wheel-flanges against the rails. As this is very 
objectionable; the outer rail is elevated so that the reaction of 
the rails against the wheels shall 
contain a horizontal component 
equal to the required centripetal 
force. In Fig. 35, if oh represents 
the reaction, oc will represent the 
weight G, and ao will represent the 
required centripetal force. From 
similar triangles we may write 
sn : sm :: ao : oc. Call g = 32.17. 
Call R= 5730-^0, which is suffi- 
ciently accurate for this purpose (see 
§ 46). Call t; =52807-- 3600, in which V is the velocity in miles 
per hour, mn is the distance between rail centers, which, for 
an 80-lb. rail and standard gauge, is 4.916 feet sm is slightly 
less than this. As an average value we may call it 4.900, which 
is its exact value when the superelevation is 4| inches. Calling 
sn=e, measured in feet, we have 

e=sm-=4 9^'l= 4.9X5280^F^D 

oc ' gR G 32.17 X 36002 XSTSO"' 

e = . 0000572 F'D (30) 

It should be noticed that, according to this formula, the re- 
quired superelevation varies as the square of the velocity, which 
means that a change of velocity of only 10% would call for a 
change of superelevation of 21%. Since the velocities of trains 
over any road are extremely variable, it is impossible to adopt 






\ 



Fig. 35. 



§72. 



ALINEMENT. 



83 



any superelevation which will fit all velocities even approx- 
imately. The above fact also shows why any over-iefinement 
in the calculations is useless and why the above approximations, 
which are really small, are amply justifiable. For example, the 
above formula contains the approximation that R = 57S0-7-D. 
In the extreme case of a 10° curve the error involved would be 
about 1%. A change of about | of 1% in the velocity, or say 
from 40 to 40.2 miles per hour, would mean as much. The error 
in e due to the assumed constant value of sm is never more than 
a very small fraction of 1%. The rail-laying is not done closer 
than this. Table XIX is based on Eq. (30) : 

Table XIX. superelevation of the outer rail (in feet) 

FOR various velocities AND DEGREES OF CURVATURE. 



Velocity in 

Miles per 

Hour. 


Degree of Curve. 


1° 


2° 


3° 


4° 


5° 


6° 


7° 


8° 


9° 


10° 


30 


.05 
.09 
.14 
.20 


.10 
.18 
.29 
.41 


.15 
.27 
■ 43 


.20 
.37 


.26 

■ 46 


■ 31 


■ 36 


■ 41 


■ 46 
1.29 


|.51 


40 


.86 


■ 64 
1.00 


1.14 


.92 


50 
60 


. .82 


.71 
1.03 





72. Practical rules for superelevation. A much used rule for 
superelevation is to " elevate one half an inch for each degree of 
curvature." The rule is rational in that e in Eq. 30 varies 
directly as D. The above rule therefore agrees with Eq. 30 
when V is about 27 miles per hour. However applicable the 
rule may have been in the days of low velocities, the elevation 
thus computed is too small now. The rule to elevate one inch 
for each degree of curvature is also used and is precisely similar 
in its nature to the above rule. It agrees with Eq. 30 when 
the velocity is about 38 miles per hour, which is more nearly 
the average speed of trains. 

Another (and better) rule is to "elevate for the speed of the 
fastest trains." This rule is further justified by the fact that a 
four-wheeled truck, having two parallel axles, will always tend 
to run to the outer rail and will require considerable flange pres- 
sure to guide it along the curve. The effect of an excess of super- 
elevation on the slower trains will only be to relieve this flange 
pressure somewhat. This rule is coupled with the limitation 



84 



RAILROAD CONSTRtJCTION. 



§ T2- 



that the elevation should never exceed a limit of six inches— 
sometimes eight inches. This limitation implies that locomo- 
tive engineers must reduce the speed of fast trains around shnrp 
curves until the speed does not exceed that for which the actua,! 
superelevation used is suitable. The heavy line in Table XIX 
shows the six-inch limitation. 

Some roads furnish their track foremen with a list of the super- 
elevations to be used on each curve in their sections. This 
method has the advantage that each location may be separately 
Studied, and the proper velocity, as affected by local conditions 
(e.g., proximity to a stopping-place for all trains), may be de- 
termined and applied. 

Another method is to allow the foremen to determine the 
superelevation for each curve by a simple measurement taken 
at the curve. The rule is developed as follows: By an inversion 
of Eq. 19 we have 

x = chord^-i-8R (31) 

Putting X equal to e In Eq. 30 and solving for "chord,!' we 

have 

chord ? = .0000572 V^D: P, 
-2.621F2. 

chord = 1. 62V (32) 

To apply the rule, assume that 50 miles per hour is fixed as 
the velocity from which the superelevation is to be computed. 
Then 1.627 = 1.62X50=81 feet, which is the distance given t6 
the trackmen. Stretch a tape (or even a string) with a length 
of 81 feet between two points on the concave side of the head of 
Cither the inner or the outer rail. The ordinate at the middle 
point then equals the superelevation. The values of this chord 
length for varying velocities are given in the accompanying 
tabular form. 



Velocity in miles per hour. 
Chord length in feet 



20 
32.4 



25 
40.5 



30 
48.6 



35 
56.7 



40 
64.8 



45 
72.9 



50 
81.0 



55 

89.1 



60 
97.2 



The following tabular form shows the standard (at one time) 
on the N. Y., N. H. & H. R. R. It should be noted that the 
elevations do not increase proportionately with the radius, and' 
that t>A,ey »,re higher for descending grades than for level or 



§ 73. 



ALINEMENT. 



8S 



ascending grades. This is on the basis that the velocity on curveg 
and on ascending grades will be less than on descending grades. 
For example, the superelevation for a 0° 30' curve on a de- 
scending grade corresponds to a velocity of about 54 miles per 
hour, while for a 4° curve on a level or ascending grade the super- 
elevation corresponds to a velocity of only about 38 miles per 
hour. 



TABLE OF THE SUPERELEVATION OF THE OUTER RAIL ON CURVES. 

N. Y., N. H. & H. R. R. 



Degree of 
curve. 


Level or as- 


Descending 


cending grade. 


grade. 




inches. 


inches. 


0° 30' 


Of 


1 


1 00 


n 


H 


1 15 


u 


2 


1 30 


2 


2i 


1 45 


2i 


2i 


2 00 


21 


2f 


2 15 


2f 


3 


2 30 


2^ 


3t 


2 45 


3 


31 


3 00 


3i 


31- 


3 15 


3i 


31 


3 30 


3f 


4 


3 45 


31 


4* 


4 00 


4 


4i 



73. Transition from level to inclined track. On curves the 
track is inclined transversely; on tangents it is level. The tran- 
sition from one condition to the other must be made gradually. 
If there is no transition curve, tiiere must be either inclined 
track on the tangent or insufficiently inclined track on the curve 
or both. Sometimes the full superelevation is continued through 
the total length of the curve and the "run-off (having a length 
of 100 to 400 feet) is located entirely on the tangents at each 
©hd. In other practice it is located partly on the tangent and 
partly on the curve. Whatever the method, the superelevation 
is correct at only one point of the run-off. At all other points 
it is too great or too small. This (and other causes) produces 
©bjectionable lurches and resistances when entering and leav- 
ing curves. The object of transition curves is to obviate these 
Resistances. 

. "On the Lehigh Valley R. R. the run-off is made in the form 
of a reversed vertical curve, as shown in the accompanying 
Sgure. Accordirig to- this system the length of run-off varies 



86 



EAILROAD CONSTRUCTION. 



§74. 



from 120 feet, for a superelevation of one inch, to 450 feet> 
for a superelevation of ten inches. Such a superelevation 
as ten inches is very unusual practice, but is successfully \ 
operated on that road. The curve is concave upward for two- 
thirds of its length and then reverses so that it is convex upward. 

TABLE FOR RUN-OFF OF ELEVATION OF OUTER RAIL OF CURVES. 
Drop in inches for each 30-foot rail commencing at theoretical point of curve. 



«.2 


\" 


\" 


r 


¥ 


\" 


¥ 


¥ 


1" 


ir 


U" 


\¥ 


1" 


¥ 


¥ 


¥ 


¥ 


¥ 


¥ 


1 




•3 

o 


^^ 












































H 


1" 




30 


30 






























30 




30 




120 


r 




30 








30 
















— 


- 


30 


30 






30 






150 


3" 




30 








30 














30 




30 




30 






30 






180 


4" 




30 




30 






30 












30 




30 


30 


, , 


30 




30 






240 


5" 




30 




30 








30 








30 




3(J 


30 


30 




30 




30 






270 


6" 




30 




30 






30 




30 






30 




30 


30 


30 




30 




30 






300 


7" 




30 




30 






30 




30 




30 


30 




30 


,30 




30 


30 




30 






330 


8" 




30 




30 




30 






30 


30 


30 


30 




30 


30 




30 




30 




30 


360 


9" 


30 






30 




30 


. 


30 


30 




30 


30 


30 


30 


30 


30 


30 




30 




30 


420 


10" 


30 


• • 


30 




30 






30 


30 


30 


30 


30 


30 


30 


30 


30 


30 




30 




30 


450 




The figure (and also the lower line of the tabulated form) 
shows the drop for each thirty-foot rail length. For shorter 
lengths of run-off, the drop for each 30 feet is shown by the cor- 
responding lines in the tabular form. Note in each horizontal 
line that the sum of the drops, under which 30 is found, equals 
the total superelevation as found in the first column. For 
example, for 4 inches superelevation, length of curve 240 feet, 
the successive drops are \" , \" , \'\ l"\ I", \" , \" , and \" 
whose sum is 4 inches. Possibly the more convenient form 
would be to indicate for each 30-foot point the actual super- 
elevation of the outer rail, which would be for the above case 
(running from the tangent to the curve) \" ^ |", ^", IV', 2f", 

74. Fundamental principle of transition curves. If a curve 



§ 75. ALINEMENT. 87 

has variable curvature, beginning at the tangent with a curve 
of infinite radius, and the curvature gradually sharpens until it 
equals the curvature of the required simple curve and there 
becomes tangent to it, the superelevation of such a transition 
curve may begin at zero, at the tangent, gradually increase to 
the required superelevation for the simple curve, and yet have 
at every point the superelevation required by the curvature at 
that point. Since in Eq. (30) e is directly proportional to D, 
the required curve must be one in which the degree of curve 
increases directly as the distance along the curve. 

75. Varieties of Transition Curves. A theoretically exact 
transition curve is very complicated and its mathematical 
solution very difficult.., - A committee of the Amer. Rwy. Eng. 
Assoc, investigated the many systems which have been proposed 
and reported that all of them seemed to be objectionable for 
one or more of the following reasons: "(1) If simple approximate 
formulas were used, they were not sufficiently accurate. (2) 
Accurate formulas were too complex. (3) The curve could not 
be expressed by formulas. (4) Formulas were of the endless 
series class. (5) Complex field methods were required to make 
the field-work agree with formulas- with spirals of large angles." 
The committee then developed a method which gives results 
whose accuracy is beyond that of the most careful field-work and 
yet which is sufficiently simple for practical use. The mathe- 
matical development is so elaborate that it will not be detailed 
here, but the working formulas and a condensation of the table 
together with an explanation of their practical use and applica- 
tion, will be given, with numerical examples. 

The general form of these curves, whatever their precise 
mathematical character, is shown in Fig. 36. AVB are two 
tangents, joined by the simple circular curve AMB, having the 
center 0. Assume that the entire curve is moved in the direc- 
tion MO a distance 00' = MM' = BB' = AA'. At some point TS 
on the tangent, the spiral begins and joins the circular curve 
tangentially at SC. The other spiral runs from CS to ST. The 
significance of these symbols may be readily remembered from 
the letters; T, S, and C signify tangent, spiral and circular curve; 
TS is the point of change from tangent to spiral, SC, the point 
of change from spiral to curve, etc. At the other end of the 
circular curve the letters are in reverse order, the station numbers 
increasing from A to B, The meaning of the various symbols is 



88 



RAILROAD CONSTRUCTION. 



§76. 



indicated in Fig. 36. The student should appreciate the fact of 
the necessary distortion of the figure in order to make it plain. 
Based on the figures of the following numerical problem, the 
distance MM' is about fourteen times its proper amount. Another 
effect of the distortion is that the dimension U, instead of being 




Fig. 36. 



Aearly twice V, which is usual, as given in Table IV, Part B, is 
only a little longer than V. 

76. Proper length of spiral. This can only be computed oil 
the basis of certain assumptions as to the desired rate of tipping 
the car, so as to avoid discomfort to passengers, and, of course, 
this depends on the expected velocity. There is also a maximum 
Umitation, since the sum of the two spiral angles cannot exceed 
the total central angle of the curve. The minimum lengths 
recommended are as follows: 'i 



§ 77. ALINEMENT. 89 

On curves which limit the speed: 

6° and over, 240 feet; 

Less than 6°, 5iXspeed in m.p.h. for elevation of 8 inches. 
On curves which do not limit the speed : 

30 times elevation in inches, or 

fXultimate speed in m.p.h. Xelevation in inches. 

For example. (1) 5° curve which limits speed; speed limit 
48 m.p.h. by interpolation in table, § 71; 48X5^ = 256 feet 
minimum length, (2) 3° curve; maximum operating speed 60 
m.p.h.; superelevation, .62 feet = 7.44 inches; 30X7.44 = 223.2 
feet; or, 1X60X7.44 = 297.6 feet. Of course the higher value 
should be used, or say 300 feet as the minimum length. 

While it is generally true that the longer transition curves 
give easier riding, the spiral must not reach the center point of 
the curve. Since it is approximately true that the spiral extends 
for equal distances on each side of the original point of curve, it is 
nearly true that two spirals, each having the same length as the 
original curve, would just meet at the center. The length of a 
spiral should in general be very much less than the length of the 
original curve. 

77. Symbols. Beside the symbols whose significance is 
clearly indicated in Fig. 36, the following are defined: 

a The angle between the tangent at the TS and the chord from 

the TS to any point on the spiral; ai is the angle to the 

first chord point. 
A The angle between the tangent at the TS and the chord 

from the TS to the SC. 
D The degree of the central circular curve. 
A The central angle of the original circular curve, or the angle 

between the tangents. 
The total central angle of the spiral. 
k The increase in degree of curve per station on the spiral. 
L The length of the spiral in feet from the TS to the SC. 
S The length of the spiral in stations from the TS to the SC. 
s The length of the spiral in stations from the TS to any given 

point. 

78. Deflections. The field formulas for deflections are based 
on the following two equations: 

a = 10 ks^ minutes, 
A = 10 kS^ minutes. 



90 RAILROAD CONSTRUCTION. § 78. 

The first deflection ai = 10 ksi^ minutes. But k is the increase in 
degree of curve per station, and since the degree of curve in- 
creases as the length, k = D-^S, S being expressed in stations. 

/ D\ 
For point 1, since S = 10s, ai = 10l — jSi^ = Dsi, which may be 

expressed as the degree of the curves times the length of the chord 
in stations. For example, if the spiral is 400 feet long (which 
means that L = 400 and 8 = 4) and runs on to a 5° curve (then 
D = 5), one chord is 40 feet long and s = A station. Then ai = 5 
X0.4 = 2 minutes of arc for the deflection for the first chord point. 
And since the deflections are as the square of the number of sta- 
tions, the deflections from TS to succeeding stations will be 4, 9, 
16, 25, 36, 49, 64, 81, and 100 times 2 minutes, these factors being 
those given in the second vertical column of Part A of Table 
IV. The last deflection = A = 100X2' = 200' = 3"=' 20' = ^ (10*) 
= l4>, (f) being the total central angle of the spiral. Although 
it is always nearly true that A = !</>, and the error is inappreciable 
for small angles, the error amounts to 30 seconds of arc when 
^=21° 30', an unusually large angle. 

The deflection from any other point of the spiral to any other 
point, either forward or backward, may be found by multiplying 
the value of ai (in this case 2'), by the coefficients in the proper 
vertical column of that table. 

The spiral angle 



\lso, 



kS^ kL^ DL 5X400 ^„ 
w,= — = = — = =10'. 

2 20000 200 200 



kS^ DS 5X4 
= —- = -—=—— = 10' 



The values of the ratios U-^-L and V-^L for even degrees, and 
for A, C-7-L, X-^L, and F-v-L for half degrees are given in Parts 
B and C of Table IV. When it is desired to temporarily omit 
locating the intermediate points of the spiral, the jump from the 
TS to the SC may be made by measuring the distance U from the 
TS along the tangent. At that point a deflection and a 
measured distance V will give not only the position of SC but 
also the direction of the tangent at the beginning of the circular 
curve. Another method of locating the SC without locating 
the intermediate points is to make the deflection A at the TS 



§ 79 ALINEMENT. 91 

and measure the long chord G. In the above numerical problem 
this equals 400 X. 998664 = 399.47, a little over 6 inches short of 
the full 400 feet. By setting wp the transit at the SC, back- 
sighting at the TS, and turning off the angle (<^— A), which in 
the above case is 10° -3° 19' 57" = 6° 20' 03", the direction of 
the tangent at the SC is obtained. In this case, the three sec- 
onds variation from the approximate value is utterly negligible. 
The other dimensions are easily determined from the tables if 
desired;- 

X =. 996975 X400 = 398 . 79, 
r=. 058053X400= 23.22, 
C7=. 667742X400 = 267. 10 
7= .334313X400 = 133.73. 

For greater convenience of notation, the points TS, SC, CS, 
and ST, in Fig. 36 are also indicated by the letters Q, Z, Z' and 
Q' respectively. The same letters are used for the corresponding 
points in Figs. 37 and 38. 

79. Location of spirals and circular curve with respect to 
tangents. See Fig. 36. Let AV and BV be the tangents to be 
connected by a D° curve, ;having a suitable spiral at each end. 
If no spirals were to be used, the problem would be solved as in 
simple curves giving the curve AMB. Introducing the spiral has 
the effect of throwing the curve away from the vertex a distance 
MM' and reducing the central angle of the D° curve by 20. 
Continuing the curve beyond Z and Z' to A' and B', we will 
have AA' = BB'=MM'. ZK= the Y ordinate and is therefore 
known. Call MM' = m. A'N = Y-R vers </>. Then 

A'N Y-R vers <t> 

m = MM' = AA'-= — = -. , , . . . (33) 

cos f A cos ^A 

NA =AA' sin |A = (F-i2 vers 0) tan ^A. 

YQ = QK-K^^NA^AY 

=X — R sin ^-\-{Y—R vers 0) tan iA+ R tan |A 

=X — E sin <^ + F tan^A+E cos tan ^A. . . . (34) 

When A'N has already been computed, it may be more con- 
venient to write 

VQ=^X^-R (tan ^A- sin 0)+A'iV tan ^A (35) 



92 



RAILROAD CONSTRUCTION. 



§ 79.' 



= R exsec ^A-\ 



R vers </> 



(36) 



cos |A cos ^A 
AQ=VQ-AV 

= X -R sin (f) + {Y-R vers (l>) t&n I A (37) 

Example. To join two tangents making an angle of 34° 20' 
by a 5° 40' curve and suitable spirals. Assume that the spiral I 
is 300 feet long. Then 



ct> 



DS 5.67X3 



= 8.5° = 8° 30'. 



2 2 

Since, from Table IV, Part A, F-iv = . 049374 for <^ = 8° 30', 
F = 14.812; similarly, we find Z = 299.344 and C = 299.71. 



[Eq. 33] 



R 

vers 



3.00497 
8.04076 



[Eq. 361 



[Eq. 351 



[Eq. 37] 



m = MM' = 


11.110 

y = 14.812 

A'N= 3.702 

=AA' = 3.875 

7M = 47.164 

m= 3.875 

FM' = 51.039 


cos |A 

R 

exsec ^A 

= .30891 
= .14781 

.16110 
R 

A'N 
tan ^A 

AN 

R 

tan f A 


1.04573 

0.56843 
9.98021 

0.58822 




3.00497 
8.66863 
1.67360 


Z = 299. 344 
162.954 


nat. tan ^A = 
nat. sin = 

[See above] 


9.20709 
3.00497 
2.21206 


1.144 


0.56843 

9.48984 

0.05827 


FQ = 463.442 


3.00497 
9.48984 



312.471 
AQ = 150.971 



AV 2.49481 



§ 80. ALINiJMENT. 93 

It should be noted that AQ Is within a foot of equaling one-half 
the length of the spiral, which illustrates the general fact that a 
spiral begins at approximately one-half its length from the 
P.C. of the simple curve. All approximate systems of spirals 
assume this to be exactly true. 

80. Field-work. When the spiral is designed during the 
original location, the tangent distance VQ should be computed 
and the point Q located. It is hardly necessary to locate all of 
the points of the spiral until the track is to be laid. The extrem- 
ities should be located, and as there will usually be two or more 
full station points on the spiral, these should also be located. 
Z may be located by setting off QK = X and KZ = Y, or else by 
the tabular deflection for Z from Q and the distance ZQ, which 
is the long chord c. Setting up the instrument at Z and sighting 
back at Q with the proper deflection, the tangent at Z may be 
found and the circular curve located as usual, its central angle 
being A — 2(^. A similar operation will locate Q' from Z'. 

To locate points on the spiral. Set up at Q, with the plates 
reading 0° when the telescope sights along VQ. Set off from 
Q the deflections computed from Table IV for the instrument at 
Q, using a chord length of L^-IO, the process being like the 
method for simple curves except that the deflections are variable. 
If a full station-point occurs within the spiral, interpolate 
between the deflections for the adjacent spiral-points. For exam- 
ple, a 400-foot spiral running on to a 3° 31' curve begins at Sta. 
56 + 15. The spiral points are 40 feet apart. Sta. 57 comes 
5 feet beyond the second spiral point. The first deflection ai 
= Ds = 3.5X.4 = 1.4 min. The deflection to point 2 is 4X1.4 
= 5.6 min. and that to point 3 is 9X1.4 = 12.6 min. Then the 
deflection to Sta. 57 is 4% X (12.6 ~ 5.6) +5.6 = 6.47 min. 

This method is not theoretically accurate, but the error is small. 
Arriving at Z, the forward alinement may be obtained by sights 
ing back at Q (or at any other point) with the proper deflection 
for that point from the station occupied. Then when the plates 
read 0° the telescope will be tangent to the spiral and to the 
succeeding curve. All rear points should be checked from Z, 
If it is necessary to occupy an intermediate station, use the de- 
flections given for that station, orienting as just explained for Z, 
checking the back points and locating all forward points up to Z 
if possible. 

After the center curve has been located and Z- is reached, the 



M 



RAILROAD CONSTRUCTION. 



§81. 



other spiral must be located but in reverse order, i.e., the sharp' 
curvature of the spiral is at Z' and the curvature decreases 
toward Q'. 

8i. To replace a simple curve by a curve with spirals. Thisj 
may be done by the method of § 79, but it involves shifting the 
whole track a distance m, which in the given example equals 
3.87 feet. Besides this the track is appreciably shortened, 




Fig. 37. 



which would require rail-cutting. But the track may be kept 
at^practically the same length and the lateral deviation from the 
old track may be made very small by slightly sharpening the 
ieurvature of the old track, moving the new curve so that it is 
wholly or partially outside oi the old curve, the remainder of it 
with ike spirals being inside of the old curve. It is found by 
experience that a decrease in radius of from 5% to 10% will 
answer the purpose. The larger the central angle the less the 
change. The solution is as indicated in Fig. 37. 

O'N^R' cos <t>-\-Y. 

0'F = 0'iV sec |A % 

=R' cos <j> sec lA+F sec ^A. 



i 

§ 81. ALINEMENT. 95 

mm = MM'=MV-M'V 

ir =RexsecU-{0'V-R') 

= R exsec ^A—R' cos ^ sec |A — 7 sec lA+JK'. . , (38) 
AQ=QK-KN+NV-VA 

=X-R' sin + (/2' cos <f> + Y) tan ^A -E tan |A 
=Z-/2'sin0+/2'cos0tan|A-(i2-7) tan^A. . . (39) 

A 
The length of the old curve from Q to Q' = 2 AQ +100 -. 

A-2<^ 
The length of the new curve from Q to Q' = 2L+100 — — — , 

in which L is the length of each spiral. 

Example. Suppose the old curve is a 7° 30' curve with a 
central angle of 38° 40'. As a trial, compute the relative length 
of a new 8° 20' curve with spirals 240 feet long. |A = 19° 20'; 
R (for the 7° 30' curve) =764.49; R' (for the 8° 20' curve) = 
688.16; .^ = 10° 0';T = 13.933; Z=239.274. 

[Eq. 38] 



[Eq. 39] 



45.687 




R 

exsec jA 


2.88337 

8.77642 

. 1 . 65979 


B' =688. 16 
733.847 


718.200 
14.766 




R' 

cos <j> 
sec iA 




2.83768 
9.99335 
0.02521 

, 2.85624 




• 


r 

sec iA 

• • • 

R' 
sin ^ 






1 . 14405 
0.02.521 

1.16926 


732.966 


732.966 
119.497 




m= 0.881 


2.83768 
9.23967 

2.07735 


X =239.274 






i2' 

cos <t> 
tan fA 




237.770 . 


2.83768 
9.99335 
9.54512 

2.37615 






R 
Y 


= 764.49 
=13.93 

750.56 
tan |A 






2.87538 
9.54512 



477.044 



263.333 2.42050 



382.830 382.830 
AQ= 94.214 



06 RAILROAD CONSTRUCTION. §82. 



The length of the old Qurve from Q to Q' is 



100^=100?|^|I= 515.556 

JJ 7.5 

2AQ = 2X94.214= 188.428 



703.984 
2L =2X240 =480.000 



XT ir^n^-2<A ,„^ 38. 667 -20.000 „^, ^^^ 

New curve: 100 , =100 —— =224.000 

U o . oo 



704.000 704.000 
Difference in length = 0.016 

Considering that this difference may be divided among 21 
joints (using 33-foot rails) no rail-cutting would be necessary. 
If the difference is too large, a slight variation in the value of 
the new radius R' will reduce the difference as much as neces- 
sary. A truer comparison of the lengths would be found by 
comparing the lengths of the arcs. 

82. Applicatioii of ^ transition curves to compound curves. 
Since compound curves are only employed when the location is 
limited by local conditions, the elements of the compound curve 
should be determined (as in §§68 and 69) regardless of the 
transition curves, depending on the fact that the lateral shifting 
of thf^ curve when transition curves are introduced is very small. 
If the limitations are very close, an estimated allowance may be 
made for tht'm. * 

Methods have been devised for inserting transition curves 
between the bran nes of a compound curve, but the device is 
complicated and usually needless, since when the train is once 
on H f'urve the wheels press against the outer rail steadily and 
a change in curvature will not produce a serious jar even though 
the superelevation is temporarily a little more or less than it 
should b^ 

If the easier curve of the compound curve is less than 3° or 
4**, there may be no need for a transition curve off from that 
branch. This problem then has two cases according as transition 
curves are used at both ends or at one end only. 

a. With imnsiiion curves at both ends. Adopting the method 
of § 79, calHng Ji = JJ, we may compute mi=^MM/. Similarly, 
calling J.^ = ^J, we may compute m.^=MM2. But M/ and i1// 
must be made to coincide. This may be done by moving the 
curve Z'M/ and its transition curve parallel to Q'V a distance 
ilf/Af.,, and the other curve parallel to QF a distance M/M^, 



§82. 



ALINEMENT. 



97 



In the triangle M/MgM/, the angle at M/ = 90° — J,, the angle 
at ikf/ = 90° — //a, and the angle at M^^J. 



cos Jj 
sin J 



Then ilf/ilfo^Tli/Ma' -.—,- ^^ =(rn^-m;) . 

^ "* sm J SI 

^. .1 1 T.^ /^^ Ti^ /i^ ,sin (90° — //,) . . cos J, 

Similarly M/M. =M/M./ ——r—^ = {m^-m^j-. — }. 

J ^ i y i sin J sm J 



K40) 




02 



J^-Vl^^ 



Era. 38. 

b. With a transition curve on the sharper curve only. Com- 
pute nil = MM I as before; then move the curve Z^M^ parallel 
to Q'F a distance of 

M^'M^^m^ ^?— .\ ,.,... (41) 

sin J " 



98 RAILROAD CONSTRUCTION. § 83. 

The simple curve MA is moved parallel to VA a distance of 

MM,=m,''-.^ (42) 

sm J > 

If Ji and ^2 are both small, ilf /Af^ and MM^ may be more 
than mj, but the lateral deviation of the new curve from the old 
will always be less than m^. 

83. To replace a compound curve by a curve with spirals. 
The numerical illustration given below employs another method. 
We first solve for fn^ for the sharper branch of the curve, plac- 
ing ^1 = 1^ in Eq. 38. A value for R^ may be found whose 
corresponding value of m^ will equal mj. Solving Eq. 38 for B', 
we obtain 

R vers ^A— m cos lA — Y 
R = ~ . . . . (43) 



cos — COS |A 



Substituting in this equation the known value of mi (=m2) 
and calUng R' = R2'f R = R2, and A2 = |A, solve for R2'. Obtain 
the value of AQ for each branch of the curve separately by Eq. 
39, and compare the lengths of the old and new lines. 

Example. Assume a compound curve with A = 8°, A = 4**, 
Ai = 36°, and A2 = 32°. Use 240-foot spirals at each end. Assume 
that the sharper curve is sharpened from 8° 0' to 8° 15'. 

E(4. 381 







Ri 
exaec 36° 


2.85538 
9.37303 


169.21 


2 .22842 


695.09 


^ 8.25X240 
=9.'»9 =9°54' 

Fi =240 X. 05747 
= 13.79 


Ri' (8° 15') 
cos <t>i 
sec At 

846.39 . . . . 

Yi 

sec Ai 

17.05 .. . . . 




2.84204 
9.99348 
0.09204 


864.30 




2.92757 




1.13969 
0.09204 




1.23173 


863.44 


863.44 





mi= 0.86 



I §83. 



ALINEMENT. 



99 



[Ea.431 *,=t^!«|<^ «' 3.15615 

^- vers 32 9.18170 

=4°.86=4°51'.6 = 

217.700 2.33785 

Y2= .02826X240. mi =0.86 9.93450 

= 6.782 cos 32° 9.92842 

0.729 ..... 9.86292 
72= 6.782 

7.511 7.511 

210.189 2.32261 

nat. cos <^2 = . 99f40 
nat. cos Az = . 84805 

. 14835 9.17129 

iJa' =1416.84 14° 2' 41"] 3.15132 

Eq. 39] Xi= 239.286 Zi = .997024 X240 . ^=^ 

= 239.286 Ri' 2.84204 

sin 4>i 9.23535 

119.505 2.07739 

Ri' 2.84204 

coS(j!>i 9.99348 

tan ^A[Ai =36°] 9 .86126 

497.489 . . • 2.69678 

721=716.78 
Yi= 13.70 

703.08 2.84700 

tan JA 9.86126 

736.775 ~ 

630.325 510. S20 2.70826 

ilQi = 106.450 630.325 

[Eq. 391 Ri' 3.15132 

X'2 = . 999284X240 sin 02 8.92799 

=239.828 120.035 2.07931 

Ri' 3.15132 

cos <^2 9.99843 

tan |A(A2=32°) 9.79579 

882.145 . . .~".T. 2.94554 

722=1432.7 
Y2= 6.8 

1425.9 3.1540D 

tan |A 9.79579 

891.00 2.94988 

1121.973 1011.03 
1011.03 

AQi= 110.94 



100 RAILROAD CONSTRUCTION. §84. 

For the length of the old track we have: 

100-^=100^= 450. 

100-^^=100^- 800. 

AQi= 106.45 
AQ2= 110.94 

= 1467.39 

For the length of the new track we have: 

100 ^' = 100 f^- 671.11 

Spiral on 8° 15' curve = 240.00 

Spiral on 4° 02' 41 ' curve = 240,00 

Length of new track = 1467 . 47 
Length of old track =■■ 1467 . 39 

Excess in length of new track = . 08 feet. 

Since the new track is slightly longer than the old, iu shows 
that the new track runs too far outside the old track at the 
P.C.C. On the other hand the offset m is only 0.86. The 
maximum amount by which the new track comes inside of the 
old track at two points, presumably not far from Z' and Z, is 
very difficult to determine exactly. Since it is desirable that 
the maximum offsets (inside and outside) should be made as 
nearly equal as possible, this feature should not- be sacrificed to 
an effort to make the two lines of precisely equal length so that 
the rails need not be cut. Therefore, if it is found that the offsets 
inside the old track are nearly equal to m (0.86), the above 
figures should stand. Otherwise m may be diminished (and the 
above excess in length of track diminished) by increasing Ri 
very slightly and making the necessary consequent changes. 

VERTICAL CURVES 

84. Necessity for their use. Whenever there is a change in 
the rate of grade, it is necessary to eliminate the angle that 
would be formed at the point of change and to connect the two 
grades by a curve. This is especially necessary at a sag between 
two grades, since the shock caused by abruptly forcing an up- 
ward motion to a rapidly moving heavy train is very severe both 
to the track and to the rolling stock. The necessity for vertical 
curves was even greater in the days when link couplers were in 
universal use and the "slack" in a long train was very great. 



§ 85. ALINEMENT. 101 

tinder such circumstances, when a train was moving down a 
heavy grade the cars would crowd ahead against the engine. 
Reaching the sag, the engine would begin to pull out, rapidly 
taking out the slack. Six inches of slack on each car would 
amount to several feet on a long train, and the resulting jerk on 
the couplers, especially those near the rear of the train, has fre- 
quently resulted in broken couplers or even derailments. A 
vertical curve will practically eliminate this danger if the curve 
is made long enough. 

85. Required length. Theoretically the length should de- 
pend on the change in the rate of grade and on the length of the 
longest train on the road. A sharp change in the rate of grade 
requires a long curve; a long train requires a long curve; but 
since the longest trains are found on roads with light grades and 
small changes of grade, the required length is thus somewhat 
equahzed. The A.RtE.A. rule is: "On class A roads (see § 234) 
rates of change of 0.1 per cent per station on summits and 0.05 per 
dent per station in sags should not be exceeded. On minor roada 
0.2 per cent per station on summits and 0.1 per cent per station 
in sags may be used." When changing from a down grade to an 
up grade (or vice versa) the change of grade equals the numerical 
sum of the two rates of grade. For example, if a 0.5 per cent 
down grade is followed by a 0.7 per cent up grade, the road being 
a "minor" road, then, by the above rule the length of the curve 
should be at least [0.5 -(-0.7)] ^0.1 =12 stations or 1200 feet. 
Added. length increases the amount of earthwork required both 
in cuts and fills, but the resulting saving in operating expenses 
will always justify a considerable increase. 

86. Form of curve. In Fig. 39 assume that A and C, equi- 



m 




d -J. -\ 'J^^^ 

level_line ^ b' 

Fig. 39. 

distant from B, are the extremities of the vertical curve. Bisect 
AC at e; draw Be and bisect it at h. Bisect AB and BC at k 
and I. The Une kl will pass through h. A parabola may be 
drawn with its vertex at h which will be tangent to AB and BC 
at A and C. It may readily be shown * from the properties of 

* See note at end of this chapter. 



102 RAILROAD CONSTRUCTION. § 87. I 

' I 

a parabola that if an ordinate be drawn at any point (as at n) 
we will have 

sn : eh (or hB) \ : Arn^ : Ae^ 
OP sn=^eh (44> 

In Fig. 39 the grades are necessarily exaggerated enormously. 
With the proportions found in practice we may assume that 
ordinates (such as mt, eB, etc.) are perpendicular to either 
grade, as may suit our convenience, without any appreciable 
error. In the numerical case given below, the variation of 
these ordinates from the vertical is 0° 07', while the effect of 
this variation on the calculations in this case (as in the^ most 
extreme cases) is absolutely inappreciable. It may easily be- 
shown that the angle CA^^half the algebraic difference of the 
rates of grade. Call the difiference, expressed in per cent ef 
grade, r; then CAB = ^r. Let Z=length (in "stations" of lOQ 
feet) of the line AC, which is practically equal to the horizontal 
measurement. Since the angle CAB is one-half the total charigeT 
of grade at B, it follows that Be = \lX ^r Therefore : 

Bh^\lr. ....... (45) 

Since Bh (or eh) and Ae are constant for any one curve, the cor- 
rection sn alt any point (see Eiq. 44) equals a constant times Am^. 
87. Numerical example. Assume that B is located at Stal 
16+20; that the grade of AB is -0.5%, and of BC +0.7%; 
also that the elevation of B above the datum plane is 162.6.'^ 
Then the algebraic difference of the grades, r, =0.7 — (— 0.5) ==' 
1.2; Z = 12. 5/i = |Zr = ^Xl2X1.2 = l,8. A is at Sta. 10+20 
and its elevation is 162.6+(6X0.5) = 165.6; C is at Sta. 22+20; 
and its elevation is 162.6 + (6X0.7) = 166.8. The elevation of- 
Sta. 11 is found by adding sn to the elevation of s on the 
straight 'grade line. The constant] (e^-j-Ae ) equals in this case 
1.8-j-6002 = ^ijjji^^^. Therefore the curve elevations are 

A, Sta. 10+20, 162. 6+ (6. 00x0. 5) =165.60 

11 165.6-(0.80X0.5) + 3tjsW 802=165.23 

12 165.6-(1.80X0.5) + jij|jWi5l802=164.86 

13 165 . 6 - (2 . 80 XO . 5) + snxrW 280^ = 164 .59 

14 165.6-(3.80X0.5)+ac?5W3802=164.42 

15 165.6 -(4.80X0.5) + 2nj5W 4802 =164.35 

16 165.6-(5.80X0.5) + jsoW5 5802=164.38 



§87. 



ALINEMENT. 103 

B, 16+20,162.6+1.80 =164.40 

17 166.8-(5.20X0.7) + Wo?yo 5202=164.51 

18 166 . 8 - (4 . 20 X . 7) + ^^oWs 420^ = 164 . 74 

19 166.8-(3.20X0.7) + W(jTj(j320i = 165.07 

20 166.8 -(2.20 X0.7) + Woot220« =165.50 . 

21 166.8-(1.20X0.7) + lij^W 120^ = 166.03 

22 166.8 -(0.20X0.7) + 5ff3\jw 202=166.66 

C, 22+20, 162. 6+(6. 00X0. 7) =166.80 



DEMONSTRATION OF EQ. 44. 

The general equation of a parabola passing through the point n (Fig. 36) 
may be written 

2/' + y,!^ = 22?(a; + a;„), 



y2 y,^ 



from which ir„ =— — \- -rr— — x. 

^ 2p 2p N . 

When X =• x^ y '" Va ^^^ ^^ have 

The general equation of a tangent passing through the point A may be 
written 

from which 
When X = iXg,y = 







X = 


vva 
p 


^A. 




■■Va 


[ = 


= 2/^], and 


we have , 








^a = 


Vn^A 
P 


■ ^A, 




o/n 


= 


x^- x^== 


yA^ + 


Vn- 


^VriVj 


OJh 




2v 








= 


(VA-Vvy 
2p 


Art? 
2p ' 






2p = 


va' 

XA 


Ae' 

. • 

eh 








.*. »n = 


A 2 

eh -, 

Ae' 







Tliis proves the general proposition that if secants are drawn parallel to 
the axis of x, intersecting a parabola and a tangent to it, the intercepts be- 
tween the tangent and the parabola are proportional to the square of the 
distances (measured parallel to y) from the tangent point. 



CHAPTER III. 

EARTHWORK. 
FORM OF EXCAVATIONS AND EMBANKMENTS. 

88. Usual form of cross-section in cut or fill. The normal 
form of cross-section in cut is as shown in Fig. 40, in which 
g . . ,g represents the natural surface of the ground, no matter 



e \ 




how irregular; ah represents the position and width of the re- 
quired roadbed; ac and bd represent the ''side slopes" whicl; 
begin at a and 6 and which intersect the natural surface at such 




\ 



Fia. 41. 



points {c and d) as will be determined by the required slope 
angle (/?). 

The normal section in fill is as shown in Fig. 41. The points 
c and d are likewise determined by the intersection of the re- 

104 



§89. 



EARTHWORK. 



106 



quired side slopes with the natural surface. In case the required 
roadbed {ah in Fig. 42) intersects the natural surface, both cut 




Fig. 42. 

and fill are required, and the points c and d are determined as 
before. Note that /? and /?' are not necessarily equal. Their 
proper values will be discussed later. 

89. Terminal pyramids and wedges. Fig. 43 illustrates the 
general form of cross-sections when there is a transition from 
cut to fill. a,.,g represents the grade line of the road which 




Fig. 43. 

passes from cut to fill at d. sdt represents the surface profile, 
A cross-section taken at the point where either side of the road- 
bed first cuts the surface (the point m in this case) will usually 
be triangular if the ground is regular. A similar cross-section 
should be taken at o, where the other side of the roadbed cuts 
the surface. In general the earthwork of cut and fill terminates 



106 KAILROAD CONSTRUCTION. § 90. | 

in two pyramids. In Fig. 43 the pyramid vertices are at n 
and k, and the bases are Jhm and opq. The roadbed is generally, 
wider in cut than in f]ll, and therefore the section Ihm and the; 
altitude In are generally greater than the section opq and thel 
altitude pk. When the line of intersection of the roadbed and 
natural surface {nodkm) becomes perpendicular to the axis of 
the roadbed (ag) the pyramids become wedges whose bases are 
the nearest convenient cross-sections. 

90. Slopes, a. Cuttings. The required slopes for cuttings 
vary from perpendicular cuts, which may be used in hard rock 
which will not disintegrate b}^ exposure, to a slope of perhaps 
4 horizontal to 1 vertical in a soft material like quicksand or in 
a clayey soil which flows easily when saturated. For earthy 
materials a slope of 1 : 1 is the maxim.um allowable, and even 
this should only be used for firm material not easily affected by 
saturation, A slope of IJ horizontal to 1 vertical is a safer 
slope for average earthwork It is a frequent blunder that 
slopes in cuts are made too steep, and it results in excessive work 
in clearing out from the ditches the material that slides down, 
at a much higher cost per yard than it would have cost to take 
it out at first, to say nothing of the danger of accidents from 
possible landslides. 

b. Embankments. The slopes of an embankment vary from 
1 : 1 to 1.5 : 1. A rock fill will stand at 1 : 1, and if some care 
is taken to form the larger pieces on the outside into a rough 
dry wall, a much steeper slope can be allowed. This method is 
sometimes a necessity in steep side-hill work. Earthwork em- 
bankments generally require a slope of 1^ to 1. If made 
steeper at first, it generally results in the edges giving way, re- 
quiring repairs until the ultimate slope is nearly or quite 1^ : 1. 
The difficulty of incorporating the added material with the old 
embankment and preventing its sliding off frequently makes 
these repairs disproportionately costly. 

91. Compound sections. When the cut consists partly of 
earth and partly of rock, a compound cross-section must be 
made. If borings have been made so that the contour of the 
rock surface is accurately known, then the true cross-section may 
be determined. The rock and earth should be calculated sepa- 
rately, and this will require an accurate knowledge of where the 
rock "runs out" — a difficult matter when it must be deter- 



§ 92. EARTHWORK. 107 

mined by boring. During construction the center part of the 
earth cut would be taken out first and the cut widened until a 
sufficient width of rock surface had been exposed so that the 
rock cut would have its proper width and side slopes. Then the 
earth slopes could be cut down at the proper angle. A"berni" 
of about three feet should be left on the edges of the rock cut as 




Fig. 44. 

a margin of safety against a possible sliding of the earth slopes. 
After the work is done, the amount of excavation that has been 
made is readily computable, but accurate preliminary estimates 
are difficult. The area of the cross-section of earth in the figure 
must be determined by a method similar to that developed for 
borrow-pits (see § 120). 

02. Width of- roadbed. Ow'ng to the large and often dis- 
proportionate addition to volume of cut or fill caused by the 
addition of even one foot to the width of roadbed, there is a 
natural tendency to reduce the width until embankments become 
unsafe and cuts are too narrow for proper drainage. The cost 
of maintenance of roadbed is so largely dependent on the drain- 
age of the roadbed that there is true economy in making an 
ample allowance for it. The practice of some of the leading 
railroads of the country in this respect is given in the following 
table, in which are also given some data belonging more properly 
to the subject of superstructure. 

It may be noted from the table that the average width 
for an earthwork cut, single track, is about 24.7 feet, with a 
minimum of 19 feet 2 inches. The widths of fills, single track, 
average over 18 feet, with numerous minimums of IG feet. 
The widths for double track may be found by adding the distance 
between track centers, which is usually 13 feet. 



108 



RAILROAD CONSTRUCTION. 



§92. 



M 









m 



O 

M 

PJ 

o 



o 

o 
6 



g 

Tan 

§ 

< 
O 

o 



^ 



O cj 


, 1 
























1 


ipfe 1 




!. 


iS ^ 


rs-i; 


^OOM COfOM CONCO cc 


•^ 


.Itfl^g 1 


Mr^lH »-^»HtH fHjHjH r-l 




Pffl 


o 








« 


rH i-H i-H iH »-( -J i-l rH tH iH i-H i-H 

1 - * 


THr-t 


^ 


^■M 


** ■►•••*•••••!• ••»» «*•• 


' 


in 


£ 


lO IC lO lO lO "? lO "5 »0 >0 U5 lO ! 


io»o 




r-l rH 1-1 i-H tH i-H rH 1-1 1-1 rH rH i-H | 


THrH 


ei 








rt 














0) 








P. 




*-H «-! i-l iH i-l rH rH rH r-l 


rH 


o 


H^ 


THrH rH _,rH ,_ ._ ..♦"''^ 


..^ 


CQ 


3 


•■ ■■io»otoio>o "»o "icwj^o " ■ 


lo •■ 




o 


rH F+* r-l • rH •• • rH^r^M • r-t | 






rHi-Hr-lTHrH tH rH r-i rH 


rH 








'. I> 


^ 




^ 






p3 




•O CM 




coc^' '■ 


\ 






s 




•CO -MCO 


^ COM 


CO o; ^ ■ 


rH 










■ CO 


^ 


CO 1 


CO 






^ 


























^ 




^ 

3 






•lO©^ 


^S 


^ a ^ 


X 




o 


,i 




:xxx 


^X^ 


Xo « ^ 

(Mcoss 


+ 






o 




"OOrHCO 


M + ^ 


^ (N 

CO ^ OS 
« CO (N 


rji 

Xl 
CO 






• 




; "* % 






I 






^ 


O 




CD 




^ : 


C^ll^ 




g 


(N 


• rH -M C^l rH V ^^ 


1—1 




l5 ; 


b-.os'"' 

rH r^ 


,M 


























• 














o 






rt ;5^^^ 3 


U3 




Sm 








^2^kOCOr^ . 






5s "^ 


fl 


bo 

a 
eg 


•♦3 




X 




*£3 c3^ 


3 


•^-. "^ "T'^^^rsi* c<i 


IN 




(M 


6f SW 




U 


^_^^rHrH(M ^ 
.tH 


+ 

CO 
r-l 




^ CD 
IN rH 


ININtji 
^ V. rH 

t-hC< 








• • 






• d 








'T' 










* ,__| 






• u 




















>3 






■ (U 




















<yi 






CO . 


















^=a 






: ii 












tj 






6 0) 3 • 




. tf.T5 




P) 








1 










■t/3 












CI o 

-s 






. ^.j:1 r;^ Ih aj d C-- • O 


(U M 






< 


ooc 


K 


w 


h:1h:i 


^ 


S 


!?; 


12; 


PM 


P 


1 






, =« 2 

o ca 

-" Jh 

o C <u 

O (B ^ 



CO 53 

3 



03 



a> 






0) 



O O O GQ 



CO fl 

o 



-1^ C ''' 

TJ <+H 43 53 

o §-« 

TO ' — 1 .^ 

go -fJ 



rs -« 



«3 

M tgfo 5 

43 ^!N 
•-^ O 0) CQ 

•♦^ o3 coZ3 

(B j3 C3 0) 

CO e9 OT) 

w " ai O 
2 <c rt 

X'<o:3 

CJ<*5 O) 

**! rap; 
1 

o 

tH 



* 



p^ 



s 



93. EARTHWORK. 109 

93. Form of subgrade. Specifications (or the cross-section 
drawings) formerly required that the subgrade should have a 
curved form, convex upward, or that it should slope outward 
from a slight ridge in the center, with the evident purpose of 
draining to the sides all water which might percolate through the 
ballast. If the subsoil were hard and impenetrable by the ballast, 
the method might answer, but experience has shown that, with 
ordinary subsoils, the ballast immediately under each rail is 
forced a little deeper into the subsoil by the passage of each train. 
Periodical retamping of ballast under the ends of the ties, and 
httle or no tamping under the center, only adds to the accumula- 
tion under each rail. A cross-section of a very old roadbed will 
frequently show twice as much depth of ballast under the rails 
as there is under the center. This method of tamping quickly 
obliterates the original line of demarcation between ballast and 
subsoil and any expected improvement in drainage due to sloping 
subsoil is 'not realized. Therefore the A.R.E.A. specifications 
call iorflat subgrades. 

94* Ditches. "The stability of the track depends upon the 
strength and permanence of the roadbed and structures upon 
which it rests; whatever will protect them from damage or pre- 
vent premature decay should be carefully observed. The worst 
enemy is water, and the further it can be kept away from the 
track, or the sooner it can be diverted from it, the better the 
track will be protected. Cold is damaging only by reason of 
the water which it freezes; therefore the first and most impor- 
tant provision for good track is drainage," (Rules of the Road 
Department, Illinois Central R. R.) 

The form of ditch generally prescribed has a flat bottom \2" 
to 24" wide and with sides having a minimum slope, except in 
rock-work, of 1 : 1, more generally 1.5 : 1 and sometimes 2 : 1. 
Sometimes the ditches are made V-shaped, which is objection- 
able unless the slopes are low The best form is evidently that 
which will cause the greatest flow for a given slope, and this 

will evidently be the form in which the 
ratio of area to wetted perimeter is the 

v:-.-, -■;:> ^// largest. The semicircle fulfills this con- 

mMJ/fl' ,. r , , , ^ 11 

^ dition better, than any other iorm, but the 

Fig. 45. . . 

nearly vertical sides would be difficult to 

maintain. (See Fig. 45.) A ditch, with a flat bottom and such 




110 RAILROAD CONSTRUCTION. § 95. 

slopes as the soil requires, which approximates to the circular 
form will therefore be the best. 

When the flow will probably be large and at times rapid it 
will be advisable to pave the ditches with stone, especially if the 
soil is easily washed away. Six-inch tile drains, placed 2' under 
the ditches, are prescribed on some roads. (See Fig. 46.) No 
bettor method could be devised to insure a dry subsoil. The 
ditches through cuts should be led off at the end of the cut so 
that the adjacent embankment will not be injured. 

Wherever there is danger that the drainage from the land 
above a cut will drain down into the cut, a ditch should be made 
near the edge of the cut to intercept this drainage, and this 
ditch should be continued, and paved if necessary, to a point 
where the outflow will be harmless Neglect of these simple 
and inexpensive precautions frequently causes the soil to be 
loosened on the shoulders of the slopes during the progress of a 
heavy rain, and results in a landslide which will cost more to 
repair than the ditches which would have prevented it for all 
time. 

Ditches should be formed along the bases of embankments; 
they facilitate the drainage of water from the embankment, 
and may prevent a costly slip and disintegration of the em- 
bankment, 

95- Effect of sodding the slopes, etc. Engineers are unani- 
mously in favor of rounding off the shoulders and toes of em- 
bankments and slopes, sodding the slopes, paving the ditches, 
and providing tile drains for subsurface drainage, all to be put 
in during original construction. (See Fig. 46.) Some of the 
higliest grade specifications call for the removal of the top layer 
of vegetable soil from cuts and from under proposed fills to 
some convenient place, from which it may be afterwards spread 
on the slopes, thus facilitating the formation of sod from grass- 
seed. But while engineers fayor these measures and their 
economic value may be readily demonstrated, it is generally 
impossible to obtain the authorization of such specifications 
from railroad directors and promoters. The addition to the 
original cost of the roadbed is considerable, but is by no means 
as great as the capitalized value of the extra cost of mainte- 
nance resulting from the usual practice. Fig. 46 is a copy of 



§95. 



earthwokk:. 



Ill 



designs * presented at a convention of the Ameiican Society of 
Civil Engineers by Mr. D. J. Whittemore, Past President of 
the Society and Chief Engineer of the Chi., Mil. & St. Paul 




PROPOSED SECTION OF ROAPBEO IN EXCAVATION 




PROPOSED SECTION OF ROADBED ON EMBANKMENT. 




Fig. 46. — "Whittemore on Railway Excavation and Embankments " 
Trans. Am. Soc. C. E., Sept. 1894. 

R. R. The " customary sections " represent what is, with some 
variations of detail, the practice of many railroads. The " pro- 

* Trans. Am. Soc. Civil Eng., Sept. 1894. 



112 EAILROAD CONSTRUCTION. § 96 

posed sections" elicited unanimous approval. They should be 
adopted when not prohibited by financial considerations. 



EARTHWORK SURVEYS. 

96. Relation of actual volume to the numerical result. It 

should be reali/ied at the outset that the accuracy of the result 
of computalions of the volume of any given mass of earthwork 
has but little relation to the accuracy of the mere numerical 
work. The process of obtaining the volume consists of two 
distinct parts. In the first place it is assumed that the volume 
of the earthwork may be represented by a more or less com- 
plicated geometrical form, and then, secondly, the volume of 
such a geometrical form is computed. A desire for simplicity 
(or a frank willingness to accept approximate results) will often 
cause the cross-section men to assume that the volume may be 
represented by a very simple geometrical form which is really 
only a very rough approximation to the true volume. In such 
a case, it is only a waste of time to compute the volume with, 
minute numerical accuracj''. One of the first lessons to be 
learned is that economy of time and effort requires that the 
accuracy of the numerical work should be kept proportional to 
the accuracy of the cross-sectioning work, and also that the 
accuracy of both should be proportional to the use to be made 
of the results. The subject is discussed further in § 125. 

97. Prismoids. To compute the volume of earthwork, it is 
necessary to assume that it has some geometric form whose vol- 
ume is readily determinable. The general method is to consider 
the volume as consisting of a series of prismoids, which are 
solids having parallel plane ends and bounded by surfaces which 
may be formed by lines moving continuously along the edges of 
the bases These surfaces may also be considered as the sur- 
faces generated by lines moving along the edges joining the cpr- 
respondirg points of the bases, these edges being the directrices, 
and the lines being always parallel to either base, which is a 
plane director. The surfaces thus developed may or ma}'- not 
be planes. The volume of such a prismoid is readily determin- 
able (as explained in § 108 et seq.), while its definition is so very 
general that it may be applied to very rough ground. The 
"two plane ends" are sections perpendicular to the axis of the 
road. The roadbed and side slopes (also plane) form three of 



§98. 



earthwork: 



113 



the side surfaces. The only approximation lies in the degree of 
accuracy with which the plane (or warped) surfaces coincide with 
the actual surface of the ground between these two sections. 
This accuracy will depend (a) on the number of points which 
are taken in each cross-section and the accuracy with which the 
Hnes joining these points coincide with the actual cross-sections; 
(6) on the skill shown in selectiDg places for the cross -sections so 
that the warped surfaces shall coincide as nearly as possible with 
the surface of the ground. In fairly smooth country, cross- 
sections every 100 feet, placed at the even stations, are suf- 
ficiently accurate, and such a method simplifies the computations 
greatly; but in rough country cross-sections must be inter- 
polated as the surface demands. As will be explained later, 
carelessness or lack of judgment in cross-sectioning will introduce 
errors of such magnitude that all refi.nements in the computa- 
tions are utterly wasted. 

98. Cross-sectioning. The process of cross-sectioning con- 
sists in determining at an}^ place the intersection by a vertical 
plane of the prism of earth lying between the roadbed, the side 



slope 



p, and the natural surface. 



The intersection with the road- 




FiG. 47. 



bed and side slopes gives three straight lines. The intersection 
with the natural surface is in general an irregular line. On 
smooth regular ground or when approximate results are accept- 
able this )m^. is assumed to be straight. According to the irreg- 



114 RAILROAD CONSTRUCTION. § 99. 

ularity of the ground and the accuracy desired more and more 
*' intermediate points" are taken. 

The distance (d in Fig. 47) of the roadi^ed below (or above) 
the natural surface at the center is known or determined from 
the profile or by the computed establishment of the grade line. 
The distances out from the center of all "breaks " are deter- 
mined with a tape. To determine the elevations for a cut, set 
up a level at any convenient point so that the line of sight is 
higher than any point of the cross-section, and take a rod read- 
ing on the center point. This rod reading added to d gives the 
height of the instrument (H. I.) above the roadbed. Sub- 
tracting from H. I. the rod reading at any ''break" gives the 
height of that point above the roadbed (hi, ki, hr, etc.). This 
is true for all cases in excavation. For fill, the rod reading at 
center minus d equals the H. I,, which may be positive or nega- 
tive. When negative, add to the "H. I." the rod readings of 
the intermediate points to get their depths below "grade"; 
when positive, subtract the " H. I." from the rod readings. 

The heights or depths of these intermediate points above or 
below grade need only be taken to the nearest tenth of a foot, 
and the distances out from the center will frequently be suffi- 
ciently exact when taken to the nearest foot. The roughness of 
the surface of farming land or woodland generally renders use- 
less any attempt to compute the volume with any greater accu- 
racy than these figures would imply unless the form of the ridges 
and hollows is especially well defined. The position of the slope- 
stake points is considered in the next section. Additional dis- 
cussion regarding cross-sectioning is found in § 118. 

99. Position of slope-stakes. The slope-stakes are set at the 
intersection of the required side slopes with the natural surface, 
which depends on the center cut or fill (d). The distance of 
the slope-stake from the center for the lower side is x = ^h 
+ s(d+y); for the up-hill side it is x' = ^b + s(d—y'). s is the 
"slope ratio" for the side slopes, the ratio of horizontal to ver 
tical. Tn the above equation both x and y are unknown. There- 
fore some position must be found by trial which will satisfy "the 
equation. As a preliminary, the value of x for the point a = hb 
+ sd, which is the value of x for level cross-sections. In the 
case of fills on sloping ground the value of x on the down-hill 
eide is greater than this ; on the up-hill side it is less. The differ- 
ence in distance is s times the difference of elevation. Tak€ f 



§99. EARTHWOEK. 115 

numerical case corresponding with Fig. 48. The rod reading 
on c is 2.9; d=4.2: thercfoie the telescope is 4.2—2.9 = 1.3 
below grade, s = 1.5 : 1, 6 = 16. Hence for the point a (or for 
level ground) a: = |X 16 + 1.5X4.2 = 14.3. At a distance out 
of 14.3 the ground is seen to be about 3 feet lower, which will 
not only require 1.5X3=4.5 more, but enough additional dis- 
tance so that the added distance shall be 1,5 times the additional 
drop. As a first trial the rod ma}^ be held at 24 feet out and a 
reading of, say, 8.3 is obtained. 8.3 + 1.3=9.6, the depth of 
the point below grade. The point on the slope line (n) which 
has this depth below grade is at a distance from the center 



Fig. 48. 

rc =8 + 1.5X9.6 =22.4. The point on the surface (s) having 
that depth is 24 feet out. Therefore the true point (m) is 
nearer the center, A second trial at 20.5 feet out gives a rod 
reading of, say, 7.1 or a depth of 8.4 below grade. This corre- 
sponds to a distance out of 20,6, Since the natural soil (espe- 
cially in farming lands or woods) is_generally so rough that a 
difference of elevation of a tenth or so may be readil}'^ found by 
slightly varying the location of the rod (even though the dis- 
tance from the center is the same), it is useless to attempt too 
much refinement, and so in a case like the above the combina- 
tion, of 8.4 below grade and 20.6 out from center may be taken 
to indicate the proper position of the slope-stake. This is 
usually indicated in the form of a fraction, the distance out being 
the denominator and the height above (or below) grade being 
the numerator; the fact of cut or fill may be indicated by C or F» 
Ordinarily a second trial will be sufficient to determine with 
sufficient accuracy the true position of the slope- stake. Ex- 
perienced men will frequently estimate the required distance 



116 RAILROAD CONSTRUCTION. § 100. 

out to within a few tenths at the first trial. The left-hand pages 
of the note-book should have the station number, surface eleva- 
tion, grade elevation, center cut or fill, and rate of grade. The 
right-hand pages should be divided in the center and show the 
distances out and heights above grade of all points, as is illus- 
trated in § 109. The notes should read up the page, so that when 
looking ahead along the line the figures are in their proper 
relative position. The "fractions" farthest from the center 
line represent the slope-stake points. 

100. Setting slope-stakes by means of " automatic " slope- 
stake rods. The equipment consists of a specially graduated tape 
and a specially constructed rod. The tape may readily be prepared 
by marking on the hack side of an ordinary 50-foot tape which is 
graduated to feet and tenths. Mark "0" at "^b " from the tape- 
ring. Then graduate from the zero backward, at true scale, to 
the ring. Mark off "feet" and "tenths" on a scale propor- 
tionate to the slope ratio. For example, with the usual slope 
ratio of 1.5:1 each ''foot" would measure 18 inches and each 
" tenth " in proportion. 

The rod, 10 feet long, is shod at each end and has an endless 
tape passing within the shoes at each end and over pulleys — to 
reduce friction. The tape should be graduated in feet and 
tenths, from to 20 feet — the and 20 coinciding. By moving 
the tape so that is at the bottom of the rod — or (practically) 
so that the 1-foot mark on the tape is one foot above the bottom 
of the shoe, an index mark may be placed on the back of the 
rod (say at 15 — on the tape) and this readily indicates when the 
tape is "set at zero." 

The method of use may best be explained from the figure and 
from the explicit rules as stated. The proof is given for two 
assumed positions of the level. 

(1) Set up the level so that it is higher than the "center" 
and (if possible) higher than both slope-stakes, but not more 
than a rod-length higher. On very steep ground this may be 
impossible and each slope-stake must be set by separate positions 
of the level. 

(2) Set the rod-tape at zero (i.e., so that the 15-foot mark 
on the hack is at the index mark) . 

(3) Hold the rod at the center-stake (B) and note the read- 
ing (n^ or n^). Consider n to be always plus; consider d to be 
plus for cut and minus for fill. 



§100. 



EAETHWORK. 



117 



(4) Raise the tape on the face side of the rod (n + d). Applied 
Hterally (and algebraically) , when the level is helow the roadbed 
(only possible for fill), (n + d)^ (^2 + {—dj))=^n2 —df. This being 
numerically negative, the tape is lowered {df—7i^. With level 
at (1), for fill, {n + d)= {n^ + ( —dj)) = (jii—df) ; this being positive, 
the tape is raised. With level at (1), for cut, the tape is raised 
(ni + dc) . In every case the effect is the same as if the telescope 
were set at the elevation of the roadbed. 




Fig. 49 



(5) With the special distance-tape, so held that its zero is |6 
from the center, carry the rod out until the rod reading equals 
the reading indicated by the tape. Since in cut the tape is 
raised (n + d), the zero of the rod-tape is always higher than the 
level (unless the rod is held at or below the elevation of the road- 
bed — ^which is only possible on side-hill work), and the reading 
at either slope-stake is necessarily negative. The reading for 
slope-stakes in fill is always positive. 

(6) Record the rod-tape reading as the numerator of a frac- 
tion and the actual distance out (read directly from the other 
side of the distance-tape) as the denominator of the fraction. 

Proof. Fill. Level at (i). Tape is raised (n^—df). When 
rod is held. at C/, the jod reading is +x, which =rfi — {n^—df). 
But the reading on the back side of the distance-tape is also x. 

Fill. Level at (2). Tape is raised {n2—df), i.e., it is lowered 
{df—n-i). When rod is held at C/, the rod reading is -\-x, which 
similarly = rf^—in^—df) = r/2 + {df—n^. Distance-tape as be- 



118 EAILROAD CONSTRUCTION. § 101. 

Cut Level at (i). Tape is raised (n^-hdc).^ When rod is 
held at Co the rod reading is— 2;, which = rci—(ni+(ic), i.e., 
z = (ui+dc) —Tci. The distance-tape Avill read z. 

Side-hill work. It is easily demonstrated that the method, 
when followed literally, may be applied to side-hill work, al- 
though there is considerable chance for confusion and error, 
when, as is usual, ^b and the slope ratio are different for cut and 
for fUl. 

The method appears complicated at first, but it becomes 
mechanical and a time-saver when thoroughly learned. The 
advantages are especially great when the ground is fairly level 
transversely, but decrease when the difference of elevation 
of the center and the slope-stake is more than the rod length. 
By setting the rod-tape * ' at zero," the rod may always be used 
as an ordinary level rod and the regular method adopted, as in 
§ 99. Many engineers who have thoroughly tested these rods 
are enthusiastic in their praise as a time-saver. 

COMPUTATION OF VOLUME 

§ 1 01. Simple approximations. The principles developed in 
§§96 and 97 show that, except where the ground is abnormally 
smooth and level, the earthwork to be excavated has a geometrical 
form whose volume cannot be accurately computed by any simple 
rule. The usual method is to consider that the volume is approx- 
imately measured by the product of the mean of the areas of 
two consecutive sections and the distance between those sec- 
tions. When the ground is so regular that the error of such 
an approximation may be tolerated, or when only a rough approx- 
imation is necessary, such a computation may be accepted 
without correction. In any case, the " volume by averaging 
end areas " is computed as a first approximation and then 
correction is computed if desired. It should, therefore, be 
remembered that this approximate method, which is so common 
that it is often accepted without correction as the true volume, 
is never mathematically correct except under conditions which 
practically never exist. Whether a correction should be com- 
puted depends on the percentage of accuracy required, on the 
irregularity of the ground, and on the differences in the depth 
of adjacent center cuts — or fills. Experience gives the engineer 
such an idea of the probable amount of this correction under 




§102. EAETHWORK. 119 

any given conditions that he may judge when it is necessary to 
compute the correction in order to obtain the true volume with 
any desired degree of accuracy. The methods of computing 
this correction will be given later. 

102. Approximate volume, level sections. When the coun- 
try is very level or when only approximate preliminary results 

r— r46 — -*! 

Fig. 5o. 

are required, it is sometimes assumed that the cross-sections are 
level. The area of the cross-section may be written 

{a^dys-"^ (46) 

in which a, 6 and d are dimensions as indicated by the figure and 
s is the " slope ratio " or the ratio of the horizontal projection 
of the slope to the vertical. A table is very readily formed 
giving the area in square feet of a section of given depth and for 
any given width of roadbed and ratio of side slopes. Usually 
these tables give a number which equals that area times 100 and 
divided by 27, which is the volume in cubic yards of a prism 100 
feet long and with that cross-sectional area. Table XVII is 
such a table. 

The volume may also be readily determined (as illustrated in 
the following example), without the use of such a table; a table 
of squares will facilitate the work. Assuming the cross-sections 
at equal distances ( = I) apart, the total approximate volume for 
any distance will be 

-[Ao+2(Ai+A2+...A„_i)+A„] (47) 

103. Numerical example : level sections. Given the following 
center heights for the same number of consecutive stations 100 
feet apa^ width of roadbed 18 feet; slope 1| to 1. 



120 



RAILKOAD CONSTRUCTION. 



§104. 



The products in the fifth column rhay be obtained very 
readily and with sufficient accuracy by the use of the slide-rule 
described in § 106. The products should be considered as 

{a-\-d)(a-jrd)-. — . In this problem s = l|, — = .6667. To apply 
s s 

the rule to the first case above, place 6667 on scale B over 89 

on scale A, then- opposite 89 on scale B will be found 118.8 on 

scale A. The position of the decimal point will be evident from 

an approximate mental solution of the problem. 



Sta. 


Center 
Height. 


a-\-d 


ia-Vd)^ 


ia+dy-s 


Areas. 


17 
18 
19 
20 
21 
22 


2.9 
4.7 
6.8 
11.7 
4.2 
1.6 


8.9 
10.7 
12.8 
17.7 
10.2 

7.6 


79.21 
114.49 
163.84 
313.29 
104.04 

57.76 


118.81 
171.741 
245.76 i 
469.93 f 
156.06 J 
86.64 


118.81 

{ 343 . 48 

y^_ 1491.52 

^"^ "1939.86 

1312.12 

86.64 



ah 6X18 



= 54 



1752.43X100 
2X27 



2292.43 
10X54= 540 

1752.43 
=3245 cub. yards =approx. vol. 



104. Equivalent sections. When sections are very irregular 
the following method may be used, especially if great accuracy 




Fig. 51. — Equivalent Section. 



is not required. The sections are plotted to scale and then a 
uniform slope linfe is obtained by stretching a thread so that the 
undulations are averaged and an equivalent section is obtained. 
Measure the distances {xi and Xr) from the center. The area 



§105. 



EARTHWORK. 



121 



may then be obtained independent of the center depth as follows : 
Let s = the slope ratio of the side slopes = cot /3 =— . (See Fig. 



51.)" Then the 



1 (Xi + Xr 

Area = — 

. xiXr ah 



{xi-\-Xr) 



2a 



Xr Xr XI Xi ob 

s 2~ s 2~2" 



(48) 



These approximate methods are particularly useful for rapidly 
making up monthly estimates, realizing that the inaccuracies, 
plus and minus, will be wiped out when the final cornputation 
is made by a more accurate method. 

105. Three-level sections. The next method of cross-section- 
ing in the prder of complexity, and therefore in the order of 




Fig. 52. 

accuracy, is the method of three-level sections. The area of 
the section is ^{a-^d){wr-\-w{) , which may be written 

\{a-\-d)w , in which w = Wr+wi. If the volume is com- 

puted by averaging end areas, it will equal 



'-\{si\d')w' -a})^{a^d")w" -ah\. 



(49) 



122 



RAILROAD CONSTRUCTION. 



§i05. 



_o 

U 
ID 

(H 
t-l 

o 
O 



c3 
> 
u 

o 



d 

.2 
+3 

u 

o 
u 

"^ 

-a 
"o 

a 



* 



ID 

a 

_3 
"o 

o 
u 

o. 



o 



> 


* 




■* 


3^ 




+ 


^0 






^-s. 






ki. 








i 




I-l 


CO 


CO 


+ 


+ 


v.^ 








t^ 









.4 u 



I 






+ 



Pi 



1-1 






a 
o 

03. 



+ 



+ 



+ 



+ 



1> 



GO 



CO 



OJ 



o 



CO 

I 



+ 



00 

+ 



CO 









CO 

+ 



"3 



CO 



O 
1—1 
C<J 



o 



-1< 



CO 



CO 



00 






U5 

1—1 



00 
CO 



CO 

1> 



00 









1^ 



00 



CD 



00 



O 



OS 



+ 



+ 



o6 



K) 



+ 



(M OS 

O TTl 

CO Tf 



05 



CO 



00 



k 


M 


fc, 


»-( 


fc< 


iM 


fe, 


1—1 


&. 


V) 


GO 


TtH 


(N 


00 


TfH 


rH 


O 


(N 


d 


CO 


T-l 


rj< 


l-l 


IM 


r— ( 


O 



fe. 


3S 


fc,r^ 


fe, 


CO 


6^ 


o 


fe, 


ct 




00 • 


Ol 




o 




or, 




(M 


• o 




r^ 




00 




o 


(N 


»0 CO 


c 


CO 


■<f 


c< 


K> 


r-t 




rH 1 


CM 




I— 1 










ci 



CO 

T-l 

+ 



T(4 l^ 

OS Tj< 

o 

(N 

II II 

"o 

> f 



, 


n 


M 


(■5 


O 




ft 


in 


ft 
< 






» 






.-I o 

V. -I" 

^ T-l 

03 a 

o o 

Pi % 



3 

a 



03 



0) 
m 

a 
o 

u 
a> 

o 

o 



Ti .2 



0) 

-♦J 

'-I3 

3 
ft 



o 

O 

u 

(4 

3 
-p 

> 

3 
o 





O 


03 
> 


O 


J3 


<N 


■♦^ 


II 


0) 


^ 


a 


4) 
73 




v 


0) 




J3 


Si 


^^ 


+> 


*i 


O 

> 


l-t 
o 


o 


<u 


(^ 


(^ 






-]« ^- 






§ 106. EARTHWORK. . 12^ 

If we divide by 27 to reduce to cubic yards, we have, when 
Z = 100 

Vol (/ . . . .;) =^(a-\-dW-U('b+^(a+d")w''-^ab. 

For the next section 

Vol (., . . /,,) =^-{a+d")w"-^ah+^^a+d'")w'"-^ab. 

For a partial station length compute as usual and multiply 

length in feet 

result by . 

^ 100 

The following example is given to illustrate the method of 

three-level sections. 

In the first column of yards 

210 = ^-{a+d)w=^X7.3XSl.l; 
507, 734, etc., are found similarly; 
595 = 210-61+507-61; 

448 = yVo(507-61+734-61); 
602 =x%V734-61 +392-61); 

449 = 392-61 + 179-61. 

The " F " in the columns of center heights, as well as the 

columns of " right " and " left " are inserted to indicate fill for 

all those points. Cut would be indicated by " C" 

25 
io6. Computation of products. The quantities —(a-\-d)w 

^i 

25 

and — db represent in each case the product of two variable 

terms and a constant. These products are sometimes obtained 
from tables which are calculated for all ordinary ranges of the 
variable terms as arguments. A similar table computed for 

25 

— {d'—d"){w"—'w') will assist similarly m computmg the 

81 

prismoidal correction, see § 114. Prof. Charles L. Crandall, of 
Cornell University, is believed to be the first to prepare such a set 
of tables, which were first published in 1886 " Tables for the 
Computation of Railway and Other Earthwork." Another 
easy method of obtaining these products is by the use of a slide- 
rule. Any slide-rule, from which may be read directly three 
significant figures and from which the fourth may be read by 
estimation, can be utiUzed for this purpose. The Thacher or 



124 EAILROAD CONSTRUCTION. § 107. 

the Stanley cylindrical rules are still more accurate. To illus- 
trate its use, suppose {a-\-d) =28.2, and w = Q2A; then 

25, , 28.2X62.4 

— (a-j-djw = . 

27 1.08 

Set 108 (which, being a constant of frequent use, may be specially 1 
marked) on the sliding scale (B) opposite 282 on the other scale 
{A), and then opposite 624 on scale B will be found 1629 on I 
scale A, the 162 being read directly and the 9 read by estima- 
tion. Although strict rules may be followed for pointing off 
the jBnal result, it only requires a very simple mental calculation 
to know that the result must be 1629 rather than 162.9 or 
16290. For products less than 1000 cubic yards the result 
may be read directly from the scale; for products between 1000 
and 5000 the result may be read directly to the nearest 10 
yards, and the tenths of a division estimated. Between 5000 and 
10,000 yards the result may be read directly to the nearest 20 
yards, and the fraction estimated; but prisms of such volume 
will never be found as simple triangular prisms — at least, an 
assumption that any mass of ground was as regular as this would 
probably involve more error than would occur from faulty esti- 
mation of fractional parts. Facilities for reading as high as 
idjOOO cubic yards would not have been put on the scale except 
for the necessity of finding such products as -1^(9.1X9.5), for 
example. This product would be read off from the same part 
of the rule as |-|^(91X95). In the first case the product (80.0) 
could be read directly to the nearest .2 of a cubic yard, which 
is unnecessarily accurate. In the other case, the product 
(8004) could only be obtained by estimating -^^ of a division. 

The computation for the prismoidal correction (see § 114), 
may be made similarly except that the divisor is 3.24 instead of 

5 5X11 7 

1.08. For example, 1-1(5.5X11.7)=-^ '-. Set the 324 on 

^ ' ^^ 3.24 

scale B (also specially marked like 108) opposite 55 on scale A, 
and proceed as before. 

107. Approximate volume. Irregular sections. In cross- 
sectioning irregular sections, the distance from the center and 
the elevation above " grade " of every " break " in the cross- 
section must be observed. The area of the irregular section 
may be obtained by computing the area of the trapezoids {five, 
in Fig. 53) and subtracting the two external triangles. For Fig. 
53 the area would be 



§ 107. 



EARTHWORK. 



125 



hi + ki. . , ki + d , d+jr , jr-hh 



(yr-Zr) 



, kr + h.r, 



, hi/ b\ hrf h\ 




Fig. 53. 



Expanding this and collecting terms, of which many will 
cancel, we obtain 



4[ 



Area = — xiki-\-yi(d—hi) +Xrkr+yr(jr—hr) 



■{-Zr(d-kr)+ (hi + hr) 



\ ... 



(50) 



An examination of this formula will show a perfect regii- 
larity in its formation which will enable one to write out n 
similar formula for any section, no matter how irregular or how 
many points there are, without any of the preliminary work. 
The formula may be expressed in words as follows : 

A RE A equals one^-half the sum of products obtained as follows: 

the distance to each slope-stake times the height above grade of 
the point next inside the slope-stake; 

the distance to each intermediate point in turn times the height of 
the point just inside minus the height of the point just outside; 

finally, one-half the width of the roadbed times the sum of the 
slope-stake heights. 



i25 RAILROAD CONSTRUCTION. § 108. 

If one of the sides is perfectly regular from center to slope- 
stake, it is easy to show that the rule holds literally good. The 
" point next inside the slope-stake " in this case is the center; 
the intermediate terms for that side vanish. The last term 
must always be used. The rule holds good for three-level sec- 
tions, in which case there are three terms, which may be reduced 
to two. Since these two terms are both variable quantities for 
each cross-section, the special method, given in § 105, in which 
one term {^ah) is a constant for all sections, is preferable for 
three-level sections. In the general method, each intermediate 
" break " adds another term. 

1 08. Volume of an irregular prismcid. This is obtained by 
computing first the approximate volume by " averaging end 
areas " or by multiplying the length by the half sum of the end 
areas, as computed from Eq. (50). In other words, the Approx. 

volume = X— (area' + area")* But since each area equals 

27 2 

one-half the sum of products of width times height (see Eq. (50)) 

we may say that 

25 
Approx. volume = — (summation of width times height) . (51) 

^1 

the terms of width times height being like those found within 

the bracket of Eq. (50). 

As before, for partial station lengths, multiply the result by 

(length in feet -^100). There will be no constant subtractive 

25 
term, — ab, as in § 105. 

27 

109. Numerical example; approximate volume; irregular 

sections. Assume the earthwork notes as given below where 

the roadbed is 18 feet wide in cut and the slope is 1| to 1. Note 

that the stations read up the page and that when the surveyor 

is looking ahead along the line the several combinations of heights 

and distances out have approximately the same relative position 

on the notebook as they have on the ground. For example, 

, . . 8.9c 
beginning at the bottom line (Sta. 16), the combination — — 

means that the extreme left-hand point of that section (the 
" slope-stake ") is 22.4 feet horizontally from the center and that 
it is 8.9 feet above the required roadbed. The cut (c) would he 
8.9 feet to reach the roadbed, but of course the actual cutting is 



§109. 



EARTHWORK. 



127 



zero at the slope stake. The next point ig 12.0 feet horizontally 
from the center and 7.6 feet above the roadl)ed. The cut at 
the center is 6.8 feet. The combinations of dimensions on the 
right-hand side are to be interpreted similarly. 



Sta. 



19 
18 
17 

+ 42 
16 



( cut 

Centers or 
fill. 



0.6c 
2.3c 
7.6c 
10.2c 
6.8c 



Left. 



22.4 



12.0 



3.6c 
14.4 






0.1c 

4.2 


0.4c 
9.6 


4.2c 
15.3 


6.8c 

8.4 


3.2c 
5.2 


■ 


1.2c 

10.8 


8.2c 
21.3 


10.2c 
17.4 


8.0c 
6.1 




4.2c 
15.3 


12.2c 
27.3 




12 6c 

8.2 


6.2c 
7.5 


8.4c 
21.6 


8.9c 




7.6c 


3.2c 


2.6c 



Right. 



4.1 



12.9 



The numerical computation is greatly facilitated by a sys- 
tematic form as given below. For Sta. 16, the first term ia 
"the distance to the left slope stake" (22.4) times ''the height 
above grade of the point next inside" (the height being 7.6), 
and we place this pair of figures in the columns of "width" 
and "height." The "distance to the point next inside" is 
12.0 and the "height of the point just inside (6.8) minus the 

height of the point just outside" (8.9) equals (—2.1) and these 

25 

are the next pair of widths and heights. Taking ^ of the 

product of each pair of numbers we have the numbers in the 

first column of "yards." The sum of all these numbers in the 

42 
first and second groups multiplied by y^ (that section being 

only 42 feet long) equals 378 cubic yards, the volume by averag- 
ing end areas. The determination of center heights and total 
widths and the application of Eq. (54), to obtain the approxi- 
mate prismoidal correction (see § 114), is self-evident. 

no. Prismoidal correction. The foregoing methods of. cal- 
culation have been called approximate, although under many 



128 KAILROAD CONSTRUCTION. § 110: 

■ 
VdLtJME OF IRKEGULAR PRISMOID, WITH APPROXIMATE PRtSMOIDAL 

CORRECTION. 


Sta. 


W'th 


H'ght 


Yards. 


Cen. 
Height. 


Total 
width 


d'-d" 


iv" - w' 


Approx. 
pris.corr. 


16 


22.4 

12.0 

12.9 

4.1 

9.0 


7.6 

-2.1 

3.2 

4.2 

11.5 


158 

-23 

40' 

16 

96 




+ 6.8 


35.3 










27.3 

8.2 


12.6 

-2.0 

6.2 . 

1.8 

20.6 


319 

-15 

124 

13 

172 


378 


+ 10.2 


48.9 


-3.4 


+ 13.6 


-11 


+ 42 


21.6 
7.5 
9.0 


(-6) 


17 


21.3 
17.4 

6.1 
15.3 

9.0 


10.2 

-0.2 

-2.6 

7.6 

12.4 


201 

- 3 

-14 

107 

103 


584 


+ 7.6 


36.6 


+ 2.6 


-12.3 


-10 
(-6) 


18 


15.3 
8.4 
5.2 

10.8 
9.0 


6.8 

-1.0 

-4.5 

2.3 

5.4 


95 

- 7 

-22 

23 

45 


528 


+ 2.3 


26.1 


+ 5.3 


-io.5 


-17 
(-17) 


19 


14.4 
9.6 
4.2 
9 


0.6 
0.1 
0.2 
4.0 


8 

1 

1 

33 


177 


+ 0.6 


24.0 


+ 1.7 


-2.1 


-1 

(-1) 



Approx. volume =»1667 
Approx. pris; corr. = — 30 



-30 



Corrected volume = 1637 cubic yards 

conditions such results are considered to be sufficiently accurate 
to serve as final. In any case the approximate result is first 
ebmputed and then the '' prismoidal correction " is computed 
if necessary. The mathematical necessity for a correction may 
be at once appreciated from the consideration that the volume 
of a prismoid having dissimilar and unequal ends is NOT equal 
to the length times the average of the end areas but is usually 
somewhat less. In an extreme case the correction is one-third 
of the approximate volume, or one-half of the true volume. The 
amount of the prismoidal correction for a triangular prism will 
be first determined and from tha;t the correction for any kind of 
prism may be deduced. 

Let Fig. 54 represent a triangular prismoid. The two tri- 
angles forming the ends lie in parallel planes, but since the angles 
of one triangle are not equal to the corresponding angles of the 



I§ 110. 



EARTHWORK. 



129 



other triangle, at least two of the surfaces must be iuarped. If 
a section, parallel to the bases, is made at any point at a dis- 




-bj 



-^— 



Fig. 54. 



tance x from one end, the area of the section will evidently be 



■ci-x — ibxhix — 



X~ 2^X"'X~ 2 



ik 



+{h-h) 



X 



h + ihi—hi)- 



The volume of a section of infinitesimal length will be Axdx, 
and the total volume of the prismoid will be * 



'[' 



2 I hihix-\-Q)2—hiL)h-^:^-\-hi{}h— 111)21 



+ {h2-h,){h,-h)^^^ 



= i|MiZ+[(62-&i)/M+&i(A2-/ii)]|+(&2-6:)(/i2-7ii)|| 

= ;^]Ai+4A^+A2], (52) 

6 



* Students unfamiliar with the Integral Calculus may take for granted the 

fundamental formulae that i dx=x, that I xdx = ^x^, and that | xHx = \x^; 

also that in integrating between the limits of I and (zero), the value 
of the integral may be found by simply substituting 2 for x after 
integration. 



130 RAILROAD CONSTRUCTION. § 111, 

in which Ai, A2, and Am are the areas respectively of the twc 
bases and of the middle section. Note that Am is not the mean 
of Ai and A2, although it does not necessarily differ very greatlji 
from it. 

The above proof is absolutely independent of the values, ah^ 
solute or relative, of 61, 62, K, or ^2- For example, /ij '"^^Y ^^ 
zero and the second base reduces to a line and the prisnioid be 
comes wedge-shaped; or 62 and h2 may both vanish, the second 
base becoming a point and the prismoid reduces to a pyramid. 
Since every prismoid (as defined in § 97) may be reduced to a 
combination of triangular prismoids, wedges, and pyramids, and 
since the formula is true for any one of them individually, it is 
true for all collectively; therefore it may be stated that * 

The volume of a prismoid equals one sixth of the perpendicular 
distance between the bases multiplied by the sum of the areas of 
the two bases plus four times the area of the middle section. 

While it is always possible to compute the volume of any 
prismoid by the above method, it becomes an extremely compli 
eated and tedious operation to compute the true value of the 
middle section if the end sections are complicated in form. It 
therefore becomes a simpler operation to compute volumes by! 
approximate formulae and apply, if necessary, a correction. 
The most common methods are as follows : 

III. Correction for triangular prismoid. The volume of the 
triangular prismoid (Fig. 54), computed by averaging end areas, is 

I 

~[2bihi-\-^b2h2]. Subtracting this from the true volume (as 

given in the equation above Eq. 52), we obtain the correction 

-[{h-h)(h2-h)] (53) 

This shows that if either the h's or 6's are equal, the correc- 
tion vanishes; it also shows that if the bases are roughly similar 
and b varies roughly with h (which usually occurs, as will be 
seen later), the correction will be negative, which means that the 
method of averaging end areas usually gives too large results. 

If the " base " at one end vanishes to a point, making a trian- 

* The student should note that the derivation of equation (52) does not 
complete the proof, but that the statements in the following paragraph 
are logically necessary for a general proof, 



§ 112. EARTHWORK.' 131 

! ■ ■ 

gular pyramid, then 61 and hi each equal zero and the correction 

reduces to 

I Ibihz 

But the volume of a triangular prismoid is one-third of the alti- 
tude times the area of the base or ^^(162/^2) = ^^62/^2- The approx- 
imate volume, by averaging end areas, applying the rule strictly, 
is |Z(^&2^2+0) =1^62/^2. The correction is therefore one-third of 
the approximate volume, or one-half of the true volume, in this 
extreme case. Therefore, when computing the volume of ter- 
minal pyramids and wedges (see § 89 and Fig. 43), by the method 
of averaging end areas, it must be remembered that, although 
the gross volume is comparatively small, the prismoidal correc- 
tion is relatively very large. 

112. Correction for level sections. Absolutely level sections 
are practically unknown, and the error involved in assuming any 
given sections as truly level will ordinarily be greater than the 
computed correction. If greater accuracy is required, more 
points should, be obtained in the cross-sectioning, which wiQ 



generally show that the sections are not truly level. But it 
may be easily computed that the correction equals 

I h 
IZ a 

The squares of the differences of center depth of consecutive 
sections are always positive, regardless of whether the differences 
are positive or negative. Therefore the correction is always 
negative, showing that the method of averaging end areas, when 
the sections are level, always gives too large results. 

113. Prismoidal correction for " equivalent sections." It is 
a simple although tedious problem in mathematics to compute 
algebraically the true and approximate volumes of a prismoid 
when the areas are determined on the basis of " equivalent 
sections," § 104, and from thence to derive a formula for the 
prismoidal correction, but it is generally true that the errors 
due to such an approximate method of getting the area are so 
great that it is a needless refinement to compute the correction. 

114. Prismoidal correction for three-level sections. The 
prismoidal correction may be obtained by applying Eq. 53 to 
each side in turn. For the left side we have 



132 KAILROAP CONSTRUCTION. § 115V: 

—[ia+d')-ia+d")](wi"-w/), which equals 

For the right side we have, similarly, 
The total correction therefore equals 

-^{d'-d")[{wr+wn-iw/-hw/)] 

==—{d'-d"){w"-w'). 

Reduced to cubic yards, and with I = 100, 

Vvh.CovT.=^{d'-d"){w"-w'). . . . (54) 

Applying this formula to the numerical problem worked out in 
§ 105, the several values of {d' — d") and w" —w') are computed 
p,s given in the first two columns under Prismoidal Correction. 
Then, for example, 

-20 = |f(d'-d")K'-w;')=ff(2-6-8.1)(42.8— 31.1) 
■=ff(-5.5)( + 11.7). 

For the next line, -3=y%o_[2 5(_2.6)(_]_8.7)], and similarly 
for the rest. For this typical case, the correction is over 2 % of the 
volume and is, as usual, negative, or in other words, the approx- 
imate method, if used without correction, allows a contractor 
jn this case 2% too much. 

115. Prismoidal correction; irregular sections. For reasons 
given in the next article, the correction ^s computed as if th(^ 
sections were " three-level " sections. This method was used 
ill the numerical problem worked, out in § 109. Instead of con- 
sidering the heights and widths of the separate triangles, the 
center height and total width for each section is recorded in two 
columns and the differences {d' — d") and {w" —w') are computed. 
,(-3.4)X( + 13.6)-j-3.24=-14, which would be the correction 
for a section 100 feet long. For 42 feet the correction is 42% 
of —14 or —6. Note that the total prismoidal correction for 
this stretch of 300 feet is negative, as is usual, and that it is a 
Jijbtle less than 2%, about the same as the numerical problem of 
§105. 



I 116. EARTHWORK. 133 

ii6. Magnitude of the probable error of this method. In 
Drevious editions of this work, methods were given for com- 
puting the mathematically exact volume of a prismoid whose 
snds coincide with the " irregular sections " as measured, and 
whose upper surfaces are assumed to coincide with the actual 
surface of the ground. As in the previous methods, the "ap- 
proximate volume" is computed by averaging end areas and 
then a correction is applied. If the end sections have the samiB 
number of intermediate points on each side, and if it can be as- 
sumed that the corresponding lines in each section are connected 
by plane or warped surfaces, which coincide with the surface of 
the ground, then the mathematically exact or *Hrue" correc- 
tion may be obtained by dividing the volume into elementary 
triangular prismoids, finding the correction for each and adding 
the results. Although such a method appears very complicated, 
it is readily possible to develop a law by means of which the 
true prismoidal correction may be written out (similarly to 
writing out the formula for the area, Eq. (50)) without any 
preliminary calculation. Such a law has a mathematical 
fascination, but it should be remembered that when the groiuid 
surface is so broken up that the cross-sections are 'irregular" 
it is in general correspondingly rough and irregular betweeh 
the cross-sections, especially when those sections are 100 feet 
apart. It is also true that the cross-sections do not usually 
have the same number of intermediate points on corresponding 
sides of the center. In such a case, unless the actual form of 
the ground between the cross -sections is observed and measured, 
the exact method cannot be used. An extra point in one cross- 
section implies an extra ridge (or hollow) which " runs out " 
or disappears by the time the adjoining section is reached. 
Theoretically a cross-section should be taken at the point where 
such a ridge or hollow runs out. In general this point will not 
be at an even 100-foot, station. The attempt to compute the 
exact prismoidal correction usually gives merely a false appear- 
ance of extreme accuracy to the work which is not justified 
by the results. It should not be forgotten that it is readily 
possible to spend an amount of time on the surveying and 
computing which is worth more than the few cubic yards of 
earth which represents the additional accuracy of the more 
precise method. The accuracy of the office computation should 
be kept proportionate to the accuracy of the cross-sectioning 



134 



RAILROAD CONSTRUCTION. 



§ 117.; 



in the field. The discussion of the magnitude of the prismoida] 
correction in §§ 110-115 shows that it is small except when th§ 
two ends of the prismoid are very dissimilar. The dissimilarity^ 
between the two eiids of a prismoid vvould be substantially th^ 
same whether the ends were actually "irregular'* or had "three 
level" sections, which for each end had the same slope stakes 
and center heights as the irregular sections. Experience proves 
that the approximate prismoidal correction, computed by 
considering the ground as three-level, is so nearly equal to the 
true prismoidal correction that the difference is perhaps no. 
greater than the probable difference between the true volume 
of earth and the volume of the geometrical prismoid which isi 
assumed to represent that volume. The experienced surveyor 
will take his cross-sections at such places and so close together 
that the warped surfaces joining the sections will lie very nearly 
in the surface or at least will so average the errors that they 
will substantially neutralize each other. 

117. Numerical illustration of the accuracy of the approxi- 
mate rule. The " true " prismoidal correction for the numerical 
case given in § 109 was computed by the method outlined above, 
and on the basis of certain figures as to the vanishing of the 
ridges and valleys found in one section and not found in the 
adjacent sections. The various quantities for the volumes 
between the cross-sections have been tabulated as shown. 





1 


2 


3 


4 


5 


6 


7 




J M 




?j 


i/J-ST) 


(N 


t-< U -i 


n 


Sections. 


Approx. vo 

by averagin 

end areas. 


True pris- 
moidal 
correction. 


> 

(0 

3 

M 


Approx. pri 

corr. on bas 

of three-lev 

ground. 


1— J 

3 


Approx. vo 

computed 

from cente 

and side 

heights onli 


" 
b 1 

6 


16 16 + 42 


378 


- 5 


373 


- 6 


-1 


396 


-23 


16+42.. 17 


584 


- 3 


581 


- 6" 


-3 


577 


+ 4 


17 18 


528 


-16 


512 


-17 


-1 


463 


+ 49 


IS 19 


177 


- 3 


174 


- 1 


+2 


147 


+ 27 




1667 


-27 


1640 


-30 


-3 


1583 


+ 57 



There has also been shown in the last two columns the error 

involved if the "intermediate points" had been ignored in 

the cross-sectioning. From the tabular form we may learn that 

1. The differences between the "true" and approximate 



118. 



EAETHWORK. 



135 



corrections Is so small that it Is probably swallowed up by errors 
resulting from inaccurate cross-sectioning. 

2. The error which would have been involved in ignoring 
the intermediate points is so very large in comparison with 
the other corresponding errors that (although it proves nothing 
;absolLitely definite, being an individual case) the probabilities 
of the relative error from these sources are clearly indicated. 

ii8. Cross-sectioning irregular sections. The slope stake 
should preferably be determined first, and then the "breaks" 
betwesn the slope stake and the center. When, as is usual, 
the ground is not even between the cross-section just taken 
and the section at the next 100-foot station, a point should be 
selected for a cross-section such that the lines to the previous 
section should coincide with the actual surface of the ground as 
closely as the accuracy of the work demands. § 125 gives 
a numerical illustration of the magnitude of some of these 
Trors. Although it is possible for a skillful surveyor to so 
choose his cross-sections in rough and irregular ground that 
;he positive and negative errors will nearly balance, it requires 
exceptional skill. Frequently the work may be simplified by 
computing separately the volume of a mound or pit, the 
existence of which has been ignored in the regular cross- 
sectioning, 

119. Side-hill work. When the natural slope cuts the roadbed 
there is a necessity for both cut and fill at the same cross-section. 




Fig.' 55. 



When this occurs the cross-sections of both cut and fill are often 
so nearly triangular that they may be considered as such without 
great error, and the volumes may be computed separately as 
triangular prismoids without adopting the more elaborate form 



136 



EAILROAD CONSTRUCTION. 



§ 119, 



of computation so necessary for complicated irregular sections. 
When the ground is too irregular for this the best plan is tc 
follow the uniform system. In computing the cut, as in Fig. 55^ 
the left side would be as usual; there would be a small center 
cut and an ordinate of zero at a short distance to the right of the 
center. Then, ignoring the fill, and applying Eq. 56 strictly, 
we have two terms for the left side, one for the right, and the 
term involving f6, which will be \hhi in this case, since hr=0, 
and the equation becomes 

Area(cut) = 2[^ih+yi(d-hi) -j-x^d+^bhi]. 

The area for fill may also be computed by a strict application 
of Eq. 50, but for Fig. 56 all distances for the left side are zero 
and the elevation for the first point out is zero, d also must be 



f^7777m^ 




Fig. 56. 



considered as zero. Following the rule, § 107, literally, the equa- 
tion becomes 

Area(FiU) ='^Xrkr^yr{o—li-;) -^Zr{o—hr) ^-i-&(o + /lr)], 
which reduces to 

{l>^ote that Xt, hr, etc., have different significations and values 
in this and in the preceding paragraphs.) The ''terminal 
pyramids" illustrated in Fig. 43 are instances of side-hill work 
for very short distances. Since side-hill work always implies 
both cut and fill at the same cross-section, whenever either the 
cut or fill disappears and the earthwork becomes wholly cut or 
wholly fill, that point marks the end of the "side-hill work," 
and a cross-section should be taken at this point. 



§ 120. EARTHWORK. 137 

120. Borrow-pits. The cross-sections of borrow-pits will vary 
not only on account of the undulations of the surface of the 




Fig. 57. 

ground, but also on the sides, according to whether they are 
made by widening a convenient cut (as illustrated in Fig. 57) 
or simply by digging a pit. The sides should always be prop- 
▼rly sloped and the cutting made cleanly, so as to avoid un- 
sightly roughness. If the slope ratio on the right-hand side 
(Fig. 57) is 5, the area of the triangle is ^sm^. The area of the 
section is ^[ug-{-(g-\-h)v-{-{h-\-j)x-\-{j-\-k)y-\-{k-\-'m)z — snP\. If 
all the horizontal measurements were referred to one side as 
an origin, a formula similar to Eq. 50 could readily be devel- 
oped, but little or no advantage would be gained on account of 
any simplicity of computation. Since the exact yolume of the 
earth borrowed is frequently necessary, the prismoidal correc- 
tion should be computed; and since such a section as Fig. 57 
does not even approximate to a three-level section, the method 
suggested in § 115 cannot be employed. It will then be neces- 
sary to employ the more exact method of dividing the volume 
into triangular prismoids and taking the summation of their 
corrections, found according to the general method of § 111. 

121. Correction for curvature. The volume of a solid, gen- 
erated by revolving a plane area about an axis lying in the 
plane but outside of the area, equals the product of the given 
area times the length of the path of the center of gravity of the 
area. If the centers of gravity of all cross-sections lie in the 
center of the road, where the length of the road is measured, 
there is absolutely no necessary correction for curvature. If all 
the pross-sections in any given length Wjsre exactly the same and 
therefore had the same eccentricity, the correction for curvature 
would be very readily computed according to the above prin-r 
ciple. But when both the areas and the eccentricities vary 
from point to point, as is generally thp case, a theoretically exact 



138 EAILROAD CONSTRUCTION. § 121. 

solution is quite complex, both in its derivation and application. 
Suppose, for simplicity, a curved section of the road, of uniform; 
cross-sections and with the center of gravity of every cross- 
section at the same distance e from the center line of the road 
The length of the path of the center of gravity will be to the 
length of the center line as R±e:R. Therefore we have 

True vol.: nominal vol. :: R±e ', R. .'. True vol.=lA for 

K 

a volume of uniform area and eccentricity. For any other area 

R±e' 
and eccentricity we have, similarly, True vol.' = I A' — ^— . This 

shows that the effect of curvature is the same as increasing (or 
diminishing) the area by a quantity depending on the area and 
eccentricity, the increased (or diminished) area being found by 

multiplying the actual area by the ratio „ . This being 

R 

independent of the value of Z, it is true for infinitesimal lengths. 

If the eccentricity is assumed to vary uniformly between two 

sections, the equivalent area of a cross-section located midway 

between the two end cross-sections would be Am— ^ » 

Ji 

Therefore the volume of a solid which, when straight, would be 
—(A' + 4:Am+A"), would then become 

rrwewZ.=g^rA'(i2±eO+4A^(i2±^^WA''(i2±o1. 

Subtracting the nominal volume (the true volume when the 
prismoid is straight), the 



Correc<ion = ±g^r(A' + 2A„»)e'-h(2A^+A'0e" L 



(55) 



Another demonstration of the same result is given by Prof. 
C. L. Crandall in his "Tables for the Computation of Railway 
and other Earthwork," in which is obtained by calculus methods 
the summation of elementary volumes having variable areas 
with variable eccentricities. The exact application of Eq. 55 
requires that ^^ be known, which requires laborious computa- 



§ 122. EARTHWORK. 139 

tions, but no error worth considering is involved if the equation 
is written approximately 

Curv.corr.=^(A'e' + A"e"), .... (56) 

which is the equation generally used. The approximation con- 
sists in assuming that the difference between A' and Am equals 
the difference between A m and A" but with opposite sign. The 
error due to the approximation is always utterly insignificant. 
122. Eccentricity of the center of gravity. The determination 
of the true positions of the centers of gravity of a long series of 
irregular cross-sections would be a very laborious operation, 
but fortunately it is generally sufficiently accurate to consider 
the cross-sections as three-level ground, or, for side-bill work, to 




\ 1 / 



^;^ 



Fig. 58. 



be triangular, for the purpose of this correction. The. eccentricity 
of the cross-section of Fig. 58 (including the grade triangle) may 
be written 

(a + d)xiXi (a-\-d)XrXr 

^_ 2 ^~ 2 T IXi'-X,.' 1. _ . fnj. 

(a + d)xi (a + d)xr ~Z xi + xr 'a^""' '^^''- • ^^ 



The side toward xi being considered positive in the above 
demonstration, if Xr>xi, e would be negative, i.e., the center 
of gravity would be on the right side. Therefore, for three-level 



140 RAILROAD CONSTRUCTION. § 122. 

ground, the correction for curvature (see Eq. 56) may be written 

Correction = J^{A'{xi' -x/) + A"(a;/' -x/')]. 
bit 

Since the approximate volume of the prismoid is 

in which V arid V" represent the number of cubic yards corre- 
sponding to the area at eacli station, we may write 

Corr. in cub. yds. = ~JV'(xi'-x/) + 7"(a:/' -x/')]. (58) 

It should be noted that the value of e, derived in Eq. 57, is 
the eccentricity of the whole area including the triangle under 
the roadbed. The eccentricity of the true area is greater than 
this and equals 

true alrea + hab 

e X — 7 — = ei. 

true area 

The required quantity {A'e' of Eq. 56) equals true areaXCi 
which equals (true area-\-^ab)Xe. Since the value of e is very 
simple, while the value of €{ would, in general, be a complex 
quantity, it is easier to use the simple value of Eq. 57 and add 
^ab to the area. Therefore, in the case of three-level ground 
the subtractive. term ^ab (§ 105) should not be subtracted in 
computing this correction. For irregular ground, when com- 
puted by the method given in §§ 107 and 108, which does not 
involve the grade triangle, a term ^ab must be added at every 
station when computing the quantities V' and V" for Eq. 58. 

It should be noted that the factor l-^SR, which is constant 
for the length of the curve, may be computed with all necessary 
accuracy and without resorting to tables by remembering that 

„ 5730 



degree of curve* 



Since it is useless to attempt the computation of railroad 
earthwork closer than the nearest cubic yard, it will frequently 



§122. 



EARTHWORK. 



141 



be possible to write out all curvature corrections by a simple 

mental process upon a mere inspection of the computation sheet. 

Eq. 58 shows that the correction for each station is of the form 

V(xj—x •) 

Ln ' 2-^ ^*® generally a large quantity — for a 6° curve 

it is 2865. {xi—x^ ij generally small. It may frequently be 
seen by inspection that the product Y{xi~Xy) is roughly twice 
or three times 37?, or perhaps less than half of ZR, so that the 
corrective term for that station may be written 2, 3, or cubic 
yards, the fraction being disregarded. For much larger absolute 
amounts the correction must be computed with a correspondingly 
closer percentage of accuracy. 

The algebraic sign of the curvature correction is best deter- 
mined by noting that the center of gravity of the cross-section is 
on the right or left side of the center according as Xr is greater 
Dr less than xi, and that the correction is positive if the center of 
gravity is on the outside of the curve, and negative if on the 
inside. 

It is frequently found that xi is uniformly greater (or uni- 
formly less) than Xr throughout the length of the curve. Then 
the curvature correction for each station is uniformly positive or 
negative. But in irregular ground the center of gravity is apt 




Fig. 59. 



to be irregularly on the outside or on the inside of the curve, 
and the curvature correction will be correspondingly positive or 
negative. If the curve is to the right, the correction will be 
positive or negative according as (xi—Xr) is positive or negative;' 
if the curve is to the left, the correction will be positive or nega- 



142 RAILEOAD CONSTRUCTION. § 123. 

tive according as {xr—xi) is positive or negative. Therefore 
when computing curves to the right use the form {xi—Xr) in 
Eqs. 58 and 60; when computing curves to the left use the form 
{xr—xi) in these equations; the algebraic sign of the correction 
will then be strictly in accordance with the results thus obtained. ■ 
123. Center of gravity of side-hill sections. In computing the 
correction for side-hill work the cross-section would be treated 
as triangular unless the error involved would evidently be too^ 
great to be disregarded. The center of gravity of the triangle 
lies on the line joining the vertex with the middle of the base 
and at \ of the length of this line from the base. It is therefore 
equal to the distance from the center to the foot of this line plus 
^ of its horizontal projection. Therefore 



Vh 1/6 \~| If 

__h Xr XI h Xr 

~4~2 "^S" 12' 6 



^'-U"2i2+^>; 



h XI Xr 

=— ■^ + (xi—Xr) \. ' (59) 

By the same process as that used in § 122 the correction equation 
may be written 

Corr.lncuh.yd^.^^[v'(j^(ri'-Xr'\) + V"(j + ixi"-Xr"))']. (60) 

It should be noted that since the grade triangle is not used in 
this computation the volume of the grade prism is not involved 
in computing the quantities V and V". 

The eccentricities of cross-sections in side-hill work are never 
zero, and are frequently quite large. The total volume is gen- 
erally quite small. It follows that the correction for curvature 
is generally a vastly larger proportion of the total volume than 
in ordinary three-level or irregular sections. 

If the triangle is wholly to one side of the center, Eq. 59 can 
'still be used. For example, to compute the eccentricity of the 
triangle of fill, Fig. 69, denote the two distances to the slope- 



§ 124. EARTHWORK. 143 



stakes by yr and —yi (note the minus sign). Applying Eq. 5^ 

2 
order to make the notation consistent) we obtain 



literally (noting that — must here be considered as negative in 



1 



which reduces to 

1 






b 



(61) 



As the algebraic • signs tend to create confusion in these 
formula, it is more simple to remember that for a triangle 
lying on both sides of the center e is always numerically equal 

to--i — + (xi'^Xr) Land for a triangle entirely on one side, e is 

1 rb 
numerically equal to — k + the numerical sum of the two dis- 
tances out]. The algebraic sign of e is readily determinable as 

in § 122. 

124. Example of curvature correction. Assume that the fill in 

§ 105 occurred on a 6° curve to the right. — — = . The 

3R 2865 

quantities 210, 507, etc., represent the quantities V, V", etc, 

since they include in each case the 61 cubie yards due to the 

grade prism. Then 



F(x;^a;r-) _ 210(22.9-8.2) ^ 3101.7 _ 
ZR 2865 2865 ~ "*" 



The sign is plus, since the center of gravity of the cross-sec- 
tion is on the left side of the center and the road curves to the 
right, thus making the true volume larger. For Sta, 18 the 
correction, computed similarly, is +3, and the correction for 
the whole section is 1+3=4. For Sta. 18 + 40 the correction 
is computed as 6 yards. Thereforej for the 40 feet, the correc- 
tion is tA(3 + 6) =3.6, which is called 4. Computing the others 
similarly we obtain a total correction of + 16 cubic yards. 



144 RAILROAD CONSTRUCTION. § 125. 

I 
125. Accuracy of earthwork computations. The preceding ! 

methods give the 'precise volume (except where approximations t 
are distinctly admitted) of the prismoids which are supposed to' ! 
represent the volume of the earthwork. To appreciate the; 
accuracy necessary in cross-sectioning to obtain a given accuracy 
in volume, consider that a fifteen-foot length of the cross-section, 
which is assumed to be straight, really sags 0.1 foot, so that the 
cross-section is in error by a triangle 15 feet wide and 0.1 foot 
high. This sag 0.1 foot high would hardly be detected by the 
eye, but in a length of 100 feet in each direction it would make 
an error of volume of 1.4 cubic yards in each of the two pris- ! 
molds, assuming that the sections at the other ends were perfect. 
If the cross-sections at both ends of a prismoid were in error by 
this same amount, the volume of that prismoid would be in error 
by 2.8 cubic yards if the errors of area were both plus or both 
minus. If one were plus and one minus, the errors would 
neutralize each other, and it is the compensating character of 
these errors which permits any confidence in the results as 
obtained by the usual methods of cross-sectioning. It demon- 
strates the utter futility of attempting any closer accuracy than 
the nearest. cubic yard. It will thus be seen that if an error 
really exists at any cross-section it involves the prismoids on 
both sides of the section, even though all the other cross-sections 
are perfect. As a further illustration, suppose that cross-sec- 
tions were taken by the three-level method (§ 105), and that a 
cross-section, assumed as uniform from center to side, sags 0.4 
foot in a width of 20 feet. Assume an equal error (of same 
sign) at the other end of a 100-foot section. The error of 
volume for that one prismoid is 38 cubic yards, 

The computations further assume that the warped surface, 
passing through the end sections, coincides with the surface of 
the ground. Suppose that the cross -sectioning had been done 
with mathematical perfection; and, to assume a simple case, 
suppose a sag of 0.5 foot between the sections, which causes an 
error equal to the volume of a pyramid having a base of 20 feet 
(in each cross-section) times 100 feet (between the cross-sec- 
tions) and a height of 0.5 foot. The volume of this pyramid is 
K20X100)X0.5=333 cub. ft. = 12 cub. yds. And yet this sag 
or hump of 6 inches would generally be utterly unnoticed, or 
at least disregarded. 

When the ground is very rough and broken it is sometimes 



§126. 



EARTHWORK. 



145 



practically Impossible, even with frequent cross-sections, to 
locate warped surfaces which will closely coincide with all the 
sudden irregularities of the ground. In such cases the compu- 
tations are necessarily more or less approximate and dependence 
must be placed on the compensating character of the errors. 

126. Approximate computations from profiles. When a 
"paper location" has been laid out on a topographical map 
having contours, it is possible to compute approximately the 
amount of earthwork required by some very simple and rapid 
calculations. A profile may be readily drawn by noting the 
intersections of the proposed center line with the various con- 
tours and plotting the surface line on profile paper. Drawing 
the grade-line on the profile, the depth of cut or fill may be 
scaled off at any point. When it is only, desired to obtain 




Fig. 60. 



very quickly an approximate estimate of the amount of earth- 
work required on a suggested line, it may be done by the method 
described in § 103, or by the use of Table XVII. But the 
assumption that the surface of the ground at each cross-section 
is level invariably has the effect that the estimated volumes 
are not as large as those actually required. The difference 
between the ''level section" hkms and the actual slope section 
hknq equals the difference between the triangles mon and oqs, 
and this difference equals the shaded area mpn. The excess 
volume is proportional to the area of the triangle mpn. This 
area may be expressed by the formula, 

A OM7 , J . osoSin^'a sIn/?cos^ 

Area mpn=2{ib+d cot fiy—-^-Ji-^. 



146 KAILROAD CONSTRUCTION. § 126. 

The percentage of this excess area to the nominal area hkms 
therefore depends on the dimensions h and d and the angles a 
and /3. A solution of this equation for ninety different com- 
binations of various numerical values for these four variables 
is included in Table XVII for the purpose of making cor- 
rections. A study of this correction table points conclusively 
to the following laws, a thorough understanding of which will 
enable an engineer to appreciate the degree of accuracy which 
is attainable by this approximate method: 

(a) Increasing the ividth of the roadbed (6), the other three 
factors remaining constant, increases the percentage of error, 
but the increase is comparatively small. 

(6) Increasing the dej)th of cut or fill (d), decreases the per- 
centage of error, but the decrease is almost insignificant. 

(c) Increasing the angle of the side slopes (P) decreases the 
percentage of error, the decrease being very considerable. 

(d) Increasing the angle of the slope of the ground (a), 
■increases the percentage of error, the percentage rapidly in- 
creasing to infinity as the value of a approaches that of /?. 
This is another method of stating the fact that a must always 
be less than /? and, practically, must be considerably less, so 
that the slope stake shall be within a reasonable distance from 
the center. 

Since the above value for the corrective area is a function of 
the angle a, which is usually variable and whose value is fre- 
quently known only approximately, it is useless to attempt 
to apply the correction with great precision, and the following 
rules will usually be found amply accurate, considering the 
probable lack of precision in the data used. 

1. For embankments or cuts, having a slope of 1.5:1, and 
with a surface slope of 5° (nearly 9%) the excess of true area 
over nominal area is about 2%. There is only a slight varia- 
tion from this value for all ordinary depths (d) and widths (6) 
of roadbed. Therefore the nominal volume would be about 2% 
too small. On the other hand, the effect of the prismoidal 
correction is such that, even with truly level sections, the 
nominal volume is too large. See §§ 103 and 112. The amount 
of the prismoidal correction depends on the differences between 
successive center depths. In the very ordinary numerical case 
given in § 103, the correction was nearly 3%, which more 
than neutralizes the error due to surface slope. Therefore in 



§ 126. EARTHWORK. 147 

many cases on slightly sloping ground the error due to the 
surface slope will so nearly neutralize the prismoidal correc- 
tion that the quantities taken directly from the tables (without 
correction for either cause) will equal the true volume with as 
close an approach to accuracy as the precision of the surveying 
will permit. 

2. For a cut with a slope of 1:1, and with a surface slope of 
5° the error is about 1%. This wUl be neutralized by still 
smaller prismoidal corrections. Therefore, for surface slopes 
of 5° or less, no allowance should be made for this error unless 
the prismoidal correction is also considered. 

3. When the surface slope is 10° (nearly 18%) the error for 
a 1.5:1 slope is from 7% to 10% and for a 1:1 slope from 3% 
to 5%. 

4. For a 30° surface slope and 1.5:1 side slopes the excess 
volume is three or four times the nominal volume. Such a 
steep surface slope implies the probability of "side-hill work" 
to which the above corrective rules are not applicable. When 
the surface slopes are very steep careful work must be [done 
to avoid excessive error. For a 1:1 side slope, the errors are 
from 50% to 80%. 

A still closer approximation, especially for the steeper surface 
slopes, may be obtained by using, directly or by interpolation, 
figures from the corrective tabular form which forms part of 
Table XVII. Unless the surface slope angle is known accurately 
(especially when large) no great accuracy in the final result is 
possible. Close accuracy would also require the determination 
of the prismoidal correction. But if such close accuracy is 
deemed essential, it can be most easily obtained by ac- 
curate cross-sectioning at each station and the adoption of 
other methods of computation^ — such as are given in §§ 108 
aad 109, 

When the contours have been drawn in for a sufficient 
distance on either side to include the position of both slope 
stakes at every station, as will usually be the case, cross-sectiona 
may be obtained by drawing lines on the map at each station 
perpendicular to the center line — see Fig. 4. The intersection 
of these lines with the contours will furnish the distances for 
drawing on cross-section paper the transverse profile at each 
station. Drawing on the same cross-section the lines repre- 
senting the roadbed and the side slopes, the cross-section of 



148 RAILKOAD CONSTRUCTION. §126. 

cut (or fill) is complete and its area may be obtained by scaling 
from the cross-section paper. If the contours have been 
located on the map with sufficient accuracy, such a method 
will determine the cross-sectional area very closely. When 
cross-sections have been taken with a wye- or hand-level, as 
described in § 12, the cross-sections as plotted will probably 
be more accurate than when the contours are run in from 
points determined by the stadia method. In fact this semi- 
graphical method is frequently used, in place of the purely 
numerical methods described in previous sections, to make 
final estimates of the volume of earthwork. 

As a numerical example, an assumed location line was laid 
out on the contours given in Fig. 4. The volume of cut, as 
determined by Table XVII for a roadbed 20 feet wide, with 
side slopes of 1 : 1, was 5746 cubic yards. The surface slope 
varied from 3° to 11°. Computing the corrections by a careful 
interpolation from the corrective table, the total correction was 
found to be 128 cubic yards, or an average of a little over 2%. 
On the other hand the negative prismoidal correction amounts 
to 72 cubic yards, which leaves a net correction of 56 cubic 
yards — about 1%. It so happens that in this case a correction 
for curvature would tend further to wipe out this correction. 
These figures merely verify numerically the general conclusions 
stated above, although it should not be forgotten that in indi- 
vidual cases the figures taken from Table XVII require ample 
correction. 

The following approximate rule, for which the author is 
indebted to Mr. W. H. Edinger, is exceedingly useful when it 
is desired to rapidly determine the approximate volume of 
earthwork between two points along the road. Its great merit 
lies in the fact that it only means the memorizing of a com- 
paratively simple rule which will make it possible to make 
such computations in the field, without the use of tables. The 
rule is based on the fact that the area of any level section equals 
bd + sd^ ; and therefore, 

S(vol.) = (6S(^+sScZ2)^, 

in which L is usually 100 feet. For strict accuracy this would 
dnly be the volume provided the total length was an even num- 
fe^er of hundred feet, and the various values of d represented 



§ 127. EARTHWORK. 149 

the depths which were uniform for hundred foot sections. It 
makes no allowance for the comparatively large prismoidal 
error of the pyramidal and wedge-shaped sections usually found 
at each end of a cut or fill, but where an approximate estimate 
is desired, in which this inaccuracy may be neglected, the 
method is very useful. The method of applying this rule with- 
out tables may best be illustrated by a simple numerical ex- 
ample. Assume that the levels on a stretch of fairly level 
ground, which is about 500 feet long, have been taken, the depths 
being taken at points 100 feet apart, the first and last points 
being about 40 or 50 feet from the ends of the cut, or fill. The 
depths are as given in the first column in the tabular form 
below; the slope is 1.5:1, and the breadth (6) is 14 feet. 



d 




d2 


1.6 




2.56 


2.8 




7.84 


4.5 




20.25 


3.1 




9.61 


0.9 




.81 


Sd-12.9 




2^2 =41.07 


14 


adding 


one half =20. 53 


62(^ = 180.6 




s2d2 = 61.60 


61.60 







242.2 

24220^27 = 897 cubic yards. 

The 180.6 is the 6Sd and the 61.6 is sSd^j adding these and 
moving the decimal point two places to multiply by 100, we 
only have to divide by 27 to obtain the value in cubic yards. 
Although the above rule requires more work than the employ- 
ment of earthwork tables, yet it is a very convenient method 
of estimating the approximate volume of a short section of 
earthwork when no tables are at hand. 

FOEMATION OF EMBANKMENTg. 

127. Shrinkage of earthwork. The statistical data indicating 
the amount of shrinkage is very conflicting, a fact which is 
probably due to the following causes: 

1. The various kinds of earthy material act very differently 
as respects shrinkage. There is a great lack of uniformity in 



150 RAILROAD CONSTRUCTION. § 127. 

the classification of earths in the tests and experiments which 
have been made. 

2. Very much depends on the method of forming an embank- 
ment (as will be shown later). Different reports have been 
based on different methods — often without mention of the 
method. 

3. An embankment requires considerable time to shrink to 
its final volume, and therefore much depends on the time elapsed 
between construction and the measurement of what is supposed 
to be the settled volume. 

4. A soft subsoil will frequently settle under the weight of a 
high embankment and apparently indicate a far greater shrink- 
age than the actual reduction in volume. 

5. An embankment of very «oft material will sometimes 
" mush " or widen at the sides, with a consequent settling of 
the top, due to this cause alone, but such settlement would 
indicate that unsuitable material had been used to form the 
embankment. 

As a summary of the extensive discussion and wide range of 
shrinkage factors which might be quoted, the following facts 
may be stated: 

1. The density of natural soil increases with its depth below 
the surface. Some careful and accurate tests of some " clay, 
loam and gumbo," made on the C. B. & Q. R. R., showed an 
increase from 70 lbs. per cubic foot at the surface to 121 lbs. 
per cubic foot (an increase of over 70%) at a depth of 25 feet. 

2. Freshly excavated material of whatever character occupies 
a greater volume in a cart or other conveyor, or when loosely 
deposited, than it did in the original excavation. 

3. After being deposited it usually shrinks more or less from 
its volume as loose material. This shrinkage increases with 
age and with the amount of traffic over it. When the material 
is deposited in small increments from wagons or carts and each 
layer is subjected to compression from horses' hoofs and from 
w^heels, the contraction during construction is very great and the 
subsequent shrinkage is comparatively small. 

4. Light vegetable mould or " top soil," and, in general, all 
naturally deposited soil to a depth of 3 to 5 feet, will shrink 
until its final volume is less than its volume in its original state. 

5. On the other hand, compact earth, taken from the bottom 
of a deep excavation, and also rock, although it may shrink 



§ 127. EARTHWORK. 151 

somewhat from its volume as measured in carts, cars, or other 
conveyors, will never shrink to its volume in the original excava- 
tion, and will always occupy a larger volume in an embankment. 
6. An embankment continues to shrink with age, due to the 
pressure of superincumbent material and also due to the pressure 
and vibration caused by traffic. This law was clearly indicated 
by the following figures from the C. B. & Q. R. R. tests, where 
three embankments were: 

(a) 17 years old; no traffic shrinkage, 6.7% 

(b) 49 " " abandoned after 32 years of light traffic . " 12.9% 

(c) 17 " ." constantly under heavy traffic " 13.6% 

'■ 7. If an embankment is formed by dropping earth from a 
trestle, there is no compression during formation and the shrink- 
age will be long-drawn-out, especially if the material is light and 
the track continues for some time to be supported by the floor 
system of the trestle. 

8. The mere weight of an embankment, augmented by the 
vibrating action of heavy traffic, will compress the natural soil 
on which an embankment is placed. The depth of this com- 
pression will vary from zero for a rocky surface to an indefinite 
and unceasing settlement into a " bottomless bog." This effect, 
distinct from the shrinkage of the volume of embankment 
material, is called subsidence. It always occurs to some extent 
on ordinary grazing or agricultural land, which means under 
the majority of embankments. The percentage of subsidence 
will be greater for a low than for a high embankment, since the 
area of the base is less and the tamping action of traffic is more 
direct and effective. Investigation, by borings and the digging 
of test pits, has shown that there is sometimes as much (or more) 
deposited material below the original surface line as that in the 
visible embankment above. This means that when an embank- 
ment is to be formed on soft, or even ordinary agricultural 
ground, considerably more material must be deposited than 
is called for by the nominal cross-sections above the original 
surface lines. The extent of such subsidence cannot be accu- 
rately predicted. It is even more difficult than to predict the 
ultimate shrinkage of a volume of excavated material after 
being formed into an embankment. When subsidence is alto- 
gether ignored, the almost inevitable result is a future sag of 
the grade line on embankments, which can only be restored 
by comparatively expensive raising of the track under traffic 



152 RAILROAD CONSTRUCTION. § 128. 

conditions. Instances are not uncommon where a company has 
been compelled to change a location after having deposited on 
a seemingly bottomless bog a volume of material several time^ ;, 
the volume of the desired visible embankment. Of course such l 
cases are exceptional, but the engineer must use judgment^ 
aided perhaps by boring into a soft soil, to estimate how much 
the subsidence will prove to be. 

9. A sharp and clear distinction should be made between the 
coefficient of extra height of an embankment and the coefficient 
of shrinkage which determines how many cubic yards of settled 
embankment may. be made from a definite volume of earth or 
rock measured in the excavation. Even if the coefficient of 
volume shrinkage were accurately kno-wn, the effect of subsi- 
dence must still be allowed for, and the coefficient of extra height 
must be a composite of these two effects. The effect of the 
method of formation of the embankment must also be considered: 
If the material is compacted during construction, some of the 
shrinkage will have been accomplished and some of the sub- 
sidence will have taken place by the time the embankment is 
up to grade line and only the future shrinkage and subsidence 
must be allowed for, although more material has been used than 
the profile seemed to call for. A rock embankment will not 
shrink appreciably after formation and in such case only the 
future subsidence need be allowed for. 

10. The very serious expense of raising the grade of a track 
under traffic may be reduced if not altogether avoided by modi- 
fying the normal 

^^^ -^ ' grade line over 

'%)v '- -^ ^^^^^^ an embankment, 

^s^55^^§i^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ substantially as 

'm^ yfiTT^ shown in Fig. 62. 

^^^h^^^f^/7^0^ Whatever may 

^ be the coeffi-- 

^i^- 62. cients of shrink- 

age and subsi- 
dence, the lowering of the grade line by these combined effects will 
be greater for a high than for a low embankment, and any allow- 
ance must be in principle as shown in Fig. 62. From 8% to 15% 
is sometimes quoted as the required extra height of embank- 
ments, although it is strenuously claimed by many that 3% or 
2% is sufficient, or even that no allowance should be made. 



§ 129. EARTHWORK. 153 

128. Proper allowance for shrinkage or subsidence. It follows 
from the above considerations that no simple and set rules may 
be prescribed, either for the coefficient of shrinkage (or expan- 
sion) or for the coefficient of extra height, since the coefficients 
will depend on the kind of material, its depth in the cutting, 
the method of formation of the embankment, the time during 
which complete settlement is assumed to take place, and even 
on the intensity of the traffic which will run over the embank- 
ment. And also, since an embankment will be formed from 
materials taken from various depths of excavation, and also 
from various cuttings containing perhaps several kinds of mate- 
rial, it follows that the real coefficient will be a composite figure 
whose exact value will be impossible to determine, even if some 
of the elements could be determined with substantial accuracy. 
Therefore the allowance to be made when forming any embank- 
ment must be estimated according to judgment, after allowing 
for all of the factors involved. The following figures have the 
weight of considerable authority and may be used as a guide 
in making up a composite figure which will best suit any partic- 
ular case. 

Gravel or sand will shrink about 8% 



Clay. 

Loam 

Loose vegetable surface soil . 
Rock, large pieces. 



expand 



10% 
12% 
15% 
40% 



sniall pieces " " " 60% 

To utilize such figures we might say, for example, that, if 
some material will shrink 8%, 1000 cubic yards of it, measured 
in place, will make 1000—80 = 920 cubic yards of settled embank- 
ment. If the material is a mixture of earth and rock, for which 
a composite figure of 20% expansion is estimated to be correct, 
1000 cubic yards of such excavation will make 1200 cubic yards 
of settled embankment. Even this calculation ignores sub- 
sidence, which must be estimated separately. 

129. Methods of forming embankments. Embankments of 
moderate height are sometimes formed by scraping material 
with drag scrapers from ditches at the sides, especially if there 
is little or no cutting to be done in the immediate vicinity. 
Over a low level swampy stretch this method has the double 
advantage of building an embankment which is well above 
the general level and also provides generous drainage ditches 
which keep the embankment dry. Wheeled scrapers may be 
used economically up to a distance of 400 feet to excavate 



154 RAILROAD CONSTRUCTION. § 129. 

cuts and deposit the material on low embankments. Such 
methods have the advantage of compacting the embankments 
during construction and reducing future shrinkage. 

When carts are used, an embankment of any height may be 
formed by "dumping over the end" and building to the full 
height (or even higher to allow for shrinkage) as the embank- 
ment proceeds. The method is especially applicable when the 
material comes from a place as high as or higher than the 
grade-line, so that no up-hill hauling is necessary. Only a 
small contractor's plant is required for all of these methods. 

Trestles capable of carrying carts, or even cars and loco- 
motives, from which excavated material may be dropped, are 
found to be economical in spite of the fact that their cost is a 
construction expense. There is the disadvantage that such 
embankments require a long time to settle, but there are the 
advantages that the earth may be hauled by the train load 
from a distance of perhaps several miles, dumped from the 




Fia. 63. 

cars by train ploughs, or automatically dumped when the 
material is carried in patent dumping-cars, and all at a com- 
paratively small cost per cubic yard. The disadvantages of 
slow settlement may be obviated, although at some additional 
cost, by making the trestle sufficiently strong to support regular 
traffic until the settlement is complete. 

During recent years cableways have been utilized to fill 
comparatively narrow but deep ravines from material obtain- 
able on either side of the ravine. This method obviates the 
construction of an excessively high trestle which might other- 
wise be considered necessary. 

When an embankment is to be placed on a steep side hill 
which has a slippery clay surfacCi the embankment will some- 



§ 130, EARTHWORK. 155 

times slide down the hill, unless means are taken to prevent it. 
Some sort of bond between the old surface and the new material 
becomes necessary. This has sometimes been provided by 
cutting out steps somewhat as is illustrated in Fig. 63. It is 
possible that a deep ploughing of the surface would accom- 
plish the result just as effectively and much cheaper. The 
tendency to slip is generally due not only to the nature of the 
soil but also to the usual accompanying characteristic that the 
soil is wet and springy. The sub-surface drainage of such a 
place with tile drains will still further prevent such slipping, 
which often proves very troublesome and costly. 

COMPUTATION OF HAUL. 

130. Nature of subject. As will be shown later when analyz- 
ing the cost of earthwork, the most variable item of cost is thaf- 
dcpending on the distance hauled. As it is manifestly imprac- 
ticable to calculate the exact distance to which every individual 
cartload of earth has been moved, it becomes necessary to devise 
a means which will give at least an equivalent of the haulage of 
all the earth moved. Evidently the average haul for any mass 
of earth moved is equal to the distance from the center of grav- 
ity of the excavation to the center of gravity of the embank- 
ment formed by the excavated material. As a rough approxi- 
mation the center of gravity of a cut (or fill) may sometimes be 
considered to coincide with the center of gravity of that part of 
the profile representing it, but the error is frequently very large. 
The center of gravity may be determined by various methods, 
but the method of the "mass diagram" accomplishes the same 
ultimate purpose (the determination of the haul) with all-sufii- 
cient accuracy and also furnishes other valuable information. 

131. Mass diagram. In Fig. 64 let A'B' . . ,G' represent 
a profile and grade line drawn to the usual scales. Assume A' 
to be. a point past which no earthwork will be hauled. Such 
a point is determined by natural conditions, as, for example, a 
river crossing, or one end of a long level stretch along which 
no grading is to be done except the formation of a low embank- 
ment from the material excavated from ample drainage ditches 
on each side. Above the profile draw an indefinite horizontal line 
{ACn in Fig. 64) which may be called the "zero line." Above 
every station point in the profile draw an ordinate (above or be- 



156 



KAILROAD CONSTRUCTION. 



§ 131. 



low the zero line) which will represent the algebraic sum of 
the cubic yards of cut and fill 
(calling cut + and fill -) from 
the point A' to the point con- 
sidered. The computations of 
these ordinates should first be 
made in tabular form as shown 
below. In doing this shrinkage 
must be allowed for by consider- 
ing how much embankment 
would actually be made by so 
many cubic yards of excavation 
of such material. For example, 
we will assume that 1000 cubic 
yards of sand or gravel, measured 
in place (see § 128) will make 
about 920 cubic yards of embank- 
ment; therefore all cuttings in 
sand or gravel should be dis- 
counted in about this propor- 
tion. Excavations in rock should 
be increased in the proper 
ratio. In short, all excavations 
should be valued according to the 
amount of settled embankment 
that could be made from them. 
Place in the first column a list 
of the stations; in the second 
column,the number of cubicyards 
of cut or fill between each station 
and the preceding station; in 
the third and fourth columns, the kind of material and the proper 
shrinkage factor; in the fifth column, a repetition of the quan- 
tities in cubic yards, except that the excavations are diminished 
(or increased, in the case of rock) to the number of cubic yards 
of settled embankment which may be made from them. In 
the sixth column place the algebraic sum of the quantities in the 
fifth column (calling cuts + and fills -) from the starting- 
point to the station considered. These algebraic sums at each 
station will be the ordinates, drawn to some scale, of the mass 
curve. The scale to be used will depend somewhat on whether 




§132. 



EARTHWORK. 



157 



the work is heavy or light, but for ordinary cases a scale of 
5000 cubic yards per inch may be used. Drawing these ordi- 
nates to scale, a curve A, B,...G may be obtained by joining 
the extremities of the ordinates. 



Sta. 


Yard.| S;'l 


Material. 


Shrinkage 
factor. 


Yards, 

reduced 

for 

shrinkage. 


Ordinate 

in mass 

curve. 


46 + 70 

47 

48 

+ 60 
49 
50 
51 
52 

4- 30 
53 

+ 70 
64 

4- 42 
55 
56 
57 







4- 195 
+ 1792 
4- 614 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 177 
4- 180 

- 52 

- 71 
4- 276 
4- 1242 
+ 1302 


Clayey soil 
It <i 


- 10 per cent 

- 10 

- 10 


4- 175 
+ 1613 
4- 553 

- 143 

- 906 

- 1985 

- 1721 

- 112 
4- 283 
4- 289 

- 52 

- 71 
4- 249 
+ 1118 
+ 1172 


+ 175 

+ 1788 
4 2341 
4 2198 






+ 1292 






- 693 






- 2414 






- 2526 


Hard rock 
tt 11 


4 60 per cent 
+ 60 


- 2243 
-- 1954 

- 2006 






- 2077 


Clayey soil 
It 11 

11 (1 


— 10 per cent 

- 10 
-- 10 


- 1828 

- 710 
+ 462 



132. Properties of the mass curve. 

1. The curve will be rising while over cuts and falling while 

over fills. 

2. A tangent to the curve will be horizontal (as at B, D, E, 
F, and G) when passing from cut to fill or from fill to cut. 

3 When the curve is below the "zero line" it shows that 
material must be drawn backward (to the left) ; and vice versa, 
when the curve is above the zero line it shows that material 
must be drawn forward (to the right) . 

4. When the curve crosses the zero line (as at A and C) it 
shows (in this instance) that the cut between A' and B' will just 
provide the material required for the fill between B' and C, and 
that no material should be hauled past C, or, in general, past 
any intersection of the mass curve and the zero line. 

5. If any horizontal line be drawn (as ab), it indicates that 
the cut and fiU between a' and b' will just balance. 

6. When the center of gravity of a given volume of material 
is to be moved a given distance, it makes no difference (at least 
theoretically) how far each individual load may be hauled or 
how any individual load may be disposed of. The summation 



158 RAILKOAD CONSTRUCTION. § 132. 

of the products of each load times the distance hauled will be a 
coristant, whatever the method, and w^ill equal the total volume 
times the movement of the center of. gravity. The average 
haul, whigh is the movement of the center of gravity, will there-1| 
fore equal the summation of these products divided by the totall 
volume. If we draw two horizontal parallel lines at an infini- 
tesimal distance dx apart, as at ah, the small increment of cut 
dx at a' will fill the corresponding increment of fill at b', and 
thitf material must be hauled the distance ab. Therefore the 
product of ab and dx, which is the product of distance times 
volume, is represented by the area of the infinitesimal rectangle 
at ah, and the total area ABC represents the summation of 
volume times distance for all the earth movement between A' 
and C. This summation of products divided by the total 
volume gives the average haul. 

7. The horizontal line, tangent at E and cutting the curve 
at e, f, and g, shows that the cut and fill between e' and E' will 
just balance, and that a possible method of hauling (whether 
desirable or not) would be to "borrow" earth for the fill between 
C and e', use the material between D' and E' for the fill between 
e' and D', and similarly balance cut and fill betv.een E' and /' 
and also between /' and g\ 

8. Similarly the horizontal line hklm may be dra\Mi cutting 
the curve, which will show another possible method of hauling. 
According to this plan, the fill between C and A' w^ould be 
made by borrowing; the cut and fill between h' and k' would 
balance; also that between k' and V and betwoc^n I' and m'. 
Since the area ehDkE represents the measure of haul for the 
eaHh between e' and E', and the other areas measure the corre- 
sponding hauls similarly, it is evident that the sura of the areas 
ehDkE and ElFmf, which is the measure of haul of all the 
material between e' and /', is largely in excess of the sum of 
the areas hDk, kEl, and IFm, plus the somewhat uncertain 
measures of haul due to borrowing material for e'h' and wasting 
the material between m' and /'. Therefore to make the meas- 
ure of haul a minimum a line should be drawn which will make 
the sum of the areas between it and the mass curve a minimum. 
Of course this is not necessarily the cheapest plan, as it implies 
more or less borrowing and wasting of material, which may 
cost more than the amount saved in haul. The comparison of 
the two methods is quite simple, however. Sir.ce the amount 



I 133. EARTHWOKK. 159 

of fill between e' and ^' is represented by the difference of the 
ordinates at e and h, and similarly for m' and /', it follows that 
the amount to be borrowed between e' and h' will exactly equal 
the amount wasted between m'. and /'. By the first of the above 
methods the haul is excessive^ but is definitely known from the 
mass diagram, and all of the material is utilized; by the second 
method the haul is reduced to about one-half, but there is a 
known quantity in cubic yards wasted at one place and the same 
quantity borrowed at another. The length of haul necessary 
for the borrowed material would need to be ascertained; also 
the haul necessary to waste the other material at a place where 
it would be unobjectionable. Frequently this is best done by 
widening an embankment beyond its necessary width. The 
computation of the relative cost of the above methods will be 
discussed later (§ 148). ........w^j^. „ ^„c. :,^. .p,^, 

9. Suppose that it were deemed best, after drawing the mass 
curve, to introduce a trestle between s' and v' , thus saving an 
amount in fill equal to tv. If such had been the original design, 
the mass curve would have been a straight horizontal line between 
s and t and would continue as a curve which would be at all 
points a distance tv above the curve vFmzfGg. If the line Ef is 
to be used as a zero line, its intersection with the new curve at x 
will show that the material between E' and z' will just balance 
if the trestle is used, and that the amount of haul will be meas- 
ured by the area between the line Ex and the broken line Estx. 
The same computed result may be obtained without drawing 
the auxiliary curve ixn ... by drawing the horizontal line zy 
at a distance xz(=tv) below Ex. The amount of the haul can 
then be obtained by adding the triangular area between Es and 
the horizontal line Ex, the rectangle between st and Ex, and the 
irregular area between vFz and y . . . z (which last is evidently 
equal to the area between tx and E . . . x). The disposal of the 
material at the right of 2^. would then be governed by the indica- 
tions of the profile and mass diagram which would be found at 
the right of (/. In fact it is difficult to decide with the best of 
judgment as lo the proper disposal of material without having 
a mass diagram extending to a considerable distance each side 
of that part of the road under immediate consideration. 

133. Area of the mass curve. The area may be computed 
most readily by means of a planimeter, which is capable with 
reasonable care of measuring such areas with as great acQuracj 



160 KAILROAD CONSTRUCTION. § 134. 

as is necessary for this work. If no such instrument is obtain- 
able, the area may be obtained by an application of "Simpson's 
rule." The ordinates will usually be spaced 100 feet apart;] 
Select an even number of such spaces, leaving, if necessary, one 
or more triangles or trapezoids at the ends for separate and,:: 
independent computation. Let yo . . .ynhe the ordinates, i.e.'i] 
the number of cubic yards at each station of the mass curve, otl 
the figures of "column six" referred to in § 131. Let the uni- 
form distance between ordinates (=^100 feet) be called 1, i.e.,i 
one station. Then the units of the resulting area will be cubic | 
yards hauled one station. Then the I 

Area = i[2/o + 4(2/1 + 2/3 ■+■ • • .2/(»-l) + 2(2/2+2/4+ • • .2/(n_2)+2/n3- (62Xi 

When an ordinate occurs at a substation, the best plan is to 
ignore it at first and calculate the area as above. Then, if the 
difference involved is too great to be neglected, calculate the 
area of the triangle having the extremity of the ordinate at the 
substation as an apex, and the extremities of the ordinates at the 
adjacent stations as the ends of the base. This may be done by 
finding the ordinate at the substation that would be a propor- 
tional between the ordinates at the adjacent full stations. Sub- 
tract this from the real ordinate (or vice versa) and multiply the 
difference by ^Xl. An inspection will often show that the 
correction thus obtained would be too small to be worthy of con- 
sideration. If there is more than one substation between two 
full stations, the corrective area will consist of two triangles and 
one or more trapezoids which may be similarly computed, if 
necessary. 

When the zero line (Fig. 64) is shifted to eE, the drop from 
AC (produced) to E is known in the same units, cubic yards. 
This constant may be subtracted from the numbers ("column 
6," § 131) representing the ordinates, and will thus give, with- 
out any scaling from the diagram, the exact value of the modi- 
fied ordinates. 

134. Value of the mass diagram. The great value of the mass 
diagram lies in the readiness with which different plans for the 
disposal of material may be examined and compared. When 
the mass curve is once drawn, it will generally require only a 
shifting of the horizontal line to show the disposal of the material 
by any proposed method. The mass diagram also shows the 



!§ 135. EARTHWORK. 161 

extreme length of haul that will be required by any proposed 
method of disposal of material. This brings into consideration 
the " limit of profitable haul/' which will be f ulh" discussed in 
§ 148. For the present it may be said that with each method 
of carrying material there is some liniit beyond which the expense 
of hauling will exceed the loss resulting from borrowing and 
wasting. With wheelbarrows and scrapers the limit of profit- 
able haul is comparatively short, with carts and tram-cars it is 
much longer, while with locomotives and cars it may be several 
miles. If, in Fig. 64, eE or E] exceeds the limit of profitable 
haul, it shows at once that some such line as hklm should be 
drawn and the material disposed of accordingly. 

135. Changing the grade line. The formation of the mass 
curve and the resulting plans as to the disposal of material are 
based on the mutual relations of the grade line and the surface 
profile and the amounts of cut and fill which are thereby im- 
plied. If the grade line is altered, every cross-section is altered, 
the amount of cut and fill is altered, and the mass curve is also 
changed. At the farther limit of the actual change of the grade 
line the revised mass curve will have (in general) a different 
ordinate from the previous ordinate at that point. From that 
pomt on, the revised mass curve will be parallel to its former 
position, and the revised curve may be treated similarly to the 
case previously mentioned in which a trestle was introduced. 
Since it involves tedious calculations to determine accurately 
how much the volume of earthwork is altered by a change in 
grade line, especially through irregular country, the effect on 
the mass curve of a change in the grade line cannot therefore 
be readily determined except in an approximate way. Raising 
the grade line will evidently increase the fills and diminish the 
cuts, and vim versa^ Therefore if the mass curve indicated, for 
example, either an excessively long haul or the necessity for 
borrowing material (implying a fill) and wasting material 
farther on (implying a cut), it would be possible to diminish the 
fiU (and hence the amount of material to be borrowed) by lower- 
ing the grade line near that place, and diminish the cut (and 
hence the amount of material to be wasted) by raising the 
grade line at or near the place farther on. Whether the advan- 
tage thus gained would compensate for the possibly injurious 
effect of these changes on the grade line would require patient 
investigation. But the method outlined shows how the masa 



162 



KAILROAD CONSTRUCTION. 



136. 



curve might be used to indicate a possible change in grade line 
which might be demonstrated to be profitable. 
I 136. Limit of free haul. It is sometimes specified in con- 
tracts for earthwork that all material shall be entitled to free 
haul up to some specified limit, say 500 or 1000 feet, and that 
all material drawn farther than that shall be entitled to an i 
allowance on the excess of distance. It is manifestly imprac- 
ticable to measure the excess for each load, as much so as to 
measure the actual haul of each load. The mass diagram, also 
solves this problem very readily. Let Fig. 65 represent a pro- 




FiG. 65. 



file and mass diagram of about 2000 feet of road, and suppose 
that 800 feet is taken as the limit of free haul. Find two points, 
a and 6, in the mass curve which are on tJie same horizontal line 
and which are 800 feet apart. Project these points down to a' 
and h'. Then the cut and fill between a' and ¥ will just balance, 
and the cut between A ' and a' will be needed for the fill between 
h' and C . In the mass curve, the area between the horizontal 
line ah and the curve aBh represents the haulage of the material 
between a' and Z>', which is all free. The rectangle abmn repre- 
sents the haulage of the material in the cut A 'a' across the 800 
feet from a^ to b'. This is also free. The sum of the two areas 
A am and hnC represents the haulage entitled to an allowance, 
since it is the summation of the products of cubic yards times 
the excess of distance hauled. 
If the amount of cut and fill was symmetrical about the point 



i, § 137. EARTHWORK. 163 

' B', the mass curve would be a symmetrical curve about the 
t vertical line through B, and the two limiting lines of free haul 
would be placed symmetrically about B and B'. In general 
there is no such symmetry, and frequently the difference is con- 
siderable The area aBlmm will be materially changed accord- 
ing as the two vertical lines am and bn, alv^■ays 800 feet apart, 
are shifted to the right or left. It is easy to show that the area 
aBbnm is a maximum when ab is horizontal. The minimum 
value would be obtained either when m reached A or n reached 
C, depending on the exact form of the curve. Since the posi- 
tion for the minimum value is manifestly unfair, the best definite 
value obtainable is the maximum, which must be obtained as 
above described. Since aBbnm is made maximum, the remainder 
of the area, which is the allowance for overhaul, becomes a mini- 
mum. The areas Aam and bCn may be obtained as in § 120. 
If the whole area AaBbCA has been previously computed, it 
may be more convenient to compute the area aBbnm and sub- 
tract it from the total area. 

Since the intersections of the mass curve and the "zero line" 
mark limits past which no material is drawn, it follows that 
there will be no allowance for overhaul except where the dis- 
tance between consecutive intersections of the zero line and mass 
curve exceeds the limit of free haul. 

Frequently all allowances for overhaul are disregarded; the 
profiles, estimates of quantities, and the required disposal of 
material are shown to bidding contractors, and they must then 
make their own allowances and bid accordingly. This method 
has the advantage of avoiding possible disputes as to the amount 
of the overhaul allowance, and is popular with railroad com- 
panies on this account. On the other hand the facility with 
which different plans for the disposal of material may be studied 
and compared by the mass-curve method facilitates the adoption 
of the most economical plan, and the elimination of uncertainty 
will frequently lead to a safe reduction of the bid, and so the 
method is valuable to both the railroad company and the con- 
tractor. 

ELEMENTS OF THE COST OF EARTHWORK. 

137. Analysis of the total cost into items. The variation in 
the total cost of excavating earthwork, hauling it a greater or 
less distance, and forming with it an embankment of definite 



164 RAILROAD CONSTRUCTION. § 138. 

form or wasting it on a spoil bank, is so great that the only 
possible method of estimating the cost under certain assumed 
conditions is to separate the total cost into elementary items. 
Ellwood Morris was perhaps the first to develop such a method 
— see Journal of the Franklin Institute, September and October, ^ 
1841. Trautwine used the same general method with some^" 
modifications. The following analysis will follow the same^ 
general plan, will quote some of the figures given by Morris • 
and by Trautwine, but will also include facts and figures better 
adapted to modern conditions. Since every item of cost (excepts 
interest on cost of plant and its depreciation) is a direct function: 
of the current price of common labor, all calculations will be 
based on the simple unit of $1 per day. Then the actual cost 
may be obtained by multiplying the calculated cost under the 
given conditions by the current price of day labor. When 
possible, figures will be quoted giving the cost of all items of 
work on a loose sandy soil which is the easiest to work and also 
for the cost of the heaviest soils, such as stiff clay and hard pan. 
These represent the extremes, excluding rock, which will be 
treated separately. The cost of intermediate grades may be 
interpolated between the extreme values according to the 
judgment of the engineer as to the character of the soil. 

The possible division into items varies greatly according to 
the method adopted, but the differentiation into items given 
below (which is strictly applicable to the old fashioned simpler 
methods of work) can usually be applied to any other method 
by merely combining or eliminating some of the items. The 
items are 

1. Loosening the natural soil. 

2. Loading the soil into whatever carrier may be used. 

3. Hauling excavated material from excavation to embank- 

ment or spoil bank. 

4. Spreading or distributing the soil on the embankment. 

5. Keeping roadways or tracks in good running order. 

6. Trimming cuts to their proper cross-section (sometimes 

called "sandpapering"). 

7. Repairs, wear, depreciation, and interest on cost of plant. 

8. Superintendence and incidentals. 

138. Item I. Loosening, (a) Ploughs. Very light sandy 
soils can frequently be shovelled without any previous loosen- 
ing, but it is generally economical, even with very light material. 



§ 138. EARTHWORK. 165 

to use a plough. Morris quotes, as tlie results of experiments, 
that a three-horse plough would loosen from 250 to 800 cubic 
yards of earth per day, which at a valuation of $5 per day 
would make the cost per yard vary from 2 cents to 0.6 cent. 
Trautwine estimates the cost on the basis of two men handlin» 
a two-horse plough at a total cost of $3.87 per day, being $1 
each for the men, 75 c. for each horse, and an allowance of 
37 c. for the plough, harness, etc. From 200 to 600 cubic yards 
is estimated as a fair day's work, which makes a cost of 1.9 c. 
to 0.65 c. per yard, which is substantially the same estimate 
as above. Extremely heavy soils have sometimes been loosened 
by means of special ploughs operated by traction-engines. 

Gillette estimates that "a two-horse team with a driver and 
a man holding the plough will loosen 25 cubic yards of fairly 
tough clay, or 35 cubic yards of gravel and loam per hour." 
For ten hours per day this would be 250 to 350 cubic yards 
per day. These values are neither as high nor as low as the 
extremes above noted. It is probably very seldom that a soil 
will be so light that a two-horse (or three-horse) plough can 
loosen as much as 600 (or 800) cubic yards per day. 

It is sometimes necessary to plough up a macadamized street. 
This may be done by using as a plough a pointed steel bar 
which is fastened to a very strong plough frame. A prelimi- 
nary hole must be made which will start the bar under the 
macadam shell. Then, as the plough is drawn ahead, the shell 
is ripped up. Four or six horses, or even a traction-engine, 
are .used for such work. Gillette quotes two such cases where 
the cost of such loosening was 2 c. and 6 c. per cubic yard, 
with common labor at 15 c. per hour. Two-thirds of such 
figures will reduce them to the $1 per day basis. The cost for 
ploughing on the $1 per 10-hour-day basis may therefore be sum- 
marized as follows : 

For very loose sandy soils. 0.6 c. per cubic yard 

" *' heavy clay " 2.0 c. '* ** '* 

" hard pan and macadam, up to .. . 4.0 c. ** " " 

(b) Picks. When picks are used for loosening the earth, as 
is frequently necessary and as is often done when ploughing 
would perhaf)s be really cheaper, an estimate * for a fair day's 

* Trautwine. 



166 RAILROAD CONSTRUCTION. § 139,| 

work is from 14 to 60 cubic yards, the 14 yards being the esti- 
mate for stiff clay or cemented gravel, and the 60 yards the esti- 
mate for the lightest soil that would require loosening. At $1; 
per day this means about 7 c. to 1.7 c. per cubic yard, which is;! 
about three times the cost of ploughing. Five feet of the face, 
is estimated * as the least width along the face of a bank that ; 
should be allowed to enable each laborer to work with freedom 
and" hence economically, 

(c) Blasting. Although some of the softer shaly rocks may 
be loosened with a pick for about 15 to 20 c. per yard, yet rock 
in general, frozen earth, and sometimes even compact clay are 
most economically loosened by blasting. The subject of blast- 
ing will be taken up later, §§ 149-155. 

(d) Steam-shovels. The items of loosening and loading 
merge together with this method, which will therefore be treated 
in the next section. 

139. Item 2. Loading, (a) Hand-shovelling. Much depends 
on proper management, so that the shovellers need not wait un- 
duly either for material or carts. With the best of management 
considerable time is thus lost, and yet the intervals of rest 
need not be considered as entirely lost, as it enables the men to 
work, while actually loading, at a rate which it would be physi- 
cally impossible for them to maintain for ten hours. Seven 
shovellers are sometimes allowed for ea(!h cart; otherwise there 
should be five, two on each side and one in the rear. Economy 
requires that the number of loads per cart per day should be 
made as large as possible, and it is therefore wise to employ as 
ma^'y shovellers as can work without mutual interference and 
without wasting time in waiting for material or carts. The 
figures obtainable for the cost of this item are unsatisfactory on 
account of their large disagreements. The following are quoted 
as the number of cubic yards that can be loaded into a cart by 
an average laborer in a working day of ten hours, the lower 
estimate referring to heavy soils, and the higher to light sandy 
soils: 10 to 14 cubic yards (Morris), 12 to 17 cubic yards (Has- 
koll), 18 to 22 cubic yards (Hurst), 17 to 24 cubic yards (Traut- 
wine), 16 to 48 cubic yards (Ancelin). As these estimates are 
generally claimed to be based on actual experience, the discrep- 
ancies are probably due to differences of management. If the 

* Hurst. 



§139. EARTHWORK. - 167 

average of 15 to 25 cubic yards be accepted, it means, on the 
basis of $1 per day, 6.7 c. to 4 c. per cubic yard. These esti- 
mates apply only to earth. Rockwork costs more, not only 
because it is harder to handle, but because a cubic yard of solid 
rock, measured in place, occupies about 1.8 cubic yards when 
broken up, while a cubic yard of earth will occupy about 1.2 
cubic yards. Rockwork will therefore require about 50% more 
loads to haul a given volume, measured in place, than will the 
same nominal volume of earthwork. The above authorities give 
estimates for loading rock varying from 6.9 c. to 10 c. per cubic 
yard. The above estimates apply only to the loading of carts 
or cars with shovels or by hand (loading masses of rock). The 
cost of loading wheelbarrows and the cost of scraper work will 
be treated under the item of hauling. 

(b) Steam-shovels.* Whenever the magnitude of the work 
will warrant it there is great economy in the use of steam-shovels. 
These have a "bucket" or "dipper" on the end of a long beam, 
the bucket having a capacity varjdng from | to 2^ cubic yards. 
Steam-shovels handle all kinds of material from the softest 
earth to shale rock, earthy material containing large boulders, 
tree-stumps, etc. The record of work done varies from 200 to 
1000 cubic yards in 10 hours. They perform all the work of 
loosening and loading. Their economical working requires that 
the material shall be hauled away as fast as it can be loaded, 
which usually means that cars on a track, hauled by horses or 
mules, or still better by a locomotive, shall be used. The ex- 
penses for a steam-shovel, costing about $5000, will average 
about $1000 per month. Of this the engineer may get $100; the 
fireman $50; the cranesman $90; repairs perhaps $250 to $300; 
coal, from 15 to 25 tons, cost very variable on account of expen- 
sive hauling; water, a very uncertain amount, sometimes costing 
$100 per month ; about five laborers and a foreman^ the laborers 
getting $1.25 per day and the foreman $2.50 per day, which will 
amount to $227.50 per month. This gang of laborers is employed 
in shifting the shovel when necessary, taking up and relaying 

* For a thorough treatment of the capabilities, cost, and management 
of steam-shovels the reader is referred to " Steam-shovels and Steam-shovel 
Work," by E. A. Hermann. D. Van Nostrand Co., New York. 

This book is now out of print. " Earthwork and its Cost," by H. P. Gil- 
lette, to which the student is referred for a more elaborate exposition of tho 
subject, has used many of Hermann's cuts. 



168 EAILROAD CONSTRUCTION. § 139. 

tracks for tli6 cars, shifting loaded and unloaded cars, etc. In 
shovelling through a deep cut, the shovel is operated so as to 
undermine the upper parts of the cut, which then fall down 
within reach of the shovel, thus increasing the amount of material 
handled for each new position of the shovel. If the material is 
too tough to fall down by its own weight, it is sometimes found 
economical to employ a gang of men to loosen it or even blast it 
rather than shift the shovel so frequently. Non-condensing 
engines of 50 horse-power use so much water that the cost of 
water-supply becomes a serious matter if water is not readily 
obtainable. The lack of water facilities will often justify the 
construction of a pipe line from some distant source and the 
installation of a steam-pump. Heiice the seemingly large 
estimate of $100 per month for water-supply, although under 
favorable circumstances the cost may almost vanish. The larger 
steam-shovels will consume nearly a ton of coal per day of 10 
hours. The expense of hauling this coal from the nearest rail- 
road or canal to the location of the cut is often a very serious 
item of expense and may easily double the cost per ton. Some 
steam-shovels have been constructed to be operated by electrioity 
obtained from a plant perhaps several miles away. Such a 
method is especially advantageous when fuel and water are diffi- 
cult to obtain. 

The following general requirements and specifications were 
recommended in 1907 by the American Railway Engineering 
Association: 

Three important cardinal points should be given careful 
attention in the selection of a steam-shovel. These are in their 
order 

(1) Care in the selection, inspection and acceptance of all 
material that enters into every part of the machine. 

(2) Design for strength. 

(3) Design for production, 

GENERAL SPECIFICATIONS. 

Weight of shovel: Seventy (70) tons. 
Capacity of dipper: Two and one-half (2|) yards. 
Steam pressure: One hundred and twenty (120) pounds. 
Clear height above rail of shovel track at which dipper should 
unload: Sixteen (16) feet. 



§ 140. EARTHWOEK. 169 

Depth below rail of shovel track at which dipper should ciig 
Four (4) feet. 

Number of movements of dipper per minute from time of 
entering banls to entering bank: Three (3). 

Character of hoist: Cable. 

Character of swing: Cable. 

Character of housing: Permanent for all employes. 

Capacity of tank: Two thousand (2000) gallons. 

Capacity of coal-bunker: Four (4) tons. 

Spread of jack arm: Eighteen (18) feet. A special short arm 
should be provided. 

Form of steam-shovel track: "T" rails on ties. 

Length of rails for ordinary work: Six (6) feet. 

Form of rail joint: Strap. 

Manufacturers of steam-shovels will sometimes "guarantee" 
that certain of their shovels will excavate, say 3000 cubic yards 
of earth per day of ten hours. Even if it were possible for a 
shovel to fill a car at the rate of 5 cubic yards per minute, it is 
always impracticable to maintain such a speed, since a shovel 
must always wait for the shifting of cars and for the frequent 
shifting of the shovel itself. There are also delays due to 
adjustments and minor breakdowns. The best shovel records 
are made when the cars are large — other things being equal. 
The item of interest and depreciation of the plant is very large 
in steam-shovel work. This will be discussed further later. 
The cost of loading alone will usually come to between 3 and 
4 c. per cubic yard. The cost of shifting the cars so as to 
place them successively imder the shovel, haul them to the 
dumping place, dump them and haul them back, will generally 
be as much more. Gillette quotes five jobs on one railroad 
where the total cost for loading and hauling varied from 5.9 c. 
to 11.4 c. per cubic yard. But as these figures are based on 
car measurement, the cost per cubic yard in ylace measure- 
ment must be increased about one-fourth, or from 7.4 c. to 
14.2 c. 

140. Item 3. Hauling. The cost of hauling depends on 
the number of round trips per day that can be made by each 
vehicle employed. As the cost of each vehicle is practically the 
same whether it makes many trips or few, it becomes important 
that the number of trips should be made a maximum, and to that 
end there should be as little delay as possible in loading and 



170 EAILROAD CONSTRUCTION. § 140. 

unloading. Therefore devices for facilitating the passage of the 
vehicles have a real money value. 

(a) Carts. The average speed of a horse hauling a two- 
wheeled cart has been found to be 200 feet per minute, a little 
slower when hauling a load and a little faster when returning 
empty. This figure has been repeatedly verified. It means an 
allowance of one minute for each 100 feet (or "station") of 
"lead — the lead being the distance the earth is hauled." The 
time lost in loading, dumping, waiting to load, etc., has been 
found to average 4 minutes per load. Representing the num- 
ber of stations (100 feet) of lead by s, the number of loads 
handled in 10 hours (600 minutes) would be 600 ^(s + 4). The 
number of loads per cubic yard, measured in the bank, is differ- 
entiated by Morris into three classes, viz. : 

3 loads per cubic yard in descending hauling; 
3i " " '' " " level hauling; and 

4 " " " " " ascending hauling. 

Attempts have been made to estimate the effect of the grade 
of the roadway by a theoretical consideration of its rate, and of 
the comparative strength of a horse on a level and on various 
grades. While such computations are always practicable on a 
railway (even on a temporary construction track), the traction 
on a temporary earth roadway is always very large and so very 
variable that any refinements are useless. On railroad earth- 
work the hauling is generally nearly level or it is descending — 
forming embankments on low ground with material from cuts in 
high ground. The only common exception occurs when an 
embankment is formed from borrow-pits on low ground. One 
method of allowing for ascending grade is to add to the hori- 
zontal distance 14 times the difference of elevation for work 
with carts and 24 times the difference of elevation for work 
with wheelbarrows, and use that as the lead. For example, 
using carts, if the lead is 300 feet and there is a difference of 
elevation of 20 feet, the lead would be considered equivalent to 
300 + (14 X 20) = 580 feet on a level. 

Trautwine assumes the average load for all classes of work 
to be ^ cubic yard, which figure is justified by large experience. 
Using one figure for all classes of work simplifies the calculations 
and gives the number of cubic yards carried per day of 10 hours 

equal to , . . Dividing the cost of a cart per day by the 



§ 140. EARTHWORK. 171 

number of cubic yards carried gives the cost of hauling pel 
yard. In computing the cost of a cart per day, Trautwine 
refers to the practice of having one driver manage four carts, 
thus making a charge of 25 c. per day for each cart for the driver. 
Although this might be an economical method when the haul is 
very long, it is not economical for short hauls. A safer estimate 
is to allow not more than two carts per driver and iii many 
cases a driver for each cart. Some contractors employ a driver 
for each cart and then require that the drivers shall assist in 
loading. The policy to be adopted is sometimes dependent on 
labor union conditions, which may demand that drivers must 
not assist in loading. The supply of labor and the amount of 
work on hand have a great influence on the methods of work 
which a contractor may adopt, for a strike will often disarrange 
all plans. 

The cost of a horse and cart must practically include a 
charge for the time of the horse on Sundays, rainy days and 
holidays. The cost of repairs of cart and harness is generally 
included in this item for simplicity, but, under a strict applica- 
tion of the analysis suggested in § 137j it should properly be 
included under Item 7, Repairs, etc. 

Since the time required for loading loose rock is greater 
than for earthwork, less loads will be hauled per day. The time 
allowance for loading, etc., is estimated by Trautwine as 6 
minutes instead of 4 as for earth. Considering the great ex- 
pansion of rock when broken up (see § 128), one cubic yard of 
solid rock, measured in place, v/ould furnish the equivalent of 
five loads of earthwork of J cubic yard. Therefore, on the 
basis of five loads per cubic yard, the number of cubic yards 

handled per ten-hour day per cart would be _. .^.y 

Let C represent the daily cost of a horse and cart and of 
the proportional cost of the driver (according to the number of 
carts handled by one driver), then the cost per cubic yard, 
measured in the cut, for hauling may be given by the formula: 



Cost per cu. yd. of hauling earth in carts = ^^^r — ~ 

I . (63) 

"^^^^ 600 J 



172 RAILROAD CONSTRUCTION. §140 

(b) Wagons. For longer leads (i.e., from i to | of a mile) 
wagons drawn by two (or three) horses are more economical. 
The old-style wagons (about 0.8 cu. yd.) have bottoms of loose 
thick narrow boards. Raising them individually deposits the 
load underneath. Modern dump wagons contain from 1.0 to 
2.0 cu. yds. The daily cost may be estimated on the same prin- 
ciple as the cost of carts. 

The number of wagon tdps per 10 hours will depend some- 
what on the management of the shovellers. Too many shovel- 
lers per wagon is not economical, measured in yards shovelled 
per man, although it may reduce the time consumed in loading 
any one wagon. At an average figure of 20 cubic yards, 
measured in place, per shoveller per 10 hours, seven shovellers 
would load 14 cubic yards per hour or one cubic yard in 4.3 
minutes. This would be the allowance for a wagon with a 
capacity of about IJ yards of loose eaith. Adding time for 
unloading, waiting to load and other possible '' lost time," there 
is perhaps a total of six minutes. This figure will vary very 
considerably according to the number of shovellers per Magon, 
the capacity of the wagon, the type of wagon (whether self- 
dumping) and other details in the method of management. 
Adopting six minutes as the time used for loading, unloading, 
and other "lost time," the formula becomes. 

Cost per cubic yard of hauling in wagons = -^Tr^r — ^, . , . . (64) 

in which C is the cost of the wagon, team and driver per day 
of 10 hours; s is the distance hauled in stations of 100 feet, 
and c is the capacity of the wagon in cubic yards, place meas- 
urement, which should be about three fourths of the nominal 
capacity of the wagon for earth and about sixty per cent when 
handling rock. 

(c) Wheelbarrows. Gillette has computed from observa- 
tions that a man will trundle a wheelbarrow at the rate of 250 
feet per minute or 1.25 stations of lead per minute for the round 
trip. The time required for loading is estimated at 2^ minutes 
and for unloading, adjusting wheeling planks, short rests, etc., 
I minute, or a total of three minutes per trip for all purposes 
except hauling. Gillette allows for a load only 1/15 cubic yard, 



I 140. EARTHWOEK. 173 

naeasured in place, or about 1/11 yard, 2.5 cubic feet, on the 
wheelbarrow. With notation as before, and for a ten-hour day, 

«-, 

A . 

Cost per cubic yard of loading and \ _ CX 15(1.255 + 3) /g^N 
hauling earth in wheelbarrows / ~~" 600 ' 

;In this equation C is the cost of both loading and hauling, and 
usually includes the allowance (Item 7) for the cost, repairs 
and depreciation of the wheelbarrows, whose service is very 
short lived. Trautwine estimates this at five cents per day or 
^stotal of $1.05 for labor and wheelbariow. 

The number of wheelbarrow loads required for a cubic yard 
of rock, measured in place, is about twenty-four. The time 
required for loading should also be increased about one fourth; 
the time required for all purposes except hauling is therefore 
about 3.75 minutes, and the corresponding equation becomes 

Costpercubicyardofloadingand'i (7X24(1 .25$ +3.75) (qq) 
hauling rock in wheelbarrows J 600 



ft 



(d) Scrapers. These are made in three general ways, *'buck 
scrapers, "drag" scrapers and '' wheeled" scrapers. The buck 
scraper in its original form consisted merely of a wide plank, 
shod with an iron strap on the lower edge and provided with 
a pole and a small platform on which the driver may stand to 
weight it down. The earth is not loaded on to any receptacle 
and carried, but is merely pushed over the ground. Notwith- 
standing the apparent inefficiency of the method, its extreme 
simplicity has caused its occasional adoption for the construc- 
tion of canal embankments out of material from the bed of the 
canal. The occasions are rare when their use for railroad work 
would be practicable, and even then drag scrapers would prob-^ 
§bly be preferable. 

A drag scraper is an immense "scoop shovel" about three feet 
long and three feet wide. There are usually two handles and a 
bail in front by which it is dragged by a team of horses. The 
nominal capacity varies from 7.5 cubic feet for the largest sizes, 
down to 3 cubic feet for the "one-horse" size, but these figures 
must be discounted by perhaps 40 or 50% for the actual average 
volume (as measured in the cut) loaded on during one scoop. 
The expansion of the earth during loosening is alone response 



174 RAILROAD CONSTRUCTION. § 140. 

ible for a discount of 25%. These scrapers cost from $10 to 
$18. 

A wheeled scraper is essentially an extra-large drag scraper 
which may be raised by a lever and carried on a pair of large 
wheels. Their nominal capacity ranges from 10 to 17 cubic feet, 
which should usually be liberally discounted when estimating 
output. They are loaded by dropping the scoop so that it 
scrapes up its load. The lever raises the scoop so that the load 
is carried on wheels instead of being dragged. At the dump the 
scoop is tipped so as to unload it. The movement of the 
scraper is practically continuous. They cost from $40 to 
$75. Their advantages over drag scrapers consist (1) in their 
greater capacity, (2) in the economy of transporting the load 
on wheels instead of by dragging, and (3) in the far greater 
length of haul over which the earth may be economically 
handled. 

Morris estimated the speed of drag scrapers to be 140 feet per 
minute, or 70 feet of lead per minute. The "lead" should be 
here interpreted as the average distance from the center of the 
pit to the center of the dump. Gillette declares the speed to be 
220 feet per minute. Some of this variation may be due to dif- 
ferences in the method of measuring the distance actually trav- 
elled, especially when the lead is very short, since the scraper 
teams must always travel a considerable extra distance at each 
end in order to turn around most easily. This extra distance is 
practically constant whether the lead is long or short. Gillette 
quotes an instance where the length of lead was actually about 
20 feet, but the scraper teams travelled about 150 feet for each 
!oad carried. On this account Gillette adopts a minimum of 
75 feet of lead no matter how short the lead actually may be. 
Of course the speed depends considerably on how strictly the 
men are kept to their work and also on the care which may be 
taken to obtain a full load for each scraper. As a compromise 
between Morris's and Gillette's estimates we may adopt the con- 
venient rate of speed of 200 feet per minute, or 100 feet of lead 
per minute. There should also be allowed for the time lost 
in loading and unloading and for travelling the extra distance 
travelled by the teams in making the circuit, 1^ minutes. Allow- 
ing the average value of seven loads per cubic yard and letting 
C represent the cost of scraper team and driver per ten-hour 
day, we have for the cost as follows : 



§ 140. EARTHWORK. 175 

Cost per cubic yard of loading and I _ CX7 (s-\-l^) 
, hauling earth in drag scrapers f 600 

In this formula C should include the cost of not only the 
driver, team, and scraper, but also the proper proportion of 
the wages of an extra man, who assists each driver in loading 
his scraper, and whose wages should be divided among the two 
(or three) scrapers to which he is assigned. Scraper work 
nearly always implies ploughing, the cost of which should be 
computed as under Item 1. 

"When a low embankment is formed from borrow-pits on each 
side of the road, it may be done with scrapers, which move from 
one borrow-pit to the other, taking a load alternately from each 
side to the center and making but one half turn for each load 
carried. This reduces the time lost in turning by one third of a 
minute and reduces the constant in the numerator in Eq. (67) 
from 1^ to 1. In this case the lead willusually be not greater 
than 75 feet, and therefore, if we consider this as a minimum 
value, s will ordinarily equal .75 and the quantity in the paren- 
thesis will equal 1.75. 

When using wheeled scrapers the catalogue capacity, which 
varies from 9 or 10 feet for a No. 1 scraper to 16 or 17 feet for 
a No. 3 scraper, must be reduced to 5 loads per cubic yard 
(place measurement) for a No. 1 scraper and to 2^ loads per 
cubic yard for a No. 3, not only on account of the expansion of 
the earth during loosening, but also on account of the imprac- 
ticability of loading these scrapers to their maximum nominal 
capacity. When the haul or lead for wheeled scrapers is 300 
feet or over, it will be justifiable to employ shovellers to fill up 
the bowl of the shovel, especially when the soil is tough and 
when it is impracticable to fill the shovel even approximately 
full by the ordinary method. _A snatch team to assist in load- 
ing the scrapers it also economical, especially with the larger 
scrapers. The proportionate number of snatch teams to the 
total number of scrapers of course depends on the length of 
haul. The cost of these extra shovellers and extra snatch teams 
must be divided proportionally among the number of scrapers 
assisted, in determining the value C in the formula given below. 
The extra time to be allowed on account of turning, loading, 
and dumping is about H minutes. The speed is considered 
one station of lead per minute as before. If we call C the average 



176 RAILROAD CONSTRUCTION. § 140. 

daily cost of one scraper and n the capacity of the scraper, or 
the number of loads per cubic yard, we may write the following 
formula, on the basis of a ten-hour day : 

Cost per cubic yard of loading and 1 _CXn(s + l^) /ggx 

hauling earth in wheeled scrapers J 600 

f 

(e) Cars and horses. The items of cost by this method are 
(a) charge for horses employed, (&) charge for men employed 
strictly in hauling, (c) charge for shifting rails when necessary, 
(d) repairs, depreciation, and interest on cost of oars and track. 
Part of this cost should strictly be classified under items 5 and 
7, mentioned in § 137, but it is perhaps more convenient to 
estimate them as follows: 

The traction of a car on rails is so very small that grade 
resistance constitutes a very large part of the total resistance 
if the grade is 1% or more. For all ordinary grades it is 
sufficiently accurate to say that the grade resistance is to 
the gross weight as the rise is to the distance. If the distance 
is supposed to be measured along the slope, the proportion is 
strictly true; i.e., on a 1% grade the grade resistance is 1 lb. 
per 100 of weight or 20 lbs. per ton. If the resistance on a 
level at the usual velocityis -1^-^,^, grade of 1:120 (0.83%) will 
exactly double it. If the material is hauled down a grade of 
1:120, the cars will run by gravity after being started. The 
work of hauling will then consist practically of hauling the 
empty cars up the grade. The grade resistance depends only 
on the rate of grade and the weight, but the tractive resistance 
will be greater per ton of weight for the unloaded than for the 
loaded cars. The tractive power of a horse is less on a grade 
than on a level, not only because the horse raises his own weight 
in addition to the load, but is anatomically less capable of 
pulling on a grade than on a level. In general it will be pos- 
sible to plan the work so that loaded cars need not be hauled up 
a grade, unless an embankment is to be formed from a low 
borrow-pit, in which case another method would probably be 
advisable. These computations are chiefly utilized in design- 
ing the method of work — ^the proportion of horses to cars. An 
example may be quoted from English practice (Hurst), in which 
the cars had a capacity of 3^ cubic yards, weighing 30 cwt. 
empty. Two horses took five "wagons" | of a mile on a level 



I 140. EARTHWORK. 177 

railroad and made 15 journeys per day of 10 fiours, i.e., they 
handled 250 yards per day. In addition to those on the 
"straight road," another horse was employed to make up 
the train of loaded wagons. V/ith a short lead the straight- 
road horses were employed for this purpose. In the above 
example the number of men required to handle these cars, 
shift the tracks, etc., is not given, and so the exact cost of the 
above work cannot be analyzed. It may be noticed that the 
two horses travelled 22^ miles per day, drawing in one direction 
a load, including the weight of the cars, of about 57,300 Ibs.^ 
or 28.65 net tons. Allowing -jl-g- as the necessary tractivi^ 
force, it would require a pull of 477.5 Ib^., or 239 lbs. for each 
horse. With a velocity of 220 feet per minute this would amount 
to i^ horse-power per horse, exerted for only a short time, 
however, and allowing considerable time for rest and for draw- 
ing only the empty cars. Gillette claims that the rolling re- 
sistance for such cars on a contractor's track should be con- 
sidered as 40 lbs. per ton (the equivalent of a 2% grade) and 
quotes many figures to support the assertion. Unquestionably 
the resistance on tracks with very light rails, light ties with 
wide spacing and no tamping, would be very gfe&.t and might 
readily amount to 40 lbs. per ton. In the above case, the 
resistance could not have been much if any over i^-q- A re- 
sistance of 40 lbs. per ton would have required each horse to 
pull about 573 lbs. for nearly five hours per day, beside pulling 
the empty cars the rest of the time. This is far greater exertion 
than any ordinary horse can maintain. The cars generally used 
in this country have a capacity of 1^ cubic yards and cost about 
$65 apiece. Besides the shovellers and dumping-gang, several 
men ^nd a foreman will be required to keep the track in order 
and to make the constant shifts that are necessary. Two trains 
are generally used, one of which is loaded while the other is run 
to the dump. Some passing-place is necessary, but this is 
generally provided by having a switch at the cut and running 
the trains on each track alternately. This insures a train of 
cars always at the cut to keep the shovellers employed. The 
cost of hauling per cubic yard can only be computed when the 
number of laborers, cars, and horses employed are known, and 
these will depend on the lead, on the character of the excavation, 
on the grade, if any^ etc., and must be so proportioned that the 
shovellers need not wait for cars to fill, nor the dumping-gang 



i?8 RAILROAD CONSTRUCTION. § 140. 

for material to handle, nor the horses and drivers for cars to 
haul. Much skill is necessary to keep a large force in smooth 
running order. 

(f) Cars and locomotives. 30-lb. rails are the lightest that 
should be used for this work, and 35- or 40-lb. rails are better. 
One or two narrow-gauge locomotives (depending on the length 
of haul), costing about $2500 each, will be necessary to handle 
two trains of about 15 cars each, the cars having a capacity of 
about 2 cubic yards and costing about $100 each. Some cars 
can be obtained as low as $70. A force of about five men and 
a foreman will be required to shift the tracks. The track- 
shifters, except the foreman, may be common laborers. The 
dumping-gang will require about seven men. Even when the 
material is all taken down grade the grades may be too steep for 
the safe hauling of loaded cars down the grade, or for hauling 
empty cars up the grade. iJnder such circumstances temporary 
trestles are necessary to reduce the grade. When these are 
used, the uprights and bracing are left in the embankment — ■ 
only the stringers being removed. This is largely a necessity, 
but is partially compensated by the fact that the trestle forms a 
core to the embankment which prevents lateral shifting during 
settlement. The average speed of the trains may be taken as 
10 miles per hour or 5 miles of lead per hour. The time lost 
in loading and unloading is estimated (Trautwine) as 9 minutes 
or .15 of an hour. The number of trips per day of 10 hours 

will equal ,- — n -x-^ — rv-; — r? or . — -, j^ — rr-; — =-=. Of 

4-(miles of lead) + .15 (miles of lead) + .75 

course this quotient must be a whole number. Knowing the 

number of trains and their capacity, the total number of cubic 

yards handled is known, which, divided into the total daily cost 

of the trains, will give the cost of hauling per yard. The daily 

cost of a train will include 

(a) Wages of engineer, who frequently fires his own engine; 

(6) Fuel, about ^ to 1 ton of bituminous coal, depending on 
work done ; 

(c) Water, a very variable item, frequently costing $3 to $5 
per day ; 

(d) Repairs, variable, frequently at rate of 50 to 60% per year; 

(e) Interest on cost and depreciation, 16 to 40%. 

To these must be added, to obtain the total cost of haul, 
(/) Wages of the gang employed in shifting track. 



§ 141. EARTHWOKK. 179 

The above calculation for the number of train loads depends 
on the assumption that 9 minutes is total time lost by a 
locomotive for each round trip. If the haul is very short it 
may readily happen that a steam-shovel cannot fill one train 
of cars before the locomotive has returned with a load of empties 
and is ready to haul a loaded train away. The estimation of 
the number of train loads is chiefly useful in planning 
the work so as to have every tool working at its high- 
est efficiency. Usually the capacity, of the steam-shovel 
or the ability to promptly "spot" the cars under the 
shovel is the real limiting agent which determines the daily 
output. 

141. Choice of method of haul dependent on distance. In 
light side-hill work in which material need not be moved more 
than 12 or 15 feet, i.e., moved laterally across the roadbed, 
the earth may be moved most cheaply by mere shovelling. 
Beyond 12 feet scrapers are more economical. At about 100 
feet drag-scrapers and wheelbarrows are equally economical. 
Between 100 and 200 feet wheelbarrows are generally cheaper 
than either carts or drag-scrapers, but wheeled scrapers are 
always cheaper than wheelbarrows. Beyond 500 feet two- 
wheeled carts become the most economical up to about 1700 
feet; then four-wheeled wagons become more economical up to 
3500 feet. Beyond this cars on rails, drawn by horses or by 
locomotives, become cheaper. The economy of cars on rails 
becomes evident for distances as small as 300 feet provided the 
volume of the excavation will justify the outlay. Locomotives 
will always be cheaper than horses and mules, providing the 
work to be done is of sufficient magnitude to justify the pm*- 
chase of the necessary plant and risk the loss in selling the plant 
ultimately as second-hand equipment, or keeping the plant on 
hand and idle for an indefinite period waiting for other 
work. Horses will not be economical for distances much 
over a mile. For greater distances locomotives are more 
economical, but the question of "limit of profitable haul" 
(§ 148) must be closely studied, as the circumstances are cer- 
tainly not common when it is advisable to haul material much 
over a mile. 

142. Item 4. Spreading. The cost of spreading varies 
with the method employed in dumping the load. When the earth 



180 RAILROAD const:ptjction. § 143. 

i^ tipped over the edge of an embankment there is little if any 
r.ecessary work. Trautwine allows about ^ c. per cubic yard 
for keeping the dumping-places clear and in order. This would 
represent the wages of one man at $1 per day attending to the 
unloading of 1200 two-wheeled carts each carrying J cubic yard. 
1200 carts in 10 hours would mean an average of two per minute, 
which implies more rapid and efficient work than may be de- 
pended on. The allowance is probably too small. When the 
material is dumped in layers some levelling is required, for 
which Trautwine allows 50 to 100 cubic yards as a fair day's 
work^ costing from 1 to 2 cents per cubic yard. The cost of 
spreading will not ordinarily exceed this and is frequently 
nothing — all depending on the method of unloading. It should 
be noted that Mr. Morris's examples and Qomputations (Juur. 
Franklir^ Inst., Sept. 1841) disregard altogether any special 
charge for this item. 

143. Item 5. KEEPING Roadways in order. This feature 
is important as a measure of true economy, whatever the system 
of transportation, but it is often neglected. A petty saving in 
such matters will cost many times as much in increased labor 
in hauling and loss of time. With some methods of haul the 
cost is best combined with that of other items. 

(a) Wheelbarrows. Wheelbarrows should generally be run 
on planks laid on the ground. The adjusting and shifting of 
these planks is done by the wheelers, and the time for it is 
allowed for in the "| minute for short rests, adjusting the 
wheeling plank, etc." The actual cost of the planks must be 
added, but it would evidently be a very small addition per cubic 
yard in a large contract. When the wheelbarrows are run on 
planks placed on "horses" or on trestles the cost is very appre- 
ciable; but the method is frequently used with great economy. 
The variations in the requirements render any general estimate 
of such cost impracticable. 

(b) Carts and wagons. The cost of keeping roadways in 
order for carts and wagons is sometimes estimated merely as so 
much per cubic yard, but it is evidently a function of the lead. 
The work consists in draining off puddles, filling up ruts, pick- 
ing up loose stones that may have fallen off the loads, and in 
general doing everything that will reduce the traction as much 
a^ possible. Temporary inclines, built to avoid excessive grade 



§ 144. EAKTHWORK. 181 

at some one pointy are often measures ot true economy. Traut- 
wine suggests ^^ c. per cubic yard per 100 feet of lead for earth- 
work and -j^Q c. for rockwork, as an estimate for this item when 
carts are used. 

(c) Cars. When cars are used a shifting-gang, consisting 
of a foreman and several men (say five), are constantly em- 
ployed in shifting the track so that the material may be loaded 
and unloaded where it is desired. The average cost of this 
item may be estimated by dividing the total daily cost of this 
gang by the number of cubic yards handled in one day. 

144. Item 6. Trimming cuts to their proper cross- 
section. This process, often called "sand-papering," must 
be treated as an expense, since the payment received for the 
very few cubic yards of earth excavated is wholly inadequate 
to pay for the work involved. Gillette quotes bids of 2 cents 
per square yard of surface trimmed, and from this argues that, 
for average excavations, it adds to the cost four cents per cubic 
yard of the total excavation. The shallower the cut the greater 
is the proportionate cost. Of course the actual cost to the 
contractor will depend largely on the accuracy of outline de- 
manded by the engineer or inspector. 

145. Item 7. Repairs, wear, depreciation, and interest 
ON cost of plant. The amount of this item evidently depends 
upon the character of the soil — ^the harder the soil the worse the 
wear and depreciation. The interest on cost depends on the 
current borrowing value of money. The estimate for this item 
has already been included in the allowances for horses, carts, 
ploughs, harness, wheelbarrows, steam-shovels, etc. Trautwine 
estimates I c. per cubic yard for picks and shovels. Deprecia- 
tion is generally a large percentage of the cost of earth-working 
(tools, the life of all being limited to a few years, and of many 
tools to a few months or weeks. 

146. Item 8. Superintendence and incidentals. The 
incidentals include the cost of wat«r-boys, timekeepers, watch- 
men, blacksmiths, fences, and other precautions to protect the 
public from possible injury, cost of casualty insurance for 
workmen, etc. Although the cost of some of these sub-items 
may be definitely estimated, others are so uncertain that it is 
only possible to make a lump estimate and add say 5 to 7% 
of the sum of the previous items for this item. 



182 RAILROAD CONSTRUCTION. | 147. 

147. Contractor's profit and contingencies. The word '' con- 
tingencies" here refers to the abnormal expenses caused by 
freshets, continued wet weather, and "hard luck," as dis- 
tinguished from mere incidentals which are really normal 
expenses. They are the expenses which literally cannot be 
foreseen, and on which the contractor must "take chances." 
They are therefore included with the expected profit. The 
allowance for these two elements combined is variously esti- 
mated up to 25% of the previously estimated cost of the work,' 
according to the sharpness of the competition, the contractor's 
confidence in the accuracy of his estimates, and the possible un- 
certainty as to true cost owing to unfavorable circumstances. 
The contractor's real profit may vary considerably from this. 
He often pays clerks, boards and lodges the laborers in shan- 
ties built for the purpose, or keeps a supply-store, and has 
various other items both of profit and exjoense. His profit 
is largely dependent on skill in so handling the men. that all 
can work effectively without interference or delays in wait- 
ing for others. An unusual season of bad weather will often 
affect the cost very seriously. It is a common occurrence 
to find that two contractors may be working on the same kind 
of material and under precisely similar conditions and at the 
same price, and yet one may be making money and the 
other losing it — all on account of difference of manage- 
ment. 

148. Limit of profitable haul. Ag intimated in §§ 134 and 
141, there is with every method of haul a limit of distance 
beyond which the expense for excessive hauling will exceed the 
loss resulting from borrowing and wasting. This distance is 
somewhat dependent on local conditions, thus requiring an inde- 
pendent solution for each particular case, but the general prin- 
ciples involved will be 'about as follows: Assume that it has been 
determined, as in Fig. 64, that the cut and fill will I3xactly bal- 
ance between two points, as between e and x, assuming that, as 
indicated in § 132 (9), a trestle has been introduced between s 
and t, thus altering the mass curve to Esixn . . . Since there 
is a balance between A' and C, the material for the fill between 
C and e' must be obtained either by " borrowing " in the im- 
mediate neighborhood or by transportation from the excavation 
between s' and n', If cut anci . ^U have been approximately 



§ 148. - EARTHWOKK. l83 

balanced in the selection of grade line, as is ordinarily done, 
borrowing material for the fill CV implies a wastage of material 
at the cut z'n' . To compare the two methods, we may place 
against the plan of borrowing and wasting, (a) cost, if any, of 
extra right of way that may be needed from which to obtain 
earth for the fill C'e'\ (6) cost of loosening, loading, hauling 
a distance equal to that between the centers of gravity of the 
borrow-pit and of the fill, and the other expenses incidental to 
borrowing M cubic yards for the -fill C'e'\ (c) cost of loosening, 
loading, hauling a distance equal to that between the centers 
of gravity of the cut z'n' and of the spoil-bank, and the other 
expenses incidental to wasting M cubic yards at the cut z'n': 
(d) cost, if any, of land needed for the spoil-bank. The cost of 
the other plan will be the cost of loosening, loading, hauling (the 
hauling being represented by the trapezoidal figure Cexn), and 
the other expenses incidental to making the fill C'e' with the 
material from the cut z'n' , the amount of material being M cubic 
yards, which is represented in the figure by the vertical ordi- 
nate from e to the line" Cn. The difference between these costs 
will be the cost, if any, of land for borrow-pit and spoil-bank 
'plus the cost of loosening, loading, etc. (except hauling and 
roadways) of M cubic yards, minus the difference in cost of the 
excessive haul from Ce to xn and the comparatively short hauls 
from borrow-pit and to spoil-bank. 

As an illustration, taking some of the estimates previously 
given for operating with average material, the cost of all items, 
except hauling and roadways, would be about as follows: 
loosening, with plough, 1.2 c, loading 5.0 c, spreading 1.5 c, 
wear, depreciation, etc.,- .25 c, superintendence, etc., 1.5 c; 
total 8.95 c. Suppose that the haul for both borrowing and 
wasting averages 100 feet or 1 station. Then the cost of haul 
per yard, using carts, would be (§140, a) [125X3(1 +4)]-^-600 
=3.125 c. The cost of roadways would be about 0.1 c. per yard, 
making a total of 3.225 c. per cubic yard. Assume ilf = 10000 
cubic yards and the area Cexn = 1^0000 yards-stations or. the 
equivalent of 10000 yards hauled 1800 feet. This haul would 
cost [125X3(18 + 4)]^600 = 13.75 c. per cubic yard. The cost 
of roadways will be 18 X .1 or 1.8 c, making a total of 15.55 c. for 
hauling and roadways. The difference of cost of hauling and 
roadways will be 15.55-(2X3.225) =9.10 c, per yard or $91Q 



Borrowing 

AND 

Wasting. 





$2450. 




8.95 G. 


$895. 




8.95 c. 


895. 




3.225 


c. 322. 


50 


3.225 


c. 322. 


50 




$2435.00 
■ - -g'j 



184 EAILROAD CONSTEUCTION. § 149. 

for the 10000 yards. Offsetting this is the cost of loosening, etc., 
10000 yards, at 8.95 c, costing $895, These figures may be 
better compared as follows: 

f Loosening, etc., 10000 yards, @, 8.95 c. $ 895. 

T TT Hauling, " 10000 ' " @, 15.55 c. 1555. 

LongHadl. i 

[ 

Loosening, etc., 10000 yards (borrowed), 

" 10000 " (wasted). 
Hauling, etc., 10000 " (borrowed), 
10000 " (wasted), 

L 

These costs are practically balanced, but no allowance has 
been made for right of way. If any considerable amount had 
to be paid for that, it would decide this particular case in favor 
of the long haul. This shows that under these conditions 1800 
feet is about the limit of profitable haul, the land costing nothing 
extra. 

BLASTING. 

149. Explosives, The effect of blasting is due to the ex- 
tremely rapid expansion of a gas which is developed by the 
decomposition of a very small amount of solid matter. Blasting 
compounds may be divided into two general classes, (a) slow- 
burning and (b) detonating. Gunpowder is a type of the slow- 
burning compounds. These are generally ignited by heat; the 
ignition proceeds from, grain to grain; the heat and pressure 
produced are comparatively low. Nitro-glycerine is a type of 
the detonating compounds. They are exploded by a shock 
which instantaneously explodes the whole mass. The heat and 
pressure developed are far in excess of that produced by the 
explosion of powder. Nitro-glycerine is so easily exploded 
that it is very dangerous to handle. It was discovered that if 
the nitro-glycerine was absorbed by a spongy material like infu- 
sorial earth, it was much less liable to explode, while its power 
when actually exploded was practically equal to that of the 
amount of pure nitro-glycerine contained in the dynamite, which 
is the name given to the mixture of nitro-glycerine and infusorial 
earth. Nitro-glycerine is expensive; many other explosive 
chemical compounds which properly belong to the slow-burning 



§ 150. EARTHWORK. 185 

class are comparatively cheap. It has been conclusively demon- 
strated that a mixture of nitro-glycerine and some of the cheaper 
chemicals has a greater explosive force than the sum of the 
strengths of the component parts when exploded separately, 
Whatever the reason, the fact seems established. The reason is 
possibly that the explosion of the nitro-glycerine is sufficiently 
powerful to produce a detonation of the other chemicals, which 
is impossible to produce by ordinary means, and that this explo- 
sion caused by detonation is more powerful than^an ordinary 
explosion. The majority of the explosive compounds and 
"powders" on the market are of this character — a mixture of 
20 to 60 per cent, of nitro-glycerine with variable proportions of 
one or more of a great variety of explosive chemicals. 

The choice of the explosive depends on the character of the 
rock. A hard brittle rock is most effectively blasted by a 
detonating compound. The rapidity with which the full force 
of the explosive is developed has a shattering effect on a brittle 
substance. On the contrary, some of the softer tougher rocks 
and indurated clays are but little affected by dynamite. The 
result is but little more than an enlargement of the blast-hole. 
Quarrying must generally be done with blasting-powder, as the 
quicker explosives are too shattering. Although the results 
obtained by various experimenters are very variable, it may be 
said that pure nitro-glycerine is eight times as powerful as black 
powder, dynamite (75% nitro-glycerine) six times, and gun- 
cotton four to six times as powerful. For open work where 
time is not particularly valuable, black powder is by far the 
cheapest, but in tunnel-headings, whose progress determines the 
progress of the whole work, dynamite is so much more effective 
and so expedites the work that its use becomes economical. 

150. Prilling. Although many very compHcated forms of 
drill-bars have been devised, the best form (with slight modifi- 
cations to suit circumstances) is as shown in Fig. 66 (a), and (6). 
The width should flare at the bottom (a) about 15 to 30%. For 
hard rock the curve of the edge should be somewhat flatter and 
for soft rock somewhat more curved than shown, Fig. 66, {a). 
Sometimes the angle of the two faces is varied from that given. 
Fig. 66, (6) and occasionally the edge is purposely blunted so 
as to give a crushing rather than a cutting effect. The drills 
will require sharpening for each 6 to 18 inches depth of hole, 

and will require a new edge to be worked every 2 to 4 days. 



186 



RAILROAD CONSTRUCTION. 



§151. 



For drilling vertical holes the churn-drill is the most econom- 
ical. The drill-bar is of iron, about 6 to 8 feet long, 1^" in 
diameter, weighs about 25 to 30 lbs., and is shod with a piece 
of steel welded on. The bar is lifted a few inches between each 
blow, turned partially around, and allowed to fall, the impact 
doing the work. From 5 to 15 feet of holes, depending on the 
character of the rock, is a fair day's work — 10 hours. In very 
Boft rocks even more than this may be done. This method is 




Fig. 66. 



inapplicable for inclined holes or even for vertical holes in con- 
fined places, such as tunnel-headings. For such places the only 
practical hand method is to use hammers. This may be done 
by light drills and light hammers (one-man work), or by heavier 
drills held by one man and struck by one or two men with heavy 
hammers. The conclusion of an exhaustive investigation as to 
the relative economy of light or heavy hammers is that the light- 
hammer method is more economical for the softer rocks, the 
heavy-hammer method is more economical for the harder rocks, 
but that the light-hammer method is always more expeditious 
and hence to be preferred when time is important. ' 

The subject of machine rock-drills is too vast to be treated 
here. The method is only practicable when the amount of 
work to be done is large, and especially when time is valuable. 
The machines are generally operated by compressed air although 
steam is also used to operate the drills. Gasoline as a motive 
power is even more economical for a small-scale plant. The 
cost per foot of hole drilled is quite variable, but is usually 
somewhat less than that of hand-drilling — sometimes but a small 
fraction of it. 

151. Position and direction of drill-holes. As the cost of drill- 
ing holes is the largest single item in the total cost of blasting, 
it is necessary that skill and judgment should be used in so 



§152. 



EARTHWORK. 



187 



locating the holes that the blasts will be most effective. The 
greatest effect of a blast will evidently be in the direction of the 
''line of least resistance." In a strictly homogeneous material 
this will be the shortest line from the center of the explosive to 
the surface. The variations in homogeneity on account of 
laminations and seams require that each case shall be judged 
according to experience. In open-pit blasting it is generally 
easy to obtain two and sometimes three exposed faces to the 

rock, making it a simple matter 
to drill holes so that a blast will 
do effective work. When a solid 
face of rock must be broken into^ 
^ as in a tunnel-heading, the work 
is necessarily ineffectual and ex- 
pensive. A conical or wedge- 
shaped mass will first be blown 
out by simultaneous blasts in 
the holes marked 1, Fig. 67; 
blasts in the holes marked 2 and 
3 will then complete the cross- 
section of the heading. A great saving in cost may often be 
secured by skilfully taking advantage of seams, breaks, and irreg- 
ularities. When the work is economically done there is but little 
noise or throwing of rock, a covering of old timbers and branches 
of trees generally sufficing to confine the smaller pieces which 
would otherwise fly up. 

152. Amount of explosive. The amount of explosive required 
varies as the cube of the line of least resistance. The best 
results are obtained when the line of least resistance is | of the 
depth of the hole ; also when the powder fills about ^ of the hole. 
For average rock the amount of powder required is as follows : 




DRILL HOLES IN TUNNEL HEADING 
Fig. 67. 



Line of least resistance 


2 ft. 
1 lb. 


4 ft. 
2 lbs. 


6 ft. 
6i lbs. 


8 ft. 


Weight of powder 


16 lbs. 







Strict compliance with all of the above conditions would re- 
quire that the diameter of the hole should vary for every case. 
While this is impracticable, there should evidently be some 
variation in the size of the hole, depending on the work to be 
done. For example, a 1" hole, drilled 2' 8" deep, with its 
line of least resistance 2'. and loaded with ^ lb. of powder, would 



i^^ KAILROAD CONSTRUCTION. ""' § 153. 

be filled to a depth of 9^'', which is nearly | of the depth. A 
3" hole, drilled 8' deep, with its line of least resistance 6'y and 
loaded with 6| lbs. of powder, would be filled to a depth of over 
28", which is also nearly ^ of the depth. One pound of blasting- 
powder will occupy about 28 cubic inches. Quarrying necessi- 
tates the use of numerous and sometimes repeated light charges of 
powder, as a heavy blast or a powei'ful explosive like dynamite 
is apt to shatter the rock. This requires more powder to thte 
cubic yard than blasting for mere excavation, which may usually 
be done by the use of ^ to |^ lb. of powder per cubic yard of easy 
open blasting. On account of the great resistance offered by 
rock when blasted in headings in tunnels, the powder used per 
cubic yard will run up to 2, 4, and even 6 lbs. per cubic yard. 
As before stated, nitro-glycerine is about eight times (and 
dynamite about six times) as powerful as the same weight of 
powder. 

153. Tamping. Blasting -powder and the slow-burning ex- 
plosives require thorough tamping. Clay is probably the best, 
but sand and fine powdered rock are also used. Wooden plugs, 
inverted expansive cones, etc., are periodically reinvented by 
enthusiastic inventors, only to be discarded for the simpler 
methods. Owing to the extreme rapidity of the development 
of the force of a nitro-glycerine or dynamite explosion, tamping 
is not so essential with these explosives, although it unquestion- 
ably adds to their effectiveness. Blasting under water has been 
effectively accomplished by merely pouring nitro-glycerine into 
the drilled holes through a tube and then exploding the charge 
without any tamping except that furnished by the supermcum- 
bent water. It has been found that air-spaces about a charge 
make a material reduction in the effectiveness of the explosion. 
It is therefore necessary to carefully ram the explosive into a 
solid mass. Of course the liquid nitro-glycerine needs no ram- 
ming, but dynamite should be rammed with a wooden rammer. 
Iron should be carefully avoided in ramming gunpowder. A 
copper bar is generally used. 

154. Exploding the charge. Black powder is generally ex- 
ploded by means of a fuse which is essentially a cord in which 
there is a thin vein of gunpowder, the cord being protected by 
tar, extra linings of hemp, cotton, or even gutta-percha. The 
fuse is inserted into the middle of the charge, and the tamping 
barefully packed around it so that it will not be injured. To 



I 155. EARTHWORK. 189 

produce the detonation required to explode nitro-glycerine and 
dynamite, there must be an initial explosion of some easily 
ignited explosive. This is generally accomphshed by means of 
caps containing fulminating-powder which are exploded by 
felectricity. The electricity (in One class of caps) heats a very 
fine platinum wire to redness, thereby igniting the sensitive 
powder, or (in another class) a spark is caused to jump through 
the powder between the ends of two wires suitably separated. 
Dynamite can also be exploded by using a small cartridge of 
gunpowder which is itself exploded by an ordinary fuse, 

155. Cost. As a rough estimate, the cost of loosening and load- 
ing rock work, reduced to the uniform basis of $1.00 per 10-hour 
day, may be said to vary from 30c. for easy but brittle rock and 
increasing to 80c. per cubic yard when the cutting is .sballow, 
the rock especially tough, and the strata unfavorably placed. 
For a detailed analysis of cost, which is essential for close 
estimating, see Gillette's "Rock Excavation, Methods and Cost." 

156. Classification of excavated material. The classification 
of excavated material is a fruitful source of dispute between 
contractors and railroad companies, owing mainly to the fact 
that the variation between the softest earth and the hardest rock 
is so gradual that it is very difficult to describe distinctions 
between different classifications which are unmistakable and 
ihdisputable The classification frequently used is (a) earth, 
(6) loose rock, and (c) solid rock. As blasting is frequently 
used to Joosen "loose rock" and even "earth" (if it is frozen), 
the fact that blasting is employed cannot be used as a criterion, 
Especially as this would (if allowed) lead to unnecessary blasting 
for the sake of classifying material as rock. 

Earth. This includes clay, sand, gravel, loam, decomposed 
rock and slate, boulders or loose stones not greater than 1 cubic 
foot (3 cubic feet, P. R. R.), and sometimes even "hard-pan." 
In general it will signify material which can be loosened by a 
plough with two horses, oi* with which one picker can keep onse 
shoveller busy. 

Loose rock. This includes boulders and loose stones of more 
than one cubic foot and less than one cubic yard; stratified rock, 
not more than six inches thick, separated by a stratum of clay; 
also all material (not classified as earth) which may be loosened 
h'y pick or bar and which "can be quarried without blasting, 
although blasting may occasionally be resorted to/' 



190 RAILROAD CONSTRUCTION. § 157. 

Solid rock includes all rock found in masses of over one cubic 
yard which cannot be removed except by blasting. 

It is generally specified that the engineer of the railroad , 
company shall be the judge of the classification of the material, i 
but frequently an appeal is taken from his decisions to the . 
courts. 

157. Specifications for earthwork. The following specifica- 
tions, issued by the Norfolk and Western R R., represent the 
average requirements. It should be remembered that very 
strict specifications invariably increase the cost of the work, 
and frequently add to the cost more than is gained by improved 
quality of work. 

1. The grading will be estimated and paid for by the cubic 
yard, and will include clearing and grubbing, and all open ex- 
cavations, channels, and embankments required for the forma- 
tion of the roadbed, and for turnouts and sidings; cutting all 
ditches or drains about or contiguous to the road; digging the 
foundation-pits of all culverts, bridges, or walls; reconstructing 
turnpikes or common roads in cases where they are destroyed or 
interfered with; changing the course or channel of streams; and 
all other excavations or embankments connected with or incident 
to the construction of said Railroad. 

2. All grading, except where otherwise specified, whether 
for cuts or fills, will be measured in the excavations and will be 
classified under the following heads, viz.: Solid Rock, Loose 
Rock, Hard-pan, and Earth. 

Solid Rock shall include all rock occurring in masses which, 
in the judgment of the said Engineer Maintenance of Way, may 
be best removed by blasting. 

Loose Rock shall include all kinds of shale, soapstone, and 
other rock which, in the judgment of the said Engineer Main- 
tenance of Way, can be removed by pick and bar, and is soft and 
loose enough to be removed without blasting, although blasting 
may be occasionally resorted to ; also, detached stone of less than 
one (1) cubic yard and more than one (1) cubic foot. 

Hard-pan shall consist of tough indurated clay or cemented 
gravel, which requires blasting or other equally expensive 
means for its removal, or which cannot be ploughed with less 
than four horses and a railroad plough, or which requires two 
pickers to a shoveller, the said Engineer Maintenance of Way 
to be the judge of these conditions. 



§ 157. EARTHWORK. 191 

Earth shall include all material of an earthy nature, of what- 
ever name or character, not unquestionably loose rock or hard- 
pan as above defined. 

Powder. The use of powder in cuts will not be considered 
as a reason for any other classification than earth, unless the 
material in the cut is clearly other than earth under the above 
specifications. 

3. Earth, gravel, and other materials taken from the exca- 
vations, except when otherwise directed by the said Engineer 
Maintenance of Way or his assistant, shall be deposited in the 
adjacent embankment; the cost of removing and depositing 
which, when the distance necessary to be hauled is not more 
than sixteen hundred (1600) feet, shall be included in the price 
paid for the excavation. 

4. Extra Haul will be estimated and paid for as follows: 
whenever material from excavations is necessarily hauled a 
greater distance than sixteen hundred (1600) feet, there shall be 
paid in addition to the price of excavation the price of extra 
haul per 100 feet, or part thereof, after the first 1600 feet; the 
necessary haul to be determined in each case by the said Engi- 
neer Maintenance of Way or his assistant, from the profile and 
cross-sections, and the estimates to be in accordance therewith. 

5. All embankments shall be made in layers of such thick- 
ness and carried on in such manner as the said Engineer Mainte- 
nance of Way or his assistant may prescribe, the stone and heavy 
materials being placed in slopes and top. And in completing 
the fills to the proper grade such additional heights and fulness 
of slope shall be given them, to provide for their settlement, as 
the said Engineer Maintenance of Way, or his assistant, may 
direct. Embankments about masonry shall be built at such 
times and in such manner and of such materials as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

6. In procuring materials for embankments from without 
the line of the road, and in wasting materials from cuttings, the 
place and manner of doing it shall in each case be indicated by 
the Engineer Maintenance of Way or his assistant; and care 
must be taken to injure or disfigure the land as little as possible. 
Borrow-pits and spoil-banks must be left by the Contractor in 
regular and sightly shape. 

7. The lands of the said Railroad Company shall be cleared 
to the extent required by the said Engineer Maintenance of 



193 KAILROAD CONSTRUCTION. § 157. 

Way, or his assistant, of all trees, brushes, logs, and other perish- 
able materials, which shall be destroyed by burning or deposited 
in heaps as the said Engineer Maintenance of Way, or his assist^- 
ant, may direct, Large trees must be cut not more than two; 
ftiid one-half (2|) feet from the ground, and under embanks 
ments less than four (4) feet high they shall be cut close to the 
ground. All small trees and bushes shall be cut close to the 
ground. 

8. Clearing shall be estimated and paid for by the acre or: 
fraction of an acre. 

9. All stumps, roots, logs, and other obstructions shall be 
grubbed out, and removed from all places where embankments 
occur less than two (2) feet in height ; also, from all places where 
excavations occur and from such other places as the said Engi^ 
neer Maintenance of Way or his assistant may direct. 

10. Grubbing shall be estimated and paid for by the acre or 
fraction of an acre. 

11. Contractors, when directed by the said Engineer Main?- 
tenance of Way or his assistant in charge of the work, will deposit 
on the side of the road, or at such convenient points as may be 
designated, any stone, rock, or other materials that they may- 
excavate; and all materials excavated and deposited as above, 
together with all timber removed from the line of the road, will 
be considered the property of the Railroad Company, and the 
Contractors upon the respective sections will be responsible for 
its safe-keeping until removed by said Railroad Company, or 
until their work is finished. 

12. Contractors will be accountable for the maintenance of 
<safe and convenient places wherever public or private roads are 
in any way interfered with by them during the progress of th^ 
work. They will also be responsible for fences thrown down, 
and for gates and bars left open, and for all damages occasioned 
thereby. 

13. Temporary bridges and trestles, erected to facilitate the 
progress of the work, in case of delays at masonry structures 
from any cause, or for other reasons, will be at the expense of 
the Contractor. 

14. The line of road or the gradients may be changed in any 
manner, and at any time, if the said Engineer Maintenance of 
Way or his assistant shall consider such a change necessary or 
expedient; but no claim for an increase in prices of excavation 



§ 157. EARTHWORK. 193 

or embankment on the part of the Contractor will be allowed 
or considered unless made in writing before the work on that 
part of the section where the alteration has been made shall have 
been commenced. The said Engineer Maintenance of Way or 
his assistant may also, on the conditions last recited, increase or 
diminish the length of any section for the purpose of more nearly 
equalizing or balancing the excavations and embankments, or 
for any other reason, 

15. The roadbed will be graded as directed by the said En- 
gineer Maintenance of Way or his assistant, and in conformity 
with such breadths, depths, and slopes of cutting and filling as 
he may prescribe from time to time, and no part of the work 
will be finally accepted until it is properly completed and dressed 
off at the required grade. 



CHAPTER IV. : 

TRESTLES. 

158. Extent of use. Trestles constitute from 1 to 3% of the 
length of the average railroad. It was estimated in 1889 that 
there was then about 2400 miles of single-track railway trestle 
in the United States, divided among 150,000 structures and esti-;. 
mated to cost about $75,000,000. The annual charge for main-^ 
tenance, estimated at | of the cost, therefore amounted to about 
$9,500,000 and necessitated the annual use of perhaps 300,000,000 
ft. B. M. of timber. The corresponding figures at the present 
time must be somewhat in excess of this. The magnitude of 
this use, which is causing the rapid disappearance of forests, has 
resulted in endeavors to limit the use of timber for this purpose. 
Trestles may be considered as justifiable under the following 
conditions : 

a. Permanent, trestles. 

' 1. Those of extreme height — then called viaducts and 
frequently constructed of steel, as the Kinzua viaduct, 302 
feet high. 

2. Those across wide shallow waterways — e.g., that across 
Lake Pontchartrain, near New Orleans, 22 miles long. 

3. Those across swamps of soft deep mud, or across a river- 
bottom, liable to occasional overflow. 

b. Temporary trestles. 

1. To open the road for traffic as quickly as possible — often 
a reason of great financial importance. 

2. To quickly replace a more elaborate structure, destroyed 
by accident, on a road already in operation, so that the inter- 
ruption to traffic shall be a minimum. 

3. To form an earth embankment with earth brought from 
a distant point by the train-load, when such a measure would 
cost less than to borrow earth in the immediate neighborhood. 

4. To bridge an opening temporarily and thus allow time to 
learn the regimen of a stream in order to better proportion the 

194 



§ 159. TRESTLElS. 195 

size of tlie waterway and also to facilitate bringing suitable stone 
for masonry from a distance. In a new country there is always 
the double danger of either building a culvert too small, requir- 
ing expensive reconstruction, perhaps after a disastrous washout, 
or else wasting money by constructing the culvert unnecessarily 
large. Much masonry has been built of a very poor quality of 
stone because it could be conveniently obtained and because 
good stone was unobtainable except at a prohibitive cost for 
[transportation. Opening the road for traffic by the use of 
[temporary trestles obviates both of these difficulties. 

159. Trestles vs. embankments. Low embankments are very- 
much cheaper than low trestles both in first cost and mainte- 
inance. Very high embankments are very expensive to con- 
: struct, but cost comparatively little to maintain. A trestle of 
equal height may cost much less to construct, but will be expen- 
sive to maintain— perhaps | of its cost per year. To determine 
the height beyond which it will be cheaper to maintain a trestle 
rather than build an embankment, it will be necessary to allow 
for the cost of maintenance. The height will also depend on 
the relative cost of timber, labor, and earthwork. At the pres- 
ent average values, it will be found that for less heights than 
25 feet the first cost of an embankment will generally be less 
than that of a trestle; this implies that a permanent trestle 
should never be constructed with a height less than 25 feet except 
for the reasons given in § 158. The height at which a permanent 
trestle is certainly cheaper than earthwork is more uncertain. 
A high grade line joining two hills will invariably imply at least 
a culvert if an embankment is used. If the culvert is built of 
masonry, the cost of the embankment will be so increased that 
the height at which a trestle becomes economical will be mate- 
rially reduced. The cost of an embankment increases much 
more rapidly than the height — with very high embankments 
more nearly as the square of the height — while the cost of 
trestles does not increase as rapidly as the height. Although 
local circumstances may modify the application of any set rules, 
it is probably seldom that it will be cheaper to build an embank- 
ment 40 or 50 feet high than to permanently maintain a wooden 
trestle of that height. A steel viaduct would probably be the 
best solution of such a case. These are frequently used for 
permanent structures, especially when very high. The cost of 
maintenance is much less than that of wood, which makes the 



VQQ railroad CONSTRUCTION". § 160. 

use of steel preferable for permanent trestles unless wood fe 
abnormally cheap. Neither the cost nor the construction of 
steel trestles will be considered in this chapter. 

1 60. Two principal types. Tl.ere are two principal types or;' 
wooden trestles — pile trestles and framed trestles. The great' 
objection to pile trestles is the rapid rotting of the portion of the ■ 
pile which is underground, and the difficulty of renewal. The 
nlaximum height of pile trestles is about 30 feet, and even this i 
height is seldom reached. Framed trestles have been con- 
structed to a height of considerably over 100 feet They are : 
frequently built in such a manner that any injured piece may be ; 
readily taken out and renewed without interfering with traffic. 
Trestles consist of two parts — the supports called "bents," and 
the stringers and floor system. As the stringers and floor system . 
are the same for both pile and framed trestles, the " bents " arfe 
all that need be considered separately. 



PILE TRESTLES. 

161. Pile bents. A pile bent consists generally of four piles 
driven, into the ground deep enough to afford not only sufficient 
vertical resistance but also lateral resistance. On top. of these 
piles is placed a horizontal "cap." The caps are fastened to 
the tops of the piles by methods illustrated in Fig. 68. The 
method of fastening shown in each case should not be considered 
as applicable only to the particular type of pile bent used to illus- 
trate it. Fig. 68 (a and d) illustrates a mortise-joint with a hard- 
wood pin about ly in diameter. The hole for the pin should 
be bored separately through the cap and the mortise, and the 
holfe through the cap should be at a slightly higher level than 
that through the mortise, so that the cap will be drawn down 
tight when the pin is driven. Occasionally iron dowels (an 
iron pin about 1|" in diameter and about 8" long) are inserted 
partly in the cap and partly in the pile. The use of drift -bolts, 
shown in Fig. 68 (6), is cheaper in first cost, but renders repairs 
and renewals very troublesome and expensive. " Split caps," 
shown in Fig. 68 (c), are formed by bolting two half -size strips 
on each side of a. tenon on top of the pile. Repairs are very 
easily and cheaply made without interference with the traffic 
and without injuring other pieces of the bent. The smaller 
pieces are more easily obtainable in a sound condition; the 



§ 161. 



TRESTLESo 



197 



decay of one does not affect the other, and the first cost is but 
Httle if any greater than the method of using a single piece. For 
further discussion, see § 170. 

For very hght traffic and for a height of about 5 feet three 
vertical piles will suffice, as shown in Fig. 68 (a). Up to a height 




\wi iW 






Fig. 68. 



*m !w^ 



of 8 or 10 feet four piles may be used without sway-bracing, as 
in Fig. 68 (6), if the piles have a good bearing. For heights 
greater than 10 feet sway-bracing is generally necessary. The 
outside piles are frequently driven with a batter varying from 
1 : 12 to 1 : 4. 

Piles are made, if possible, from timber obtained in tho 
vicinity of the work. Durability is the great requisite rather 
than strength, for almost any timber is strong enough (except 
as noted below) and will be suitable if it will resist rapid decay. 
The following list is quoted as being in the order of preference 
on account of durability- 



1. 


Red cedar 


5. 


White pine 


9. 


White oak 


12. 


Black oak 


2. 


Red cypress " 


6. 


Redwood 


10. 


Post-oak 


13. 


Hemlock 


3. 


Pitch-pine 


7. 


Elm 


11. 


Red. oak 


14. 


Tamarac 


4. 


Yellow pine 


8. 


Spruce 











Red-cedar piles are said to have an average life of 27 years 
with a possible maximum of 50 years, but the timber is rather 



198 KAILROAD CONSTRUCTION. § 162. 

weak, and if exposed in a river to flowing ice or driftwood is 
apt to be injured. Under these circumstances oak is prefer- 
able, although its life may be only 13 to 18 years. 

162. Methods of driving piles. The following are the prin-s 
cipal methods of driving piles: . 

a. A hammer weighing 2000 to 3000 lbs. or more, sliding 
in guides, is drawn up by horse-power or a portable engine, and 
the " nippers " or " tongs," which hold the hammer, are released 
by a light trip-rope, which permits the hammer to fall freely. 

b. The drum of a steam hoisting engine is gripped and released 
by some form of clutch. When the hammer has been raised 
and the clutch released, the hammer falls, dragging the rope 
and turning the drum. The energy of the blow is thus some- 
what reduced, falsely increasing the apparent resistance. But 
the hammer works much faster, the number of blows per minute 
varying from 12 to 25, depending on the height of fall. The 
mechanism for both of these methods is comparatively simple 
and inexpensive, and can be easily transported into a new 
country. 

c. Steam pile-drivers. The hammer weighs 3000 to 5000 lbs., 
and has a movement of 36 to 40 inches, striking 60 to 80 blows 
per minute. The ram is raised by steam pressure. The older 
types are single-acting the ram falling by gravity. Some later 
types are double-acting, the ram being forced down by steam 
pressure, which increases both the force and the rapidity of the 
blows. Very rapid blows, which do not allow time for the soil 
to settle around the pile between consecutive blows, are more 
effective and encounter less resistance. The destructive impact 
of a weight of 5000 lbs. falling only 3 feet is but a small part of 
that of 3000 lbs. falling 20 feet and there is less danger of over- 
driving and rupturing the pile. 

d. Water jet. Whenever a sufficient supply of water is avail- 
able, and especially when the soil is sandy, pile driving is facilitated 
by forcing water through a pipe driven into the ground near the 
desired location of the pile. Two or even three pipes per pile 
may he used. The former practice was to attach the pipes to 
the pile, but the pipes were often broken when withdrawn, and 
present practice keeps the jet independent of the pile, churning 
it up and down near the pile point by means of a rope running 
through a block on the driver leads and leading to a hand-winch 
or to a nigger-head on the engine. When the soil is very soft, 



§ 163. TEESTLES. 190 

piles may be sunk, using the jet only, or with the aid of weights 
loaded on the pile, but a hammer is essential for harder ground, 
especially for driving the last few blows, the penetration of which 
will give a measure of the resisting power of the pile — ^see § 163. 
Although the jet has been employed using a hand-pump, effective 
work requires the use of a power pump, with a 2" pipe for the 
jet, a pressure up to a maximum of about 200 lbs per square inch, 
and a flow of 250 to 500 gallons per minute. Many other details 
regarding pile driving are given in § 167. 

Excessive driving frequently fractures the pile below the 
surface and thereby greatly weakens its bearing power. To 
prevent excessive "brooming" of the top of the 
pile, owing to the action of the hammer, the top 
should be protected by an iron ring fitted to the 
top of the pile. The "brooming" not only ren- 
ders the driving ineffective and hence uneconomi- 
cal, but vitiates the value of any test of the bearing 
power of the pile by noting the sinking due to a 
given weight falling a given distance. If the pile 
is so soft that brooming is unavoidable, the top 
iG. b9. ghould be adzed off frequently, and especially 
should it be done just before the final blows which are to test its 
bearing-power. 

In a new country judgment and experience will be required 
to decide intelligently whether to employ a simple drop-hammer 
machine, operated by horse-power and easily transported but 
uneconomical in operation, or a more complicated machine 
working cheaply and effectively after being transported at 
greater expense. 

163. Pile- driving formulae. If i2=the resistance of a pile, 
and s the set of the pile during the last blow, w the weight of 
the pile-hammer, and h the fall during the last blow, then we 

may state the approximate relation that Rs=wh, or i2 = — -. 

This is the basic principle of all rational formulae, but the maxi- 
mum weight which a pile will sustain after it has been driven 
some time is by no means the same as the resistance of the pile 
during the last blow. There are also many other modifying 
elements which have been variously allowed for in the many 
proposed formulae. The formulae range from the extreme of 
empirical simplicity to very complicated attempts to allow 




200 RAILROAD CONSTRUCTION. § 163, | 

properly for all modifying causes. As the simplest rule, the ! 

A. il. E. A. specifications require that the piles shall be driven ' 

until the pile will not sink more than 2^ inches under five consecu- 1 

tive blows of a 3000-lb hammer falling 15 feet. The "Engineering 

2ivh . 
News formula " * gives the safe load as — — , m which w = ' 

s + 1 

weight of hammer, A = fall in feet, s=set of pile in inches under { 

the last blow. This formula is derived from the above basic | 

formula by calling the safe load | of the final resistance, and j 

by adding (arbitrarily) 1 to the final set (s) as a compensation \ 

for the extra resistance caused by the settling of earth around 

the pile between each blow. This formula is used only for 

ordinary hammer-driving. When the piles are driven by a 

2wh 

steam pile-driver the formula becomes safe load = . In the 

s+0.1 

last formula the constant in the denominator is changed from 

s + 1 to 5+0.1. The constant (1.0 or 0.1) is supposed to allow, 

as before- stated, for the effect of the extra resistance caused by 

the earth settling around the pile between each blow. The 

more rapid the blows the less the opportunity to settle and the 

less the proper value of the constant. 

The above formulae have been given on account of their 
simplicity and their practical agreement with experience. Many 
other formulae have been proposed, the majority of which are 
more complicated and attempt to take into account the weight of 
the pile, resistance of the guides, etc. While these elements, 
as well as many others, have their influence, their effect is so 
overshadowed by the indeterminable effect of other elements — 
as, for example, the effect of the settlement of earth around the 
pile between blows — that it is useless to attempt to employ any- 
thing but a purely empirical formula. 

For the most careful work, dependence is placed on the 
actual load which may be carried, without yielding, by test 
piles, driven on the site of the work. In § 167, par. 16 -20, 
some Am. Rwy. Eng. Assoc, rules are quoted regarding the use 
of test piles. 

Examples. 1. A pile was driven with an ordinary hammer 
weighing 2500 pounds until the sinking under five consecutive 
blows was 15| inches. The fall of the hammer during the last 

* Engineering News, Nov. 17, 1892. 



§164. 



TBESTLES. 



201 



blows was 24 feet. What was the safe bearing power of the 
pile? 



2wh 2X2500X24 120000 
s + l = (|Xl5.5) + l~ 4.1 



=29300 pounds. 



2. Piles are being driven into a firm soil with a steam pile- 
driver until they show a safe bearing power of 20 tons. The 
hammer weighs 5500 pounds and its fall is 40 inches. What 
should be the sinking under the final blow? 



40000 



2wh 2X5500X3.33 



s + 0.1 

^ 36667 
^40000 



s + 0.1 
0.1 = .81 inch. 



164. Pile-points and pile-shoeSi Piles are generally sharpened 
to a blunt point. If the pile is liable to strike boulders, sunken 

logs, or other obstructions which are 
liable to turn the point, it ii necessary 
to protect the point by some form of 
shoe. Several forms in cast iron have 
been used, also a wrought-iroli shoe, 
having four "straps" radiating from 
the apex, the straps being nailed on td 
the pile, as shown in Fig. 70 (6). The 
cast-iron form shown in Fig. 70 (u) 
has a base cast aroUnd a drift-bolt. 
The recess on the top of the base re- 
FiG. 70. ceives the bottom of the pile and pre- 

vents a tendency to split the bottom of the pile or to force the 
shoe off laterally. See § 167, par. 23. 

165. Details of design. No theoretical calculations of the 
strength of pile bents need be attempted on account of the ex- 
treme complication of the theoretical strains, the uncertainty as 
to the real strength of the timber ijsed, the variability of that 
strength with time, and the insignificance of the economy that 
would be possible even if exact sizes could be computed. The 
caps are generally 14 feet long (for single track) with a cross- 
section 12"X12" or 12"X14". " Spht caps" would consist 




202 BAILEOAD CONSTRUCTION. § 166. 

of two pieces 6"X12". The sway-braces, never used for less 
heights than 6', are made of 3"X12" timber, and are spiked on 
with I" spikes 8" long. The floor system will be the same as 
that described later for framed trestles. 

i66. Specifications for timber piles (Adopted 1909 by 
Amer. Rwy. Eng. Assoc). 1. This grade [railroad heart grade] 
includes white, burr, and post oak; longleaf pine, Douglas 
fir, tamarack, Eastern white and red cedar, chestnut. Western 
cedar, redwood and cypress. 2. Piles shall be cut from sound 
trees j shall be close-grained and solid, free from defects, such as 
injurious ring shakes, large and unsound or loose knots, decay 
or other defects, which may materially impair their strength or 
durability. In Eastern red or white cedar a small amount of 
heart rot at the butt, which does not materially injure the 
strength of the pile, will be allowed. 3. Piles must be butt cut 
above the ground swell and have a uniform taper from butt to 
tip. Short bends will not be allowed. A line drawn from the 
center of the butt to the center of the tip shall lie within the body 
of the pile. 4. Unless otherwise allowed, piles must be cut when 
sap is down. Piles must be peeled soon after cutting. All 
knots shall be trimmed close to the body of the pile. 5. The 
minimum diameter at the tips of round piles shall be 9 inches 
for lengths not exceeding 30 feet; 8 inches for lengths over 30 
feet but not exceeding 50 feet, and 7 inches for lengths over 50 
feet The minimum diameter at one-quarter of the length from 
the butt shall be 12 inches and the maximum diameter at the 
butt 20 inches. 6. The minimum width of any side of the tip 
of a square pile shall be 9 inches for lengths not exceeding 30 feet; 
8 inches for lengths over 30 feet but not exceeding 50 feet and 7 
inches for lengths over 50 feet. The minimum width of any side 
at one-quarter of the length from the butt shall be 12 inches. 7. 
Square piles shall show at least 80% heart on each side at any 
cross-section of the stick, and all round piles shall show at least 
10| inches diameter of heart at the butt. 

The second grade (" Railroad falsework grade ") includes 
other woods which " will stand driving " and which ca!nnot pass 
the specification for proportion of heart; also, they are usually 
not peeled. 

167. Pile driving — principles of practice. As adopted by the 
Amer. Rwy. Eng. Assoc. 1911 and revised 1915. 

1. A thorough exploration of the soil by borings, or preliminary 



§ 167. TRESTLES. 203 

test piles, is the most important prerequisite to the design and 
construction of pile foundations. 

2. Soil consisting wholly or chiefly of sand is most favorable 
to the use of the water- jet. 

3. In harder soils containing gravel the use of the jet 
may be advantageous, if sufficient volume and pressure be 
provided. 

4. In clay it may be economical to bore several holes in the 
soil with the aid of the jet before driving the pile, thus securing 
the accurate location of the pile, and its lubrication while being 
driven. 

5. In general, the water- jet should not be attached to the pile, 
but handled separately. 

6. Two jets will often succeed where one fails. In special 
cases a third jet extending a part of the depth aids materially in 
keeping loose the material around the pile. 

7. Where the material is of such a porous character that the 
water from the jets may be dissipated and fail to come up in the 
immediate vicinity of the pile, the utility of the jet is uncertain, 
except for a part of the penetration. 

8. A steam or drop hammer should be used in connection 
with the water- jet, and used to test the final rate of penetra- 
tion. 

9. The use of the water jet is one of the most effective means 
of avoiding injury to piles by overdriving. 

10. There is danger from overdriving when the hammer 
begins to bounce. Overdriving is also indicated by the bending, 
kicking or staggering of the pile. 

11. The brooming of the head of the pile dissipates a part, 
and in some cases all, of the energy due to the fall of the 
hammer. 

12. The steam hammer is usually more effective than 
the drop hammer in securing the penetration of a wooden pile 
without injury, because of the shorter interval between 
blows. 

13. Where shock to surrounding material, is apt to prove 
detrimental to the structure, the steam hammer should always 
be used instead of the drop hamnjer. This is especially true 
in the case of sheet piling which is intended to prevent the 
passage of water. In some cases also the jet should not be 
used. 



204 EAILROAD CONSTBUCTION. § 168. 

14. In general, the resistance of piles, penetrating soft mate-, 
rial, depending solely upon skin friction, is naaterially increased 
after a period of rest. This period may be as short as fifteen 
minutes, and rarely exceeds twelve hours. 

15. Where a pile penetrates muck or a soft yielding material 
and bears upon a hard stratum at its foot, its strength should be , 
determined as a column or beam; omitting the resistance, if any, 
due to skin friction. 

16. Unless the record of previous experience at the same site 
is available, the approximate bearing power may be obtained 
by loading test piles. , The results of loading test piles should be 
used with caution, unless their condition is fairly comparable 
with that of the piles in the proposed foundation. 

17. In case the piles in a foundation are expected to act as 
columns, the results of loading test piles should not be depeAde4 
upon unless they are sufficient in number to insure their action in 
a similar manner; and unless they are stayed against lateral 
motion. 

18. Before testing the penetration of a pile in a soft material 
where its bearing power depends principally, or wholly, upor^ 
skin friction, the pile should be allowed to rest for 24 hours after 
driving. 

19. Where the resistance of piles depends mainly upon skiq 
friction it is possible to diminish the combined strength, or bear- 
ing capacity, of a group of piles, by driving additional pile^ 
within the same area. 

20. Where piles will foot in a hard stratum, investigation 
should be made to determine that this stratum is of sufficient 
depth and strength to carry the load, 

21. Timber piles may be advantageously pointed, in some 
cases, to a 4-inch or 6-inch square at the end. 

22. Piles should not be pointed when driven into soft 
material. 

23. Shoes should be provided for piles when the driving is 
very hard, especially in riprap or shale. These shoes should be 
so constructed as to form an integral part of the pile. 

24. The use of a' cap is advantageous in distributing the impact 
of the hammer more uniformly over the head of the pile, as well 
as in holding it in position during driving. 

1 68. Cost of pile trestles. The cost, per linear foot, of piling 
depends on the method of driving, the scarcity of suitable timber. 



§ 169. TRESTLES. 205 

the price of labor, the length of the piles, and the amount of 
shifting of the pile-driver required. The cost of soft-wood 
piles varies from 8 to 15 cents per lineal foot, and the cost of oak 
piles varies from 10 to 30 cents per foot, according to the length, 
the longer piles costing more per foot. The total cost of putting 
the piles in place is so dependent on other items than the cost of 
driving, such as the cost of shifting the driver, getting the piles 
into the leaders, straightening and bracing them, leveling and 
nailing guide strips for sawing them off^ and then the actual 
sawing, that there is a wide variation in the figures that 
are obtainable for the cost of such work. Of course the 
cost per pile of driving is also dependent on the total num- 
ber of piles in the job. The cost pei* pile of placing a dozen 
piles for a single foundation would be far greater than the 
cost per pile for several hundred piles in one job. Among a 
large number of obtainable figures the average figure of $1.54 
per pile for driving 1267 piles in 46 days is typical. Another 
quoted figure is $2.88 each, for driving 391 piles in 32 working 
days. On another job it cost $150 to drive thirty 30^foot piles, 
or an average of $5 each. In this case the piles cost $1.50 each 
or only 5 cents per lineal foot. The above cost figures are taken 
from Gillette's " Handbook of Cost Data " to which the student 
is referred for numerous examples of the cost of piles and pile- 
driving, as well as innumerable other cost analyses. 

Specifications generally say that the piling will be paid for 
per lineal foot of piling left in the work. The wastage of the tops 
of piles sawed off is always something, and is frequently very 
large. Sometimes a small amount per foot of piling saWed off is 
allowed the contractor as compensation for his loss. This 
reduces the contractor's risk and possibly reduces his bid by 
an equal or greater amount than the extfa amount actually 
paid him. 

FRAMED TRESTLES 

169. Typical design. A typical design for a framed trestle 
bent is given in Fig. 71. This represents, with slight variations 
of detail, the plan according to which a large part of the framed 
trestle bents of the country have been built— -i.e., of those less 
than 20 or 30 feet in height, not requiring multiple story con- 
struction. 

170. Joints, (a) The moftisfe-and-tenoii joint is illustrated in 



206 RAILROAD CONSTRUCTION. § 170. 

Fig. 71 and also in Fig. 68 (a). The tenon should be about 




Fig. 71. 



3" thick, 8' 



tt 



\:\sA 




wide, and hy long. The mortise should be cut 
a little deeper than the tenon. "Drip-holes" 
from the mortise to the outside will assist in 
draining off water that may accumulate in the 
joint and thus prevent the rapid decay that 
would otherwise ensue. These joints are very 
troublesome if a single post decays and requires 
renewal. It is generally required that the mor- 
FiG. 72. tise and tenon should be thoroughly daubed 
with paint before putting them together. This will tend to 
make the joint water-tight and prevent decay from the accu- 
mulation and retention of water in the joint. 

(b) The plaster joint. This joint is made by bolting and 
spiking a 3"Xl2" plank on 

both sides of the joint. The y^i^-wj^ ^l^ 

cap and sill should be 
notched to receive the posts. 
Repairs are greatly facili- 
tated by the use of these 
joints. This method has been 
used by the Delaware and 
Hudson Canal Co. [R. R.]. 

(c) Iron plates. An iron plate of the form shown in Fig. 74 




(]::.-^^-.::„ 



J-==^ 



FiQ. 73. 



§171. 



TRESTLES. 



^07 






o o 


b 

c 


o 
o 


o o 


o 


a 


a 



Fig. 74. — Joint Plates. 



(6) is bent and used as shown in Fig, 74 (a). Bolts passing 
through the bolt - holes .--■T"""7''l 

shown secure the plates 
to the timbers and make 
a strong joint Avhich may 
be readily loosened for re- 
pairs. By slight modifi- ^^ 
cations in the design the f"~ 
method may be used for | ,- 
inclined posts and compli- 
cated joints. 

(d) Split caps and sills. 
These are described in 
§ 161. Their advantages apply with even greater force to 
framed trestles. 

(e) Dowels and drift-bolts. These joints facilitate cheap and 
rapid construction, but renewals and repairs are very difficult, it 
being almost impossible to extract a drift-bolt, which has been 
driven its full length, without splitting open the pieces contain- 
ing it. Notwithstanding this objection they are extensively 
used, especially for temporary work which is not expected to 
be used long enough to need repairs. 

171. Multiple-story construc- 
tion. Single-story framed trestle 
bents are used for heights up 
to 18 or 20 feet and exception- 
ally up to 30 feet. For greater 
heights some such construction as 
is illustrated iji a skeleton design 
in Fig. 75 is used. By using split 
sills between each story and sepa- 
rate vertical and batter posts in 
each story, any piece may readily 
be removed and renewed if neces- 
sary. The height of these stories 
varies, in different designs, from 
15 to 25 and even 30 feet. In 
some designs the structure of each 
story is independent of the stories 
above and below. This greatly 
facilitates both the original construction and subsequent repairs. 




Fig. 75. 



208 



RA1LK0A.D CONSTRUCTION. 



§172- 



In other designs the verticals and batter-posts are made con^ 
tinuous through two consecutive stories. The structure is 
somewhat stiffer, but is much more difficult to repair. 

Since the bents of any trestle are usually of variable height 
and those heights are not always an even multiple of the uniform 
height desired for the gtoTie^, it becomes necessary to make the 




^^^^IXIXIXIXIXI 




Fig. 76, — 3keleton Elevation op Trestle. 

upper stories of uniform height p,nd let the odd amount go to the 
lowest story, as shown in Figs, 75 and 76. 

172. Span. The shorter the sp^n the greater the number of 
trestle bents; the longer the span the greater the required strength 
9f the stringers supporting the floor. Economy deniands the 
adoption of s^ span that shall inake the sum of these require- 




FiG. 77. — Knee-braces for Long-span Stbjingi^rs. 



ments a minimum. The higher the trestle the greater the cqst 
of each bent, and the greater the span that would be justifiable. 
Nearly all trestles have bents of variable height, but the advan- 
tage of employing uniform standard sizes is so grea+ that many 



§173. 



TRESTLES. 



209 




roads use the same span and sizes of timber not only for the 
panels of any given trestle, but also for all trestles regardless of 
height. The spans generally used vary from 10 to 16 feet. The 
Norfolk and Western R, R, uses a span of 12' 6" for all single- 
story trestles, and a span of 25' for all multiple-story trestles. 
The stringers are the same in both cases, but when the span is 
25 feet, knee-braces are run from the sill of the first story below 
to near the middle of each set of stringers. These knee-braces 
are connected at the top by a "straining-beam" on which the 
stringers rest, thus supporting the stringer in the center and vir- 
tually reducing the span about one-half. 

173. Foundations, (a) Plies. Piles are frequently used as a 
foundation, as in Fig. 78, particularly in soft ground, and also 
for temporary structures. These 
foundations are cheap, quickly 
constructed, and are particularly 
valuable when it is financially 
necessary to open the road for 
traffic as soon as possible and 
with the least expenditure of 
money; but there is the disad- 
yantage of inevitable decay 
Tvithin a few years unless the piles are chemically treated, as will 
be discussed later. Chemical treatment, however, increases the 
cost so that such a foundation would often cost more than a 
foundation of stone. A pile should be driven under each post 
as shown in Fig. 78. 

(b) Mud-siUs, Fig. 79 illustrates the use of mud-sills as 

built by the Louisville and 
Nashville R. R. Eight blocks 
12"X12"X6' are used under 
each bent. When the ground 
is very soft, two additional 
timbers (12" X 12'' X length of 
bent-sill), as shown by the 
dotted lines, are placed under- 
neath. The number required 

evidently depends on the na- 
Fio. 79.-MirD-sii.L Foundation, ^^^^ ^^ ^^^ ^^^^^^^ 

(c) stone foundations. Stone foundations are the best and 
the most expensive. For very high trestles the Norfolk and 



Fig. 78. — Pile Foundation. 




Ikiif 



— I 



SIUL 



r 
14- 



ap^s 



210 



RAILROAD CONSTRUCTION. 



§174. 



Western R. R. employs foundations as shown in Fig. 80, the 
walls being 4 feet thick. When the height of the trestle is 72 
feet or less (the plans requiring for 72' in height a foundation- 
wall 39' 6" long) the foundation is made continuous. The sill 



SILL OF TRESTLE 






« 13- *■ " — 8 — ^ t :13 > 

Fig. 80. — Masonky Trestle Foundation. 

of the trestle should rest on several short lengths of 3''X12" 
plank laid transverse to the sill on top of the wall. 

174. Longitudinal bracing. This is required to give the 
structure longitudinal stiffness and also to reduce the columnar 
length of the posts. This bracing generally consists of hori- 
zontal "waling-strips" and diagonal braces. Sometimes the 
braces are placed wholly on the outside posts unless the trestle 
is very high. For single-story trestles the P. R. R. employs 
the ''laced" system, i.e., a line of posts joining the cap of one 
bent with the sill of the next, and the sill of that bent with the 
cap of the next. Some plans employ braces forming an X in 
alternate panels. Connecting these braces in the center more 
than doubles their columnar strength. Diagonal braces, when 
bolted to posts, should be fastened to them as near the ends of 
the posts as possible. The sizes employed vary largely, depend- 
ing on the clear length and on whether they are expected to act 
by tension or compression. 3" X 12" planks are often used 
when the design would require tensile strength only, and 8'' X 8" 
posts are often used when compression may be expected. 

173. Lateral bracing. Several of the more recent designs of 
trestles employ diagonal lateral bracing between the caps of 
adjacent bents. It adds greatly to the stiffness of the trestle 
and better maintains its alignment. 6"X&' posts, forming 
an X and connected at the center, will answer the purpose. 

176. Abutments. When suitable stone for masonry is at 
hand and a suitable subsoil for a foundation is obtainable without 
too much excavation, a masonry abutment will be the best. 
Such an abutment would probably be used when masonry foot- 
ings for trestle bents v/ere employed (§ 173, c). 

Another method is to construct a "crib" of 10" X 12" timber^ 



§177. 



TRESTLES. 



211 



laid horizontally, drift-bolted together, securely braced and 
embedded into the ground. Except for temporary construction 
such a method is generally 
objectionable on account of 
rapid decay. 

Another method, used most 
commonly for pile trestles, and 
for framed trestles having pile 
foundations (§ 173, a), is to use 
a pile bent at such a place that 
the natural surface on the up- 
hill side is not far below the 
cap, and the thrust of the material, filled in to bring the surface 
to grade, is insignificant. 3''Xl2" planks are placed behind 
the piles, cap, and stringers to retain the filled material. 




Fig. 81. — Abutment Pile Bent. 



FLOOR SYSTEMS. 

177. Stringers. The general practice is to use two, three, 
and even four stringers under each rail. Sometimes a stringer 
is placed under each guard-rail. Generally the stringers are 
made of two panel lengths and laid so that the joints alternate. 
A few roads use stringers of only one panel length, but this prac- 
tice is strongly condemned by many engineers. The stringers 
should be separated to allow a circulation of air around them 
and prevent the decay which would occur if they were placed 
close together. This is sometimes done by means of 2" planks, 
6' to 8' long, which are placed over each trestle bent. Several 
bolts, passing through all the stringers forming a group and 
through the separators, bind them all into one solid construc- 
tion. Cast-iron "spools" or washers, varying from 4" to f" 
in length (or thickness), are sometimes strung on each bolt so 
as to separate the stringers. Sometimes washers are used 
between the separating planks and the stringers, the object of 
the separating planks then being to bind the stringers, especially 
abutting stringers, and increase their stiffness. 

The most common size for stringers is 8"X16". The Penn- 
sylvania Railroad varies the width, depth, and number of 
stringers under each rail according to the clear span. It may 
be noticed that, assuming a uniform load per running foot, both 
the pressure per square inch at the ends of the stringers (the 



212 



BAILROAD CONSTRUCTION. 



§178. 



caps having a width of 12") and also the stress due to trans- 
verse strain are kept approximately constant for the variable 
gross load on these varying spans. 



Span 


Y. P. stringers under each rail. 


c. to c. 

of 
bents. 


For H6b and E3d engines 

fMax. mom. about 200,000 

ft.-lbs.]. 


For heavier than H6b and 
E3d engines. 


10 ft. 

12 

14 


2 pes. 10" X ' 6" 

3 " 8"X.6" 
3 " 10"Xi6" 


2 pes. 10" X IG" 

3 " 10"X1G" 
Steel stringers 



178. Corbels. A corbel (in trestle-work) is a stick of timber 
(perhaps two placed side by side), about 3' to 6' long, placed 
underneath and along the stringers and resting on the cap. 
There are strong prejudices for and against their use, and a 
corresponding diversity in practice. They are bolted to the 
stringers and thus stiffen the joint. They certainly reduce the 
objectionable crushing of the fibers at each end of the stringer, 
but if the corbel is no wider than the stringers, as is generally 
the Case, the area of pressure between the corbels and the cap is 





Fig. 82. — Corbel. 

no greater and the pressure per square inch on the cap is no lessl 
than the pressure on the cap if no corbels were used. If the 
corbels and cap are made of hard wood, as is recommended by 
some, the danger of crushing is lessened, but the extra cost and 
the frequent scarcity of hard wood, and also the extra cost and 
labor of using corbels, may often neutralize the advantages 
obtained by their use. 

179. Guard-rails. These are frequently made of 6''X8'' stuff, 
notched 1" for each tie. The sizes vary up to 8''X8", and the 
depth of notch from f" to 1^". They are generally bolted to 
every third or fourth tie. It is frequently specified that they 
((Shall be made of oak, white pine, or yellow pine. The joints 
are made over a tie, by halving each piece, as illustrated in Fig. 
83. The joints on opposite sides of the trestle should be "stag- 



§ 180, TRESTLES. ^13 

gered." Some roads fasten every tie to the guard-rail, using a 
bolt, a spike, or a lag-screw. 

Guard-rails were originally used with the idea of preventing 
the wheels of a derailed truck from running off the ends of the 
ties. But it has been found that an outer guard-rail alone (with- 
out an inner guard-rail) becomes an actual element of danger, 
since it has frequently happened that a derailed wheel has caught 
on the outer guard-rail, thus causing, the truck to slew around 




Fig. 83. — Guard Timber. 

and so produce a dangerous accident. The true function of the 
QUtside guard-rail is thus changed to that of a tie-spacer, which 
keeps the ties from spreading when a derailment occurs. The 
inside guard-rail generally consists of an ordinary steel rail 
spiked about 10 inches inside of the running rail. These inner 
guard-rails should be bent inward to a point in the center of the 
track about 50 feet beyond the end of the bridge or trestle. If 
the inner guard-rails are placed with a clear space of 10 inches 
inside the running rail, the outer guard-rails should be at leas^ 
6' 10^' apart. They are generally much farther apart than thi^., 

i8q. Ties on trestles. If a car is derailed on a bridge or 
trestle, the heavily loaded wheels are apt to force their way be- 
tween the ties by displacing them unless the ties are closely 
spaced and fastened. The clear space between ties is generally 
equal to or less than their width. Occasionally it is a little more- 
than their width. 6"X8'' ties, spaced 14" to 16'' from center 
to center, are most frequently used. The length varies from 
9' to 12' for single track. They are generally notched V deep 
on the under side where they rest on the stringers. Oak ties 
are generally required even when cheaper ties are used on , the 
other sections of the road. Usually every third or fourth tie is 
bolted to the stringers. When stringers are placed imderneath 
the guard-rails, bolts are run from the top of the guard-rail to 
the under side of the stringer. The guard-rails thus hold down 
the whole system of ties, and no direct fastening of the ties to 
the stringers is needed. 

i8i. Superelevation of the outer rail on curves. The locatioa 
qi curves on trestles should be avoided if possible, especially 
when the trestle is high. Serious additional strains are intro- 



214 



RAILROAD CONSTRUCTION. 



§181. 



duced especially when the curvature is sharp or the speed high. 
Since such curves are sometimes practically unavoidable, it is 
necessary to design the trestle accordingly. If a train is stopped 
on a curved trestle, the action of the train on the trestle is 
evidently vertical. If the train is moving with a considerable 
velocity, the resultant of the weight and the centrifugal action 
is a force somewhat inclined from the vertical. Both of these 
conditions may be expected to exist at times. If the axis of 
the system of posts is vertical (as illustrated in methods a, h, c, d, 
and e) , any lat^.,ral force, such as would be produced by a mov- 
ing train, will tend to rack the trestle bent. If the stringers are 
set vertically, a centrifugal force likewise tends to tip them 
sidewise. If the axis of the system of posts (or of the stringers) 
is inclined so as to coincide with the pressure of the train on the 
trestle when the train is movitig at its normal velocity, there is 
no tendency to rack the trestle when the train is moving at that 
velocity, but there will be a tendency to rack the trestle or 
twist the stringers when the train is stationary. Since a moving 
train is usually the normal condition of affairs, as well as the 
condition which produces the maximum stress, an inclined axis 
is evidently preferable from a theoretical standpoint; but what- 
ever design is adopted, the trestle should evidently be suffi- 
ciently cross-braced for either a moving or a stationary load, 
and any proposed design must be studied as to the effect of both 
of these conditions. Some of the various methods of securing 
the requisite superelevation may be described as follows : 

(a) Framing the outer posts longer than the inner posts, so 

that the cap is inclined at the 
proper angle; axis of posts verti- 
cal. (Fig. 84.) The method re- 
quires more work in framing the 
trestle, but simplifies subsequent 
track-laying and maintenance, un- 
less it should be found that the 
superelevation adopted is unsuit- 
able, in which case it could be cor- 
rected by one of the other methods 
given below. The stringers tend 
to twist when the train is sta- 
tionaiy. 

(b) Notching the cap so that the stringers are at a different 




Fig. 84. 



§181. 



TRESTLES. 



215 




FiQ. 85. 



elevation. (Fig. 85.) This weakens the cap and requires that 
all ties shall be notched to a 
bevelled surface to fit the string- 
ers, which also weakens the ties. 
A centrifugal force will tend to 
twist the stringers and rack the 
trestle. 

(c) Placing wedges underneath 
the ties at each stringer. These 
wedges are fastened with two 
bolts. Two or more wedges will 
be required for each tie. The ad- 
ditional number of pieces required 
for a long curve will be immense, and the work of inspection and 
keeping the nuts tight will greatly increase the cost of main- 
tenance. 

(d) Placing a wedge under the outer rail at each tie. This 
requires but one extra piece per tie. There is no need of a 
wedge under the inner tie in order to make he rail normal to 
the tread. The resulting inward inclination is substantially that 
produced by some forms of rail-chairs or tie-plates. The spikes 
(a little longer than usual) are driven through the wedge into 
the tie. Sometimes "lag-screws" are used instead of spikes. 
If experience proves that the superelevation is too much or too 
little, it may be changed by this method with less work than 
by any other. 

(e) Corbels of different heights. When corbels are used (see 

§ 178) the required in- 
cHnation of the floor sys_ 
tem may be obtained by 
varying the depth of the 
corbels. 

(f) Tipping the whole 
trestle. This is done by 
placing the trestle on an 
inclined foundation. If 
very much inclined, the 
trestle bent must be se- 
cured against the possi- 
FiG. 86. bility of slipping sidewise, 

for the slope would be considerable with a sharp curve, and the 




216 RAILROAD CONSTRUCTION. § 182.] 

vibration of a moving train would reduce the coefficient ol 
friction to a comparatively small quantity. 

(g) Framing the outer posts longer. This case is identical 
with case (a) except that the axis of the system of posts ig 
inclined, as in case (/), but the sill is horizontal. 

The above-described plans will suggest a great variety ot 
methods which are possible and which differ from the above 
only in minor details. 

' 182. Protection from fire. Trestles are peculiarly subject to 
fire, from passing locomotives, which may not only destroy the 
trestle, but perhaps cause a terrible disaster. This danger is. 
sometimes reduced by placing a strip of galvanized iron along 
the top of each set of stringers and also along the tops of the 
caps. Still greater protection was given on a long trestle on the 
Louisville and Nashville R. R. by making a solid flooring of 
timber, covered with a layer of ballast on which the ties and 
fails were laid as usual. 

Barrels of water should be provided and kept near all trestles/ 
and on very long trestles barrels of water should be placed every 
two or three hundred feet along its length. A place for the bar- 
rels may be provided by using a few ties which have an extra 
length of about four feet, thus forming a small platform, which 
jhould be surrounded by a railing. The track-walker should be 
held accountable for the maintenance of a supply of water in 
these barrels, renewals being frequently necessary on account of 
evaporation. Such platforms should also be provided as refugE!- 
BAYS for track-walkers and trackmen working on the trestle. On 
very long trestles such a platform is sometimes provided with 
sufficient capacity for a hand-car, 

183. Timber. Any strong durable timber may be used when 
the choice is limited, but oak, pine, or cypress are preferred 
when obtainable. When all of these are readily obtainable, 
the various parts of the trestle will be constructed of different 
kinds of wood — the stringers of long-leaf pine, the posts and 
braces of pine or red cypress, and the caps, sills, and corbels (if 
used) of white oak. The use of oak (or a similar hard wood) 
fdt caps, sills, and corbels is desirable because of its greater 
strength in resisting crushing across the grain, which is the 
critical test for these parts. There is no physiological basis to 
the objection, sometimes made, that different species of timber, 
ill contact with each other, will rot quicker than if only one 




TIES OAPPEil 
GUARD RAILS ' 




PACKING BLOCK 



IS'o'CENTER TO CE^TER OF I 
FOR JWO STORY. BENTS OR mI 
12'6*CENTERTO CENTER 0f| 
FOR ONE STOnr BplTS OR HI 



NORFOLK & WESTERN 

8Tl\NDARD ONE AND TWO STORy TB 
(SEPT. 10,1891.) 



^VAUNQ STRI 



ONE STORY BENT 



(h -f- 2.*) + i't'= length OF SILL. 

H I'l l"= LENGTH OF iz" XJ2" TIMBER REQUIRED FOR PLUMB POSTS, SINGLE BENT* 

H — 2l'l1= « " " " " " " " IN LOWER StORY. DOUBLE " 

H — a's' == « " PLUMB POSTS BETWEEN SHOULDERS, SINGLE •• 

H — 22'ir= " " " " " " •« " " DOUBLE " 

(length OF PLUMB POST X 1.02l)-j-3'= LENGTH OF BATTER POST, EXCEPT IN 8"x 1 2°lftTEJ«(EDIATE 
BATTER POSTS WHERE ADD s' INSTEAD OF 3* 
WITH ALTERNATIVE ARRANGEMENT OF STRINGERS, ADD e' TO LENGTHS GIVEN BY ABOVE FOHMULAflt 



(ro \(JJC6 'page 216.) 



>R STRINGEH 
' TIES I 



PLATE II. 

CROSS T!E s'x 10"x Id'o" 
kl-ii 15"C.|T0C. y-J'j^ 

^ A |\ — ^T^ r ^^ 

**tsi^fi ^5 SUEGRADE _J_ 




§ 185. TRESTLES. 217 

kind of timber i3 used. When a very extensive trestle is to be 
built at a place where suitable growing timber is at hand but 
there is no convenient sawmill, it will pay to transport a port- 
able sawmill and engine and cut up the timber as desired. 

184. Cost of framed timber trestles. The cost varies widely 
on account of the great variation in the cost of timber. When 
a railroad is first penetrating a new and undeveloped region, the 
cost of timber is frequently small, and when it is obtainable from 
the company's right-of-way the only expense is felling and 
sawing. The work per M, B. M., is small, considering that a 
single stick 12'^ X 12'' X 25' contains 300 feet, B. M., and that 
sometimes two hours' work, worth perhaps $1, will finish all 
the work required on it. Smaller pieces will of course require 
more work per foot, B. M. Long-leaf pine can be purchased 
from the mills at from $27 to $45 per M feet, B. M., according 
to the dimensions. To this must be added the freight and labor 
of erection. The cartage from the nearest railroad to the trestle 
may often be a considerable item. Wrought iron will cost 
about 3 cents per pound and cast iron 2 cents, although the prices 
are often lower than these. The amount of iron used depends on 
the detailed design, but, as an average, will amount to $1.50 
to $2 per 1000 feetj B. M., of timber. A large part of the tres- 
tling of the country has been built at a contract price of about 
$30 per 1000 feet, B. M.^ erected. While the cost will frequently 
rise to $50 and even $60 when timber is scarce, it will drop to 
$13 (cost quoted) when timber is cheap, 

DESIGN OF WOODEN TRESTLES. 

185. Common practice. A great deal of trestliiig has beeii 
constructed without any rational resign except that custom and 
experience have shown that certain sizes and designs are probably 
safe. Tliis method has resulted occasionally in failures but more 
frequently in a very large waste of timber. Many railroads 
ernploy a uniform size for all posts, caps, arid sills, arid a uniform 
size for stringers, all regardless of the height or span of the 
trestle. For repair work there are practical reasoris favoring 
this. "To attempt to run a large lot of sizes would be more 
wasteful in the eftd than to maintain a few stock sizes only. 
Lumber can be bought more cheaply by giving a general order 
for ' the run of the mill for the season/ or ' a cargo lot,' specif}' 



218 RAILROAD CONSTRUCTION. ' § 186. 

ing approximate percentages of standard stringer size, or 
12 X 12-inch stuff, 10 X 10-inch stuff, etc., and a Hberal propor- 
tion of 3- or 4-inch plank, all lengths thrown in. The 12 X 12- 
inch stuff, etc., is ordered all lengths, from a certain specified 
length up. In case of a wreck, washout, burn-out, or sudden 
call for a trestle to be completed in a stated time, it is much 
more economical and practical to order a certain number of 
carloads of 'trestle stuff' to the ground and there to select piece 
after piece as fast as needed, dependent only upon the length of 
stick required. When there is time to make the necessary sur- 
veys of the ground and calculations of strength, and to wait for a 
special bill of timber to be cut and delivered, the use of differ- 
ent sizes for posts in a structure would be warranted to a certain 
extent." * For new construction, when there is generally 
sufficient time to design and order the proper sizes, such waste- 
fulness is less excusable, and under any conditions it is both 
safer and more economical to prepare standard designs which 
can be made applicable to varying conditions and which will at 
the same time utilize as much of the strength of the timber as 
can be depended on. In the following sections will be given 
the elements of the preparation of such standard designs, which 
will utilize uniform sizes with as little waste of timber as possible. 
It is not to be understood that special designs should be made 
for each individual trestle. 

i86. Required elements of strength. The stringers of trestles 
are subject to transverse strains, to crushing across the grain 
at the ends, and to shearing along the neutral axis. The strength 
of the timber must therefore be computed for all these kinds 
of stress. Cajps and sills will fail, if at all, by crushing across 
the grain; although subject to other forms of stress, these could 
hardly cause failure in the sizes usually employed. There is an 
apparent exception to this: if piles are improperly driven and 
an uneven settlement subsequently occurs, it may have the 
effect of transferring practically all of the weight to two or three 
piles, while the cap is subjected to a severe transverse strain 
which may cause its failure. Since such action is caused gener- 
ally by avoidable errors of construction it may be considered as 
abnormal, and since such a failure will generally occur by a 
gradual settlement, aU danger may be avoided by reasonable 
•_ < 

* From "Economical Designing of Timber Trestle Bridges." 



§ 187. tilESTLES. 219 

care in inspection. Posts must be tested for their columnar 
strength. These parts form the bulk of the trestle and are the 
parts which can be definitely designed from known stresses. 
The stresses in the bracing are more indefinite, depending on 
indeterminate forces, since the inclined posts take up an un- 
known proportion of the lateral stresses, and the design of the 
bracing may be left to what experience has shown to be safe, 
without involving any large waste of timber. 

187. Strength of timber. Until recently tests of the strength 
of timber have generally been made by testing small, selected, 
well-seasoned sticks of " clear stuff," free from knots or imper- 
fections. Such tests would give results so much higher than 
the vaguely known strength of large unseasoned "commercial" 
timber that very large factors of safety were recommended— 
factors so large as to detract from any confidence in the whole 
theoretical design. Recently the U. S. Government has been 
making a thoroughly scientific test of the strength of full-size 
timber under various conditions as to seasoning, etc. The work 
has been so extensive and thorough as to render possible the 
economical designing of timber structures. 

One important result of the investigation is the determina- 
tion of the great influence of the moisture in the timber and 
the law of it^^ effect on the strength. It has been also shown 
that timber soaked with water has substantially the same 
strength as green timber, even though the timber had once been 
thoroughly seasoned. Since trestles are xsxposed to the weather 
they should be designed on the basis of using green timber. 
It has been shown that the strength of green timber is very 
regularly about 55 to 60% of the strength of timber in which 
the moisture is 12% of the dry weight, 12% being the proportion 
of moisture usually found in timber that is protected from the 
weather but not heated, as, e.g., the timber in a barn. Since 
the moduli of rupture have all been reduced to this standard of 
moisture (12%), if we take one-eighth of the rupture values, it 
still allows a factor of safety of about five, even on green timber. 
In Table XX there are quoted the values taken from the U. S. 
Government reports on the strength of timber, the tests prob- 
ably being the most thorough and reliable that were ever made. 

In Table XXI are given the " working unit stresses for struc- 
tural timber, expressed in pounds per square inch," as recom- 
mended by the committee on " Wooden Bridges and Trestles," 



22Q 



RAILROAD CONSTRUCTION. 



§188. 



pf the American Railway Engineering Association- The report 
was presented at their tenth annual convention, held in Chicago, 
in March, 19.QQ. 

Table XX. moduli of rupture for various timbe^^s. 

[12% nioisture.] 

(Copd^psed from U. S. Forestry Circular, No. 15.) 









U O 


Cross-bendipg. 


Crush- 
ing 
end- 
wise. 


to 

m 

2 . 
.B ^ 

'S a) 


o . 

u 


No. 


Speqips,. 


j3 
.§1 


Modulus 

of 
Elasticity. 


1 
2 

3 
4 
5 
6 

7. 


Long-leaf pine. . . . 

Cuban " 

Short-leaf " .... 
Loblolly " .... 
White " .... 
Red " .... 
Sprupe " .... 


38 

39 

32 . 

33 

24 

31 

39 


12 600 

13 600 

10 100 

11 300 
7 900 
9 100 

10 0()0 


2 070 000 
2 370 000 
1 680 000 
. 2 050 000 
1 390 000 
1 620 000 
1 640 000 


8000 
8700 
6500 
7400 
5400 
6700 
73pO 


1180 
^220 

960 
1150 

700 
lOOO 
1200 


700 
700 
700 
700 
400 
500 
800 


8 


Bald cypress 


29 
23 
32 


7 900 

6 300 

7 900 

13 100 

11 300 

12 300 

1 1 500 

iMoq 

13 100 
10 400 

12 000 


1 290 000 

910 000 

1 680 000 


6000 
5200 
5700 


800 
700 
800 


500 


9 
10 


White cedar 

Douglas soruce. . . . 


400 
500 


11 


White oak. . , . . . 


50 
46 
50 
46 
45 
46 
45 
46 


2 090 000 

1 620 000 

2 030 000 
1610 000 
1 970 ono 
1 860 000 
1 750 000 

1 930 ooq 


8500 
7300 
7100 
7400 
7200 
8100 
7200 
7700 


2200 
1900 
3000 
1900 
2300 
2000 
1600 
1800 


1000 


12 


Overcup " . . 




1000 


13 


Post " . . 


• 


1100 


14 


Cow " . . 




900 


1"^ 


Red " . . 




1100 


i6 


Texan " . . 




900 


19 


Willow " . . 




900 


?0 


Spanish " . . 




900 








21 
27 

9^ 


Shagbark hickory. . 

Pignut " 

White elm 


51 

66 . 
34 
46 
39 


16 000 

18 70b 
10 300 
13.500 
10 800 


2 390 000 
2 730 0.00 
1 540 0(W 
1 700 000 
1 640 000 


9500 
10900 
6500 
8000 
7200 


2700 
3200 
1200 
2100 
' 1900 


1100 

120p 

800 


?9 


Cedar " 


1300 


30 


White ash 


1100 









1 88. Loading. As shown in § 172, the span of trestles is always 
small, is generally 14 feet, and is never greater than 18 feet 
except when supported by knee-braces. The greatest load that 
will ever come on any one span ^^dll be the concentrated loading 
of the drivers of a very heavy locomotive. With spans of 14 
feet or less it is impossible for even the four pairs of drivers to 
be on the same span at once. The weight of the rails, ties, and 
guard-rails should be added to obtain the total load on the string- 
ers, and the weight of these, plus the weight of the stringers, 
should be added to obtain the pressure on the caps or corbels.. 



§ 188. 



TKESTLES. 



221 





.2 "5 Mo-5 












1 « 


; 


\ 










inches, 
inins. 
25 per 
ection. 


05 
O 


l^'^ ^o S"^ a 



1—1 



I— 1 




T-H 




rH 
















rH 


Oi 


rt ii -e i^'o 


























r^ 




vj 


























d 




t- (» pi tn 


^ 


^ 


^ 


^ 


*^! 


0^ 


:^ 


^ 


-5 


i^ 


§^^ 


r- to 


o 

03 




1 



1 


CO 
1 


1 


1 
1 


■•D CD 
1 


CO 
1 


• CO 
1 


1 


CO 

1 


CO. 

1 


CD |CJ 
1 


11. II ^ 1 


CO 




§.9 u2 

(^ J- O 5r1 


r-H 


r-H 


rH 


T-H 




^H 


T-H 


y^ 




,». 


r— 


I-H 


< 
















c 











c 


c 


^ - — 

G 





• 




o o— f* 














c 











e 


c 


C 





^'^^^ 


O 




tn &■ c^ 'i^ 


OA 


CO 


t-H 





T— 


00 





cq 


c 


I— 


CT 


CO 


^ 




»-*i p^ 


'.3 u 


i-H 


T-H 


T-H 


T-\ 


1—1 




rH 


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222 



RAILROAD CONSTRUCTION. 



§189. 



This dead load is almost insignificant compared with the live 
load and may be included with it. The weight of rails, ties, 
etc., may be estimated at 240 pounds per foot. To obtain the 
weight on the caps the weight of the stringers must be added, 
which depends on the design and on the weight per cubic foot 'j 
of the wood employed. But as the weight of the stringers is '. 
comparatively small, a considerable percentage of variation in 
weight will have but an insignificant effect on the result. Dis- 
regarding all refinements as to actual dimensions, the ordinary 
maximum loading for standard-gauge railroads may be taken 
as that due to four driving-axles, spaced 5' 0" apart and giving 
a pressure of 40000 pounds per axle. This should be increased 
to 54000 pounds per axle (same spacing) for the heaviest traffic. 
On the basis of 40000 pounds per axle or 20000 pounds per wheel 
the following results have been computed: This loading is 
assumed to allow for impact. 

STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OF 20000 
POUNDS, SPACED 5' 0" APART, WITH 120 POUNDS PER FOOT 
OF DEAD LOAD. 



Span in feet. 


Max. moment, 
ft. lbs. 


Max shear. 


Max. load on 

one cap under 

one rail. 


10 
12 
14 
16 
18 


51 500 

82 160 

112 940 

123 840 

164 860 


30 600 
35 720 
39 410 
43 460 
47 747 


41 200 
49 440 
57 680 
65 920 
75 160 



Although the dead load does not vary in proportion to the 
live load, yet, considering the very small influence of the doad 
load, there will be no appreciable error in assuming the corre- 
epofiding values, for a load of 54000 lbs. per axle, to be f f of 
^ose given in the above tabulation. 

# 189. Factors of safety. The most valuable result of the gov- 
ernment tests is the knowledge that under given moisture condi- 
tions the strength of various species of sound timber is not the 
variable uncertain quantity it was once supposed to be, but that 
its strength can be relied on to a comparatively close percentage. 
This confidence in values permits the employment of lower facr- 
tors of safety than have heretofore been permissible. Stresses, 
which when excessive would result in immediate destruction, 
such as cross-breaking and columnar stresses, should be allowed 
a higher factor of safety — say 6 or 8 for green timber. Other 
stresses, such as crushing across the grain and shearing along the 



§ 190. TRESTLES. 223 

neutral axis, which will be apparent to inspection before it is 
dangerous, may be allowed lower factors — say 3 to 5. 

iQO. Design of stringers. The strength of rectangular beams 
of equal width varies as the square of the depth ; therefore deep 
beams are the strongest. On the other hand, when any cross- 
Isectional dimension of timber much exceeds 12" the cost is 
much higher per M, B. M., and it is correspondingly difficult to 
obtain thoroughly sound sticks, free from wind-shakes, etc. 
; Wind-shakes especially affect the shearing strength. Also, if 
the required transverse strength is obtained by using high nar- 
row stringers, the area of pressure between the stringers and the 
cap may become so small as to induce crushing across the grain. 
This is a very common defect in trestle design. As already in- 
dicated in § 172, the span should vary roughly with the average 
height of the trestle, the longer spans being employed when the 
trestle bents are very high, although it is usual to employ the 
yame span throughout any one trestle. 

To illustrate, if we select a span of 14 feet, the load on one 
cap will be 57680 lbs. If the stringers and cap are made of 
long-leaf yellow pine, the allowable value, according to Table 
; XXI, for " compression across the grain " is 260 pounds per 
square inch; this will require 222 square inches of surface. 
If the cap is 12" wide, this will require a width of 18.5 inches, 
or say 2 stringers under each rail, each 9 inches wide. For 
rectangular beams. 

Moment = |i2'6/i2, 

Using for R' the safe value 1300 lbs. per square inch, we have 

112940Xl2 = iXl300Xl8X/i2, 

from which ^=f 18".7. If desired, the width may be increased 
to 10" and the depth correspondingly reduced, which will give 
similarly A = 17". 7 or say 18". This shows that two beams, 
10"X18", under each rail will stand the transverse bending and 
have more than enough area for crushing. 
The shear per square inch will equal 

3 total shear 3 39410 ,„. „ . , 

2 cross-section =2 2X10X18 =^^^ ^^'- P"" '^- '^'^' 

This is higher than the recommended working value. The com- 
bination suggested in § 177, viz., 3 beams 10"X16" for 14 feet 
span, gives a far safer value. Considering that wooden beams^ 



224 RAILROAD CONSTRUCTION. § 191. 

tested to destructioti, usually fail by shearing, the three-beam; 
combination is safer. 

The deflection should be computed to see if it exceeds the 
somewhat arbitrary standard of -^^^ of the span. The deflec*! 
tion for uniform loading is 



in which I = length in inches; 

TF= total load, assumed as uniform = 57680; 
E= modulus of elasticity, given as 1610000 lbs. 

per sq. in. for long-leaf pine, according to Table XXI. Then 

__5X57680X1683__^ 
32X30X16^X1610000 

so that the calculated deflection is well within the limit. Of 
course the loading is not strictly uniform, but even with a lib- 
gral allowance the deflection is still safe. 

For the heaviest practice (65000 lbs. per axle) these stringed 
dimensions must be correspondingly increased. 

iQi. Design of posts. Four posts are generally used for 
single-track work. The inner posts are usually braced by the 
cross-braces, so that their columnar strength is largely increased ; 
but as they are apt to get more than their share of work, the ad-* 
vantage is compensated and they should be treated as unsup- 
ported columns for the total distance between cap and sill in 
simple bents, or for the height of stories in multiple-story con- 
struction. The caps and sills are assumed to have a width of \2", 
It facilitates the application of bracing to have the columns of 
the same width and vary the other dimension as required. 

Unfortunately the experimental work of the U. S. Govern- 
ment on timber testing has not yet progressed far enough to 
establish unquestionably a general relation between the strength 
of long columns and the crushing st:"ength of short blocks. The 



§192. TRESTLES. 225 

following formula has been suggested, but it cannot be consid-* 
ered as established: 

■ „ , 700 + 15c . , . , 

7^^X700 + 15^+7- ^^^^^^^ 

/= allowable working stress per sq. in for long colunms; 
F= " " " '' '' '' " short blocks; 

I 

Z= length of column in inches; 

<i=least cross-sectional dimensions in inches. 

The formula recommended by the A. R. E. A. is found in 
Table XXI. For all columns of which the length is less than 15 
times the least diameter, a uniform unit stress is recommended. 
For longer columns, a unit stress is multiplied by the factor 
(1— Z-^60(i), which is always less than unity. For the above 
case, I = 240 and d = 12, and the factor = .667, which, multiplied 
by 1300, gives a unit stress of 867 lbs. per square inch for a long- 
leaf yellow pine column of these dimensions. 

867 X 144 = 124848 lbs., the working load for each post. This is 
more than the total load on one trestle bent and illustrates the 
usual great waste of timber. Making the post 8"X12" and 
calculating similarly, we have/ ^650, and the working load per 
column is 650X93=62400 lbs. As considerable must be 
allowed for "weathering," which destroys the strength of the 
outer layers of the wood, and also for the dynamic effect of 
the hve load, 8" X 12" may not be too great, but it is certainly 
a safe dimension, considered as a column. One method of 
allowing for weathering is to disregard the outer half-inch on 
all sides of the post, i.e., to calculate the strength of a post one 
inch smaller in each dimension than the post actually employed. 
On this basis an 8" X 12" X20' post, computed as a 7" X 11" post, 
would have a safe columnar strength of 556 lbs. per square inch. 
With an area of 77 square inches, this gives a working load of 
42812 lbs. for each post, or 171248 lbs. for the four posts. Con- 
sidering that 115360 lbs. is the maximum load on one cap (14 feet 
span), the great excess of strength is apparent. 

192. Design of caps and sills. The stresses in caps and sills 
are very indefinite, except as to crushing across the grain. As 



^26 RAILROAD CONSTRUCTION. § 193. 

the stringers are placed almost directly over the Inner posts, and 
as the sills are supported just under the posts, the transverse 
stresses are almost insignificant. In the above case four posts 
have an area of 4 X 12" X 8" = 384 sq. in. The total load 1 1 5360 
lbs. will then give a pressure of 300 pounds per square inch, 
which is more than the allowable limit. This one feature will 
require the use of 12"X12" (or at least 10"X12") posts rather 
than 8"X12" posts, for the smaller posts, although probably 
strong enough as posts, would produce an objectionably high 
pressure. 

«* 193. Bracing. Although some idea of the stresses in the 
bracing could be found from certain assumptions as to wind- 
pressure, etc., yet it would probably not be found wise to de- 
crease, for the sake of economy, the dimensions which practice 
has shown to be sufficient for the work. The economy that 
would be possible would be too insignificant to justify any risk. 
Therefore the usual dimensions, given in §§ 174 and 175, should 
be employed. 



CHAPTER V. 

TUNNELS. 



SURVEYING. 

194. Surface surveys. As tunnels are always dug freiii each 
end and frequently from one or more intermediate shafts, it is 
necessary that an accurate surface survey should be made 
between the two ends. As the natural surface in a locality 
where a tunnel is necessary is almost invariably very steep and 
rough, it requires the employment of unusually refined methods 
of work to avoid inaccuracies. It is usual to run aline on the 
surface that will be at every point vertically over the center line 
of the tunncil. Tunnels are generally made straight unless 
curves are absolutely necessary, as curves add greatly to the 
cost. Fig. 87 represents roughly a longitudinal section of the 




— ^eo-' •►I -6000-'-— i -1^)00-— -H 6000^ p— 5000- 

FiG. 87. — Sketch of Section of the Hoosac Tunnbi,. 



Hoosac Tunnel. Permanent stations were located at A , B, C, 
D, E, and F, and stone houses were built at A, B, C, and D. 
These were located with ordinary field transits at first, and then 
all the points were placed as nearly as possible in one vertical 
plane by repeated trials and minute corrections, using a very 
large specially constructed transit. The stations D and F were 
necessary because E and A were invisible from C and B. Th« 
alinement at A and E having been determined with great accu- 
racy, the true alinement was easily carried into the tunnel. 

227 



228 RAILROAD CONSTRUCTION. § 195. 

The relative elevations of A and E were determined with 
great accuracy. Steep slopes render necessary many settings j 
of the level per unit of horizontal distance and require that the ' 
work be unusually accurate to obtain even fair accuracy per 
unit of distance. The levels are usually re -run many times ij 
until the probable error is a very small quantity 

The exact horizontal distance between the two ends of the 
tunnel must also be known, especially if the tunnel is on* a ; 
grade. The usual steep slopes and rough topography likewise 
lender accurate horizontal measurements very difficult. Fre- 
quently when the slope is steep the measurement is best ob- i 
tained by measuring along the slope- and allowing for grade. 
This may be very accurately done by employing two tripod$ 
(level or transit tripods serve the purpose very well), setting I 
them up slightly less than one tape-length apart and measuring j 
between horizontal needles set in wooden blocks inserted in the 
top of each tripod. The elevation of each needle is also observed. 
The true horizontal distance between two successive positions 
p| the needles then equals the square root of the difference of 
the squares, of the inclined distance and the difference of eleva- 
tion. Such measurements will probably be more accurate than 
those made by attempting to hold the tape horizontal and 
plumbing down with plumb-bobs, because (1) it is practically 
difficult to hold both ends of the tape truly horizontal; (2) on 
^teep slopes it is impossible to hold the down-hill end of a lOO- 
foot tape (or even a 25-foot length) on a level with the other 
end, and the great increase in the number of applications of the 
unit of measurenaent very greatly increases the probable error 
of the whole measurement; (3) the vibrations of a plumb-bob 
introduce a large probability of error in transferring the meas- 
urement from the elevated end of the tape to the ground, and 
the increased number of such applications of the unit of meas- 
urement still further increases the probable error. 

195. Surveying down a shaft. If a shaft is sunk, as at *S^, 
Fig. 87, and it is desired to dig out the tunnel in both directions 
from the foot of the shaft so as to meet the headings from the 
outside, it is necessary to know, when at the bottom of the 
shaft, the elevation, alinement, and horizontal distance from 
each end of the tunnel. 

The elevation is generally carried down a shaft by means of 
a steel tape. This method involves the least number of appli- 



§ 195, TUNNELS. 229 

cations of the unit of measurement and greatly increases the 
accuracy of the final result. 

The horizontal distance from each end may be easily trans- 
ferred down the shaft by means of a plumb -bob, using some of 
the precautions described in the next paragraph. 

To transfer the alinement from the surface to the bottom of 
a shaft requires the highest .skill because the shaft is always 
small, and to produce a line perhaps several thousand feet long 
in a direction given by two points 6 or 8 feet apart requires 
that the two points must be determined with extreme accuracy. 
The eminently successful method adopted in the Hoosac Tunnel 
will be briefly described: Two beams were securely fastened 
across the top of the shaft (1030 feet deep), the beams being 
placed transversely to the direction of the tunnel and as fair 
apart as possible and yet allow plumb-lines, hung from the 
intersection of each beam with the tunnel center line, to swing 
freely at the bottom of the shaft. These intersections of the 
beams with the Center line were determined by averaging the 
results of a large number of careful observations for alinement. 
Two fine parallel wires, spaced about j^" apart, were then 
stretched between the beams so that the center line of the 
tunnel bisected at all points the space between the wires. 
Flnmb-bobs, weighing 15 pounds, were suspended by fine wires 
beside each cross-beam, the wires passing between the two 
parallel alinement wires and bisecting the space. The plumb- 
bobs were allowed to swing in pails of water at the bottom. 
Drafts of air up the shaft required the construction of boxes 
surrounding the wires. Even these precautions did not suffice 
to absolutely prevent vibration of the wire at the bottom 
through a very small arc. The mean point of these vibrations 
in each case was then located on a rigid cross-beam suitably 
placed at the bottom of the shaft and at about the level of the 
roof of the tunnel. Short plumb-lines were then suspended 
from these points whenever desired; a transit was set (by trial) 
so that its line of collimation passed through both plumb-lines 
and the line at the bottom could thus be prolonged. 

Some recent experience in the ''Tamarack" shaft, 4250 feet 
deep, shows that the accuracy of the results may be affected by 
air-currents to an unsuspected extent. Two 50-lb. cast-iron 
plumb-bobs were suspended with No. 24 piano-wire in this 
shaft. The carefully measured distances between the wires' 



230 



KAILBOAD CONSTRUCTION. 



§196. 



at top and bottom were 16.32 and 16.43 feet respectively. 
After considerable experimenting to determine the cause of 
the variation, it was finally concluded that air-currents were j 
alone responsible. The variation of the bobs from a true ver- I 
tical plane passing through the wires at the top was of course ■' 
an unknown quantity, but since the variation in one direction 
amounted to 0.11 foot, the accuracy in other directions was 
viery questionable. This shows that a careful comparative i 
measurement between the wires at top and bottom should 
always be made as a test of their parallelism. 

196. Underground surveys. Survey marks are frequently 
placed on the timbering, but they are apt to prove unreliable 
on account of the shifting of the timbering due to settlement 
of the surrounding material. They should never be placed at 
the bottom of the tunnel on account of the danger of being 
disturbed or covered up. Frequently holes are drilled in the 
roof and filled with wooden plugs in which a hook is screwed 
exactly on line Although this is probably the safest method, 
even these plugs are not always undisturbed, as the material, 
unless very hard, will often settle slightly as the excavation 
proceeds. When a tunnel is perfectly straight and not too long, 
alinement-points may be given as frequently as desired from 

permanent stations located outside 
the tunnel where they are not liable 
to disturbance. This has been ac- 
complished by running the aline- 
ment through the upper part of the 
cross-section, at one side of the cen- 
ter, where it is out of the way of 
the piles of masonry material, 
debris, etc., which are so apt to 
choke up the lower part of the 
cross-section. The position of this 
line relative to the cross-section 
being fixed, the alignment: of any 
required point of the cross-section 
is readily found by means of a light 
frame or template with a fixed tar- 
get located where this line would intersect the frame when 
properly placed. A level-bubble on the frame will assist in 
setting the frr.me in its proper position. 




Fig. 88. 



§ 197. TUNNELS. 231 

In all tunnel surveying the cross-wires must be illuminated 
by a lantern, and the object sighted at must also be illuminated. 
A powerful dark-lantern with the opening covered with ground 
glass has been found useful. This may be used to illuminate a 
plumb-bob string or a very fine rod, or to place behind a brass 
plate having a narrow slit in it, the axis of the slit and plate 
being coincident with the plumb-bob string by which it is 
hung. 

On account of the interference to the surveying caused by 
the work of construction and also by the smoke and dust in the 
air resulting from the blasting, it is generally necessary to make 
the surveys at times when construction is temporarily suspended. 

197. Accuracy of tunnel surveying. Apart from the very 
natural desire to do surveying which shall check well, there is 
an important financial side to accurate tunnel surveying. If 
the survey lines do not meet as desired when the headings come 
together, it may be found necessary, if the error is of appreciable 
size, to introduce a slight curve, perhaps even a reversed curve, 
into the alinement, and it is even conceivable that the tunnel 
section would need to be enlarged somewhat to allow for these 
curves. The cost of these changes and the perpetual annoyance 
due to an enforced and undesirable alteration of the original 
design will justify a considerable increase in the expenses of the 
survey. Considering that the cost of surveys is usually but a 
small fraction of the total cost of the work, an increase of 10 or 
even 20% in the cost of the surveys will mean an insignificant 
addition to the total cost and frequent' /, if not generally, it will 
result in a saving of many times the increased cost. The 
accuracy actually attained in two noted American tunnels is 
given as follows: The Musconetcong tunnel is about 5000 feet 
long, bored through a mountain 400 feet high. The error of 
alinement at the meeting of the headings was 0'.04, error of 
levels 0'.015, error of distance 0'.52. The Hoosac tunnel is 
over 25000 feet long. The heading from the east end met the 
heading from the central shaft at a point 11274 feet from the 
east end and 1563 feet from the shaft. The error in alinement 
was Ye of an inch, that of levels " a few hundredths," error of 
distance '' trifling." The alinement, corrected at the shaft, was 
carried on through and met the heading from the west end at a 
point 10138 feet from the west end and 2056 feet from the shaft. ' 
Here the error of alinement was ^" and that of levels 0.134 foot. 



232 RAILROAD CONSTRUCTION. § 198. 

RESIGN 

198, Cross-section. Nearly all tunnels have cross-sections 
peculiar to themselves — all varying at least in the details. The 
general form of a great many tunnels is" that of a rectangle sur- 
mounted by a semi-circle or semi-ellipse. In very soft material 
an inverted arch is necessary along the bottom. In such cases 
the sides will generally be arched instead of vertical. The sides' 
are frequently battered. In very long tunnels, several forma 
of cross-section will often be used in the same tunnel, owing to 
differences in the material encountered. In solid rock, which 
will not disintegrate upon exposure, no lining is required, and 
the cross-section will be the irregular section left by the blasting, 
the only requirement being that no rock shall be left within the 
required cross-sectional figure. Farther on, in the same tunnel, 
when passing through some very soft treacherous material, it 
may be necessary to put in a full arch lining — top, sides, and 
bottom — which will be nearly circular in cross-section. For 
an illustration of this see Figs. 89 and 90. 

The cross-section recommended by the A. R. E. A. for single 
track is a rectangle 16 feet wide by 16 feet 6 inches high, sur- 
mounted by a semi-circle with a radius of 8 feet. The top of the 
tie is to be 2 feet above the bottom which is at sub-grade. If 
the surrounding material is yielding and exerts great pressure, 
the sides should be battered inward 1 foot at the bottom. For 
a double track tunnel the design is similar, except that the width 
is increased by the standard spacing between double tracks and 
the top is a compound curve made up of two 8-foot-radius 
curves at the sides which compound into a curve over the center 
which will give a clear height of 22 feet 6 inches over the center 
of each tie. The base of the roof curve is 13 feet 6 inches above 
the top of the ties. The bottom slopes to a central gutter which 
is 6 inches below the side corners, which are at sub-grade. Six- 
inch cast-iron pipes should be spaced as needed and run from 
each side to the central gutter. The width of both single and 
double track tunnels should be increased, if the tunnel is on a 
curve, and the track centers should also be displaced, so that 
the clearance on each side is as great as on a tangent. Figs. 
89, 90 and 91,* show some typical cross-sections. 

199. Grade. A grade of at least 0.2% is needed for drainage. 
If the tunnel is at the summit of two grades, the tunnel grade 

♦ Drinker's ''Tunneling.", 



§198. 



TUNNELS. 



23.*^ 




Fig. 89. — IIoosac Tunnel. Section throxtgh Sqlid Roc|k. 




FjQ. 90, — HopsAc TuNNE" Sectiqi^ TpROTJGB^ Soft Ghouni>, 



234 



RAILflOAD CONSTRUCTION. 



§200. 



should be practically level, with an allowance for drainage, the 3 
actual summit being at either end but not in the center. When ' 
the tunnel forms part of a long ascending grade, it is advisable 
to reduce the grade through the tunnel unless the tunnel is 
very short. The additional atmospheric resistance and thei 
decreased adhesion of the driver wheels on the damp rails in*; 
a tunnel will cause an engine to work very hard and still more 
rapidly vitiate the atmosphere until the accumulation of poison- 
ous gases becofnes a source of actual danger to the engineer and 




\ Fig. 91. -St. Cloud Tunnel. 

fireman of the locomotive and of extreme discomfort to the 
passengers. If the nominal ruling grade of the road were 
maintained through a tunnel, the maximum resistance would be 
found in the tunnef. This would probably cause trains to stall 
there, which would be objectionable and perhaps dangerous. 

200. Lining. It is a characteristic of many kinds of rock 
and of all earthy material that, although they may be self- 
sustaining when first exposed to the atmosphere, they rapidly 
disintegrate and require that the top and perhaps the ?ides and 
even the bottom shall be lined to prevent caving in. In this 
country, when timber was cheap, it was formerly framed as an 
arch and used as the permanent lining (see Fig. 92), but in 



§201. 



TUNNELS. 



235 



any such case the cross-section should be made extra large 
so that a masonry lining may subsequently be placed inside 
the wooden lining and thus postpone a large expense until the 
road is better able to pay for the work. In very soft unstable 
material, like quicksand, an arch of cut stone voussoirs may be 
necessary to withstand the pressure. A good quality of brick is 
occasionally used for lining, as they are easily handled and make 
good masonry if the pressure is not excessive. Only the best 
of cement mortar should be used, economy in this feature being 
the worst of folly. Of; course the excavation must include 'the 
outside line of the lining. Any excavation which is made out- 
side of this line (by the fall of earth or loose rock or by excessive 
blasting) must be refilled with stone well packed in. Occasionally 
it is necessary to fill these spaces with concrete. Of course it is 
not necessary that the lining be uniform throughout the tunnel. 
201. Shafts. Shafts are variously made with square, rectan- 
gular, elliptical, and circular cross-sections. The rectangular. 




Fig. 92, —Connection: with Shaft, Church Hili. Tunneu 



cross-section, with the longer axis parallel with the tunnel, is 
most usually employed. Generally the shaft is directly over the 
center of the tunnel, but that always implies a comphcated con- 
nection between the hnings of the tunnel and shaft, provided 



236 



RAILROAD CONSTRUCTION. 



§202. 



such linings are necessaty. It ig easier to sink a shaft near to 
one side of the tunnel and make an opening through the nearly 
vertical side of the tunnel. Such a method was employed in the 
Church Hill Tunnel, illustrated in Fig. 92.* Fig. 93 f shows 
a cross-section for a large main shaft. Many shafts have been 
built with the idea of being left open permanently for ventila- 
tion and have therefore been elaborately lined with masonry. 




Fig. 93. — Cross-segtion. Large Main Shaft. 

The general consensus of opinion now appears to be that shafts 
are worse than useless for ventilation ; that the quick passage of 
a train through the tunnel is the most effective ventilator; and 
that shafts only tend to produce cross-currents and are ineffective 
to clear the air. In consequence, many of these elaborately 
lined shafts have been permanently closed, and the more recent 
practice is to close up a shaft as soon as the tunnel is completed. 
Shafts always form 'drainage -wells for the material they pass 
through, and sometimes to such an extent that it is a serious 
matter to dispose of the water that collects at the bottom, 
requiring the construction of large and expensive drains. 

202. Drains. A tunnel will almost invariably strike veins of 
water which will promptly begin to drain into the tunnel and 
not only cause considerable trouble and expense during construc- 
tion, but necessitate the provision of permanent drains for its 
perpetual disposal. These drains must frequently be so large as 



* Drinker's "Tunneling." 

t Rziha, "Lehrbuch der Gesammten TunttelbaUkUtist." 



§203. 



TUNNELS. 



237 



to appreciably increase the required cross-section of the tunnel. 
Generally a small open gutter on each side will suffice for this 
purpose, but in double-track tunnels a large covered drain is 
often built; between the tracks. It is sometimes necessary to 
thoroughly grout the outside of the lining so that water will not 
force its way through the masonry and perhaps injure it, but 
ma,y freely drain down the sides and pass through openings in 
the side walls near their base into the gutters. 



CONSTRUCTION. 

203. Headings. The naethods of all tunnel excavation de- 
pend on the general principle that all earthy material, except 
the softest of liquid mud and quicksand, will be self-sustaining 
over a greater or less area and for a greater or less tim^e after 
excavation is made, and the work consists in excavating some 
material and immediately propping up the exposed surface by 
timbering and poling-boards. The excavation of the cross- 
section begins with cutting out a "heading/' which is a small 
horizontal drift whose breast is constantly kept 15 feet or more 
in advance of the full cross-sectional excavation. In solid 
self-sustaining rock, which will not decompose upon exposure 
to air, it becomes simply a matter of excavating the rock with 
the least possible expenditure of time and energy. In soft 
ground the heading must be heavily timbered, and as the heading 
is gradually enlarged the timbering must be gradually extended 
and perhaps replaced, according to some regular system, so that 
when the full cross-section has been ex- 
cavated it is supported by such timbering 
as is intended for it. The heading is 
sometimes made on the center line near 
the top; with other plans, on the center 
line near the bottom ; and sometimes two 
simultaneous headings are run in the two 
lower corners. Headings near the bot- 
tom serve the purpose of draining the 
material above it and facilitating the 
excavation. The simplest case of head- 
ing timbering is that shown in Fig. 94, 
in which cross-timbers are placed at in- ^la. 94. 

tervals just under the roof, set in notches 
cut in the side walls and supporting poling-boards which sus- 




238 



KAILROAD CONSTRUCTION. 



§204. 



tain whatever pressure may come on them. Cross-timbers 
near the bottom support a flooring on which vehicles for trans- 
porting material may be run and under which the drainage 
may freely escape. As the necessity for timbering becomes 
greater, side timbers and even bottom timbers must be added, 
these timbers supporting poling -boards, and even the breast 
of the heading must be protected by boards suitably braced. 




Fig. 95. — Timbering for Tunnel Heading. 



as shown in Fig. 95. The supporting timbers are framed into 
collars in such a manner that added pressure only increases 
their rigidity. 

204. Enlargement. Enlargement is accomplished by remov- 
ing the poling-boards, one at a time, excavating a greater or less 
amount of material, and immediately supporting the exposed 
material with poling-boards suitably braced, (See Figs. 95 and 
96.) This work being systematically done, space is thereby 
obtained in which the framing for the full cross-section may be 
gradually introduced, The framing is constructed with a cross- 



§205. 



TUNNELS. 



239 



section so large that the masonry lining may be constructed 
within it. 
205. Distinctive features of various methods of construction. 

There are six general systems, known as the English, German, 
Belgian, French, Austrian, and American. They are so named 




Fia. 96. 



from the origin of the methods, although their use is not con- 
fined to the countries named. Fig. 97 shows by numbers (1 to 5) 
the order of the excavation within the cross-sections. The Eng- 
lish, Austrian, and American systems are alike in excavating the 
entire cross-section before beginning the construction of the 
masonry lining. The German method leaves a solid core (5) 
until practically the whole of the lining is complete. This has 
the disadvantage of extremely cramped quarters for work, poor 
ventilation, etc. The Belgian and French methods agree in 
excavating the upper part of the section, building the arch at 
once, and supporting it temporarily until the side walls are 
built. The Belgian method then takes out the core (3), removes 
very short sections of the sides (4) immediately underpinning 
the arch with short sections of the side walls and thus gradually 
constructing the whole side wall. The French method digs out 
the sides (3), supporting the arch temporarily with timbers and 
then replacing the Limbers with masonry; the core (4) is taken 
out last. The French method has the same disadvantage as the 
German — working in a cramped space. The Belgian and French 
systems have the disadvantage that the arch, supported tempo- 
rarily on timber, is very apt to be strained and cracked by the 
slight settlement that so frequently occurs in soft material. The 
Enghsh, Austrian, and American methods differ mainly in the 



245 



KAILEOAD CONSTRUCTION. 



§ 205. 



design of the timbering. The English support the roof by lines 
of very heavy longitudinal timbers which are supported at com- 
paratively wide intervals by a heavy framework occupying the 




I 

4 


' I"* 


1 

J 


5 1 i 

1 

1 



i 



ENGLISH 



AUSTRIAN 





L.-..L,.-. 



J- 



GERMAN 



BELGIAN 





FRENCH AMERICAN 

Fig. 97. — Ordek of Workino by the Various Systems. 



whole cross-section. The Austrian system uses such frequent 
cross-frames of timber-work that poling-boards will suffice to 
support the material between the frames. The American sys- 
tem agrees with the Austrian in using frequent cross-frames 



§ 206, TUNNELS. 241 

supporting poling-boards, but differs from it in that the "cross 
frames" consist simply of arches of 3 to 15 wooden voussoirs, 
the voussoirs being blocks of 12''Xl2" timber about 2 to 8 feet 
long and cut with joints normal to the arch. These arches are 
put together on a centering which is removed as soon as the arch 
is keyed up and thus immediately opens up the full cross-section, 
so that the center core (4) may be immediately dug out and the 
masonry constructed in a large open space. The American sys- 
tem has been used successfully in very soft ground, but its ad- 
vantages are greater in loose rock, when it is much cheaper than 
the other methods which employ more timber. Fig. 92 and 
Plate III illustrate the use of the American system. Fig. 92 
shows the wooden arch in place. The masonry arch may be 
placed when convenient, since it is possible to lay the track and 
commence traffic as soon as the wooden arch is in place. The 
student is referred to Drinker's "Tunneling" and to Rziha's 
"Lehrbuch der Gesammten Tunnelbaukunst " for numerous 
illustrations of European methods of tunnel timbering. 

206. Ventilation during construction. Tunnels of any great 
length must be artificially ventilated during construction. If 
the excavated material is rock so that blasting is necessary, the 
need for ventilation becomes still more imperative. Fresh air 
is forced into the tunnel at or near the heading ("plenum proc- 
ess ") and the foul air is thereby crowded out, or the foul air 
is sucked out (" vacuum process ") and fresh air rushes into 
the tunnel at the entrance. " Compressed air wasted from 
power drills is so contaminated with oil from the cylinders that 
it cannot be taken into consideration as ventilation." The draft 
of air up a shaft will occasionally modify, and perhaps assist, 
the work of ventilation, but, in general, the work must be done 
by means of power fans. 

207. Excavation for the portals. Under normal conditions 
there is always a greater or less amount of open cut preceding 
and following a tunnel. Since all tunnel methods depend (to 
some slight degree at least) on the ca,pacity of the exposed ma- 
terial to act as an archj there is implied a considerable thickness 
of material above the tunnel. This thickness is reduced to 
nearly zero over the tunnel portals and therefore requires special 
treatment, particularly when the material is very soft. Fig. 98 * 

* Eisiha, " Lehrbuch der Gesammtea Tunnelbaukunst," 



242 



RAILROAD CONSTRUCTION. 



§208. 



illustrates one method of breaking into the ground at a portal. 
The loose stones are piled on the framing to give stability to the 
framing by their weight and also to retain the earth on the 




Fig. 98. — Timbering fok Tunnel Poetal. 



slope above. Another method is to sink a temporary shaft to 
the tunnel near the portal; immediately enlarge to the full size 
and build the masonry lining; then work back to the portaL 
This method is more costly, but is preferable in very treacherous 
ground, it being less liable to cause landslides of the surface 
material. 

2o8. Tunnels vs. open cuts. In cases in which an open cut 
rather than a tunnel is a possibility the ultimate consideration 
is generally that of first cost combined with other financial con- 



[3 




(To face page 243.) 



PLATE III. 




LONGITtTDINAI. SSCTZOK OV FOKTAIt. 



§209. 



TUNNELS. 



243 



siderations and annual maintenance charges directly or indirectly 
connected with it. Even when an open cut may be constructed 
at the same cost as a tunnel (or perhaps a little cheaper) the 
tunnel may be preferable under the following conditions: 

1. When the soil indicates that the open cut would be liable 
to landslides. 

2. When the open cut would be subject to excessive snow- 
drifts or avalanches. 

3. When land is especially costly or it is desired to run under 
existing costly or valuable buildings or monuments. When run- 
ning through cities, tunnels are sometimes constructed as open 
cuts and then arched over. 

These cases apply to tunnels vs. open cuts when the aline- 
ment is fixed by other considerations than the mere topography. 
The broader question of excavating tunnels to avoid excessive 
grades or to save distance or curvature, and similar problems, 
are hardly susceptible of general analysis except as questions of 
railway economics and must be treated individually. 

209. Cost of tunneling. The cost of any construction which 
involves such uncertainties as tunneling is very variable. It 
depends on the material encountered, the amount and kind of 
timbering required, on the size of the cross-section, on the price 
of labor, and especially on the reconstruction that may be neces- 
sary on account of mishaps. 

Headings generally cost $4 to S5 per cubic yard for excava- 
tion, while the remainder of the cross-section in the same tunnel 
may cost about half as much. The average cost of a large 
number of tunnels in this country may be seen from the follow- 
ing table:* 





Cost per cubic yard. 


Cost per 
lineal foot. 




Excavation. 


Masonry. 




Material. 


Single. 






Single. 


Double. 


Single. 


Double. 


Double. 


Hard rock 

Loose rock 

Soft ground. . . . 


$5.89 
3.12 
3.62 


$5.45 
3.48 
4.64 


$12.00 

9.07 

15.00 


$8.25 
10.41 
10.50 


$69.76 

80.61 

135.31 


$142.82 
119.26 
174.42 



* Figures derived. from Drinker's "Tunneling." 



PLATE in. 




(Jo face page 243.) 



Elevation op Portal. 
Phcenizyills Tonnbl. p. S. V. R. R. 



'-IJ-L' 

LoNQITUDtNAI, SsCTIOIt OW FoBMIi. 



244 EAILROAD CONSTRUCTION. § 209. 

The above figures are averages for tunnels constructed between 
1831 and 1877. The prices paid for labor varied from $1.00 to 
$2.75 per day for " miners " and 0.75 to $2.00 for unskilled labor. 
The lower figures were usually paid during the earlier years. 
As an approximate average, the figures of $2.00 per day for 
miners and $1.50 per day for unskilled labor may be said to cor- 
respond to the average costs given in the tabular form. On 
the basis that all other expenses (explosives, cost of equipment^ 
etc.) vary proportionately to wages, the tabular figures can 
even now be utilized by increasing them according to the present 
scale for labor. The figures are also instructive since they show 
the relative cost of tunneling through hard rock, loose rock and 
soft ground. 



CHAPTER VI. 
CULVERTS AND MINOR BRIDGES. 

210. Definition and object. Although a variable percentage 
of the rain falling on any section of country soaks into the 
ground and does not immediately reappear, yet a very large 
percentage flows over the surface, always seeking and following 
the lowest channels. The roadbed of a railroad is constantly 
intersecting these channels, which frequently are normally dry. 
In order to prevent injury to railroad embankments by the im- 
pounding of such rainfall, it is necessary to construct waterways 
through the embankment through which such rainflow may 
freely pass. Such waterways, called culverts, are also appli- 
cable for the bridging of very small although perennial streams, 
and therefore in this work the term culvert will be applied to 
all water-channels passing through a railroad embankment 
which are not of sufficient magnitude to require a special struc- 
tural design, such as is necessary for a large masonry arch or a 
truss bridge, 

211. Elements of the design. A well-designed culvert must 
afford such free passage to the water that it will not "back up" 
over the adjoining land nor cause any injury to the embankment 
or culvert. The ability of the culvert to discharge freely all the 
water that comes to it evidently depends chiefly on the area of 
the waterway, but also on the form, length, slope, and materials 
of construction of the culvert and the nature of the approach 
and outfall. When the embankment is very low and the amount 
of water to be discharged very great, it sometimes becomes 
necessary to allow the water to discharge "under a head," i e., 
with the surface of the water above the top of the culvert. 
Safety then requires a much stronger construction than would 
otherwise be necessary to avoid injury to the culvert or embank- 
ment by w^ashing. The necessity for such construction should 
be avoided if possible. 

245 



246 RAILROAD CONSTRUCTION. §212. 



AREA OF THE WATERWAY. 

212. Elements involved. The determination of the required 
area of the waterway involves such a multiplicity of indeter- 
minate elements that any close determination of its value from 
purely theoretical considerations is a practical impossibility^ 
The principal elements involved are: 

a. Rainfall. The real test of the culvert is its capacity to 
discharge without injury the flow resulting from the extraordi- 
nary rainfalls and "cloud bursts" that may occur once in many 
years. Therefore, while a knowledge of the average annual 
rainfall is of very little value, a record of the maximum rainfall 
during heavy storms for a long term of years may give a relative 
idea of the maximum demand on the culvert. 

b. Area of watershed. This signifies the total area of country 
draining into the channel considered. When the drainage area 
is very small it is sometimes included within the area surveyed 
by the preliminary survey. When larger it is frequently possi- 
ble to obtain its area from other maps with a percentage of 
accuracy sufficient for the purpose. Sometimes a special survey 
for the purpose is considered justifiable. 

c. Character of soil and vegetation. This has a large in- 
fluence on the rapidity with which the rainflow from a giA^en 
area will reach the culvert. If the soil is hard and impermeable 
and the vegetation scant, a heavy rain will run off suddenly, 
taxing the capacity of the culvert for a short time, while a 
spongy soil and dense vegetation will retard the flow, making it 
more nearly uniform and the maximum flow at any one time 
much less. 

d. Shape and slope of watershed. If the watershed is very 
long and narrow (other things being equal), the water from the 
remoter parts will require so much longer time to reach the 
culvert that the flow will be comparatively uniform, especially 
when the slope of the whole watershed is very low. When the 
slope of the remoter portions is quite steep it may result in the 
nearly simultaneous arrival of a storm- flow from all parts of the 
watershed, thus taxing the capacity of the culvert. 

e. Effect of design of culvert. The principles of hydraulics 
show that the slope of the culvert, its length, the form of the 
cross-section, the nature of the surface, and the form of the 



§ 213. CULVERTS AND MINOR BRIDGES. 247 

approach and discharge all have a considerable influence on the 
area of cross-section required to discharge a given volume of 
water in a given time, but unfortunately the combined hy- 
draulic effect of these various details is still a very uncertain 
quantity. 

213. Methods of computation of area. There are three pos- 
sible methods of computation. 

(a) Theoretical. As shown above it is a practical impossi- 
bility to estimate correctly the combined effect of the great mul- 
tiplicity of elements which influence the final result. The nearest 
approach to it is to estimate by the use of empirical formula? 
the amount of water which will be presented at the upper end 
of the culvert in a given time and then to compute, from the 
principles of hydraulics, the rate of flow through a culvert of 
given construction, but (as shown in § 212, e) such methods are 
still very unreliable, owing to lack of experimental knowledge. 
This method has apparently greater scientific accuracy than 
other methods, but a little study will show that the elements 
of uncertainty are as great and the final result no more reliable. 
The method is most reliable for streams of uniform flow, but 
it is under these conditions that method (c) is most useful. The 
theoretical method will not therefore be considered further. 

(b) Empirical. As illustrated in § 214, some formulae make 
the area of waterway a function of the drainage area, the for- 
mula bemg affected by a coefficient the value of which is esti- 
mated between limits according to the judgment Assuming 
that the formula? are sound, their use only narrows the limits of 
error, the final determination depending on experience and 
judgment in the choice of the proper coefficient. 

(c) From observation. This method, considered by far the 
best for permanent work, consists in observing the high-water 
marks on contracted channel-openings which are on the same 
stream and as near as possible to the proposed culvert. If the 
country is new and there are no such openings, the wisest plan 
is to bridge the opening by a temporary structure in wood which 
has an ample waterway (see § 158, h, 4) and carefully observe 
all high-water marks on that opening during the 6 to 10 years 
which is ordinarily the minimum life of such a structure. As 
shown later, such observations may be utilized for a close com- 
putation of the required waterway. Method (b) may be utilized 
for an approximate calculation for the required are^ for the tern- 



248 BAILROAD CONSTRUCTION. § 214 

porary structure, using a value which is intentionally excessive, 
so that a permanent structure of sufficient capacity may subse* 
quently be constructed within the temporary structure. 

214. Empirical formulae. Two of the best known empirical 
formulae for area of the waterway are the following: 

(4) Myer's formula: 
Area of waterway in square feet =CX \/drainage area in acres, 
where is a coefficient varying from 1 for flat country to 4 for 
mountainous country and rocky ground. As aii illustration, if 
the drainage area is 100 acres, the waterway area should be from 
10 to 40 square feet, according to the value of the coefficient 
chosen. It should be noted that this formula does not regard 
the great variations in rainfall in various parts of the world nor 
the design of the culvert, and also that the final result depends 
largely on the choice of the coefficient. 

(b) Talbot's formula: 

Area of waterway in square feet = CX>y (drainage area in acres) ^. 
'- For steep and rocky ground C varies from f to 1. For rolling 
agricultural Country subject to floods at times of melting snow, 
and with the length of the valley three or four times its width, C 
is about 1^; and if the stream is longer in proportion to the area, 
decrease C In districts not affected by accumulated snow, and 
f/here the length of the valley is several times the width, 3 or ^ , 
or even less, may be used. C should be increased for steep side 
slopes, especially if the upper part of the valley has a much 
greater fall than the channel at the culvert." * As an illus^ 
tration, if the drainage area is 100 acres the area of waterway 
should be CX31.6. The area should then vary from 5 to 31 
square feet, according to the character of the country. Like 
the previous estimate, the result depends on the choice of a 
coefficient and disregards local variations in rainfall, except as 
they may be arbitrarily allowed for in choosing the coeffi^ 
eient. 

215. Value of empirical formulae. The fact that these fop-» 
mulse, as well as many others of similar nature that have been 
Suggested, depend so largely upon the choice of the coefficient 
shows that they are valuable " more as a guide to the judgment 
than as a working rule," as Prof, Talbot explicitly declares in 



* Prof. A. N. Talbot, "Selected Papers of the Civil Engineers' Club of 
the Univ. of Illinois." 



§ 216. CULVERTS AND MINOR BRIDGES. 249 

eommenting on his own formula. In short, they are chiefly valu^ 
able in indicating a probable maximum and minimum between 
which the true result probably lies. 

216. Results based on observation. As already indicated in 
§ 213, observation of the stream in question gives the most 
reliable results. If the country is new and no records of the 
flow of the stream during hea^y storms has been taken, eveii 
the life of a temporary wooden structure may not be long enough 
to include one of the unusually severe storms which must be 
allowed for, but there will usually be some high-water mark 
which will indicate how much opening will be required. The 
following quotation illustrates this: "A tidal estuary may gen- 
erally be safely narrowed considerably from the extreme water 
lines if stone revetments are used to protect the bank from 
wash. Above the true estuary, where the stream cuts through 
the marsh, we generally find nearly vertical banks, and we are 
safe if the faces of abutments are placed even with the banks. 
In level sections of the country, where the current is sluggish, 
it is usually safe to encroach somewhat on the general width 
of the stream, but in rapid streams among the hills the widtlj 
that the stream has cut for itself through the soil should not b^ 
lessened, and in ravines carrying mountain torrents the open- 
ings must be left very much larger than the ordinary appear- 
ance of the banks of the stream would seem to make neces- 
sary." * 

As an illustration of an observation of a storm-flow through 
a temporary trestle, the following is quoted : " Having the flood 
height and velocity, it is an easy matter to determine the vol- 
ume of water to be taken care of. I have one ten-bent pile 
trestle 135 feet long and 24 feet high over a spring branch that 
ordinarily runs about six cubic inches per second. Last sum- 
mer during one of our heavy rainstorms (four inches in less 
than three hours) I visited this place and found by float obser- 
vations the surface velocity at the highest stage to be 1.9 feet 
per second. I made a high-water mark, and after the flood- 
water receded found the width of stream to be 12 feet and an 
average depth of 2| feet. This, with a surface velocity of 1.9 
feet per second, would give approximately a discharge of 50 



* J. P. Snow, Boston & Maine Railway. From Report to Association of 
Railway Superintendents of Bridges and Buildings. 1897. 



250 RAILROAD CONSTRUCTION. § 217 

cubic feet, or 375 gallons, per second. Having this information 
it is easy to determine size of opening required." * 

217. Degree of accuracy required. The advantages result- 
ing from the use of standard designs for culverts (as well as 
other structures) have led to the adoption of a comparatively 
small number of designs. The practical use made of a compu- 
tation of required waterway area is to determine which one of 
several standard designs will most nearly fulfill the require- 
ments. For example, if a 24-inch iron pipe, having an area of 
3.14 square feet, is considered to be a little small, the next size 
(30-inch) would be adopted; but a 30-inch pipe has an area of 
4.92 square feet, which is 56% larger, A similar result, except 
that the percentage of difference might not be quite so marked, 
will be found by comparing the areas of consecutive standard 
designs for stone box culverts. 

The advisability of designing a culvert to withstand any 
storm-flow that may ever occur is considered doubtful. Several 
years ago a record-breaking storm in New England carried 
away a very large number of bridges, etc., hitherto supposed 
to be safe. It was not afterward considered that the design of 
those bridges was faulty, because the extra cost of constructing 
bridges capable of withstanding such a flood, added to interest 
for a long period of years, would be enormously greater than the 
cost of repairing the damages of such a storm once or twice in 
a century. Of course the element of danger has some weight, 
but not enough to justify a great additional expenditure, for 
common prudence would prompt unusual precautions during 
or immediately after such an extraordinary storm. 

PIPE CULVERTS. 

218. Advantages. Pipe culverts, made of cast iron or earthen- 
ware, are very durable, readily constructed, moderately cheap, 
will pass a larger volume of water in proportion to the area than 
many other designs on account of the smoothness of the sur- 
face, and (when using iron pipe) may be used very close to 
the track when a low opening of large capacity is required. 
Another advantage lies in the ease with which they may be 
inserted through a somewhat larger opening that has been 

♦ A. J. Kelley, Kansas City Belt Railway. From Report to Association 
of Railway Superintendents of Bridges and Buildings. 1897. 



§ 219. CULVERTS AND MINOR BRIDGES. 251 

temporarily lined with wood, without disturbing the roadbed 
or track 

219. Construction. Permanency requires that the founda- 
tion shall be firm and secure against being washed out. To 
accomplish this, the soil of the trench should be hollowed out to 
fit the lower half of the pipe, making suitable recesses for the 
bells. In very soft treacherous soil a foundation-block of con- 
crete is sometimes placed under each joint, or even throughout 
the whole length. When pipes are laid through a slightly 
larger timber culvert great care should be taken that the pipes 
are properly supported, so that there will be no settling nor 
development of unusual strains when the timber finally decays 
and gives way. To prevent the washing away of material 
around the pipe the ends should be protected by a bulkhead. 
This is best constructed of masonry (see Fig. 99), although wood 
is sometimes used for cheap and minor constructions. The joints 
should be calked, especially when the culvert is liable to run 
full or when the outflow is impeded and the culvert is liable to 
be partly or wholly filled during freezing weather. The cost of 
a calking of clay or even hydraulic cement is insignificant com- 
pared with the value of the additional safety afforded. When 
the grade of the pipe is perfectly uniform, a very low rate of 
grade will suffice to drain a pipe culvert, but since some uneven- 
ness of grade is inevitable through uneven settlement or im- 
perfect construction, a grade of 1 in 20 should preferably be 
required, although much less is often used. The length of a 
pipe culvert is approximately determined as follows: 

Length = 2s (depth of embankment) + (width of roadbed), 

in which s is the slope ratio (horizontal to vertical) of the banks. 
In practice an even number of lengths should be used which will 
equal or exceed the length given by this formula. 

220. Iron-pipe culverts. Simple cast-iron pipes are used in 
sizes from 12" to 48" diameter. These are usually made in 
lengths of 12 feet with a few lengths of 6 feet, so that any required 
length may be more nearly obtained. The lightest pipes made 
are sufficiently strong for the purpose, and even those which 
would be rejected because of incapacity to withstand considerable 
internal pressure may be utilized for this work. In Fig. 99 are 
shown the standard plans used on the C. C. C. & St. L. Ry., 
which may be considered as typical plans. 






k 


~~" 






\ 














•'m.' 






\ 




















\ 


















=1 ' 










c 

_ 


1- 




^0, 


}{ 


f 


r\ 




/f 


NVH 


1- ^ 




1p 




S' 


3T J. 


3N -^ 




f 








^ 






( 
























>j 


/ 









-9,-2 > 



< — a + 




0/3 > 



j0;2- 



i 



Fig. 99. — Standard Cast-iron 
Pipe C i.vert. C. C C. & 
St. L. Ry. (May 1893.) 



Q 

4- 



252 



1221. 



CULVERTS AND MINOR BRIDGES. 



253 



Pipes formed of cast-iron segments have been used up to 12 
feet diameter. The shell is then made comparatively thin, but 
is stiffened by ribs and flanges on the outside. The segments 
break joints and are bolted together through the flanges. The 
joints are made tight by the use of a tarred rope, together with 
neat cement. 

221. Tile-pipe culverts. The pipes used for this purpose vary 
from 12" to 30" in diameter. When a larger capacity is required 
two or more pipes may be laid side by side, but in such a case 
another design might be preferable. It is frequently specified 
that " double-strength " or " extra-heavy " pipe shall be used. 




UP-STR.EAM_END. DOWN-STREAM END. DOWN-STREAtyJ £N.D. THREE PIPES. 

Fig. 100. — Standard Vitrified-pipe Ctjlvert. Plant System. (1891.) 

The author's personal experience is that tile pipe are very unreli- 
able as culvert pipe, especially if there is any subsidence of the 
original soil which supports the embankment. See § 127-8. 
When a tile pipe is laid in a sewer, the soil on which it is laid is 
usually compact and there is no subsequent settlement. But 
when a culvert pipe is laid on soft meadow soil and a high em- 
bankment is formed over it, there is almost inevitably a settle- 
ment, which is probably not uniform and the culvert settles out 
of line, even if it does not break up and collapse. If the bed of the 
stream is rocky (precluding future settlement) and the pipes are 
bedded in concrete, there is less chance of failure. In Fig. 100 
are shown the standard plans for vitrified-pipe culverts as used 



254 



RAILROAD CONSTRUCTION. 



§222. 



1 



on the " Plant system." Tile pipe is much cheaper than iron 
pipe, but is made in much shorter lengths and requires much 
more work in laying and especially to obtain a uniform grade. 

Concrete pipes, factory made, both plain and with metal rein- 
forcement, 12" to 48" in diameter, have come into use in recent 
years. They are stronger and more dependable than tile and 
there is no deterioration. 

BOX CULVERTS. 

222. Wooden box culverts. This form serves the purpose 
of a cheap temporary construction which allows the use of a 
ballasted roadbed. As in all temporary constructions, the area 
should be made considerably larger than the calculated area 
(§§213-216), not only for safety but also in order that, if the 
smaller area is demonstrated to be sufficiently large, the per- 
manent construction (probably pipe) may be placed inside with- 
out disturbing the embankment. All designs agree in using 
heavy timbers (12",X12", 10"X12", or 8"X12") for the side 
walls, cross-timbers for the roof, every fifth or sixth timber 
being notched down so as to take up the thrust of the side walls, 
and planks for the flooring. Fig. 101 shows some of the standard 
designs as used by the C, M. & St, P. Ry, 




FiG.lOl. — Standakd Timber Box Culvert. C.,M.& St. P. Ry. (Feb. 1889.) 

223. Stone box culverts. In localities where a good quality 
of stone is cheap, stone box culverts are the cheapest form of 
permanent construction for culverts of medium capacity, but 
their use is decreasing owing to the frequent difficulty in obtain- 
ing really suitable stone within a reasonable distance of the 
culvert. The clear span of the cover-stones varies from 2 to 4 
feet. The required thickness of the cover-stones is sometimes 



§223. 



CULVERTS AND MINOR BRIDGES. 



255 



calculated by the theory of transverse strains on the basis of 
certain assumptions of loading — as a function of the height of 
the embankment and the unit strength of the stone used. Such 
a method is simply another illustration of a class of calculations 
which look very precise and beautiful, but which are worse than 
useless (because misleading) on account of the hopeless uncer- 




" 1 jc 1 



PLAN 

Fig. 102. — Standard Single Stone Culvert (3'X40. N. & W. R.R. 

(1890.) 

tainty a"s to the true value of certain quantities which must be 
used in the computations In the first pla,ce the true value of 
the unit tensile strength of stone is such an uncertain and variable 



256 



EAILROAD CONSTRUCTION. 



§223. 



quantity that calculations based on any assumed value for it are 
t>f small reliability. In the second place the weight of the prism 
of earth lying directly above the stone, plus an allowance for live 
load, is by no means a measure of the load on the stone nor of 
the forces that tend to fracture it. All earthwork will tend tQ; 




n^'T'TTTv^x 








PLAN 



Fig. 103. — Standard Double Stone Culvert (3'X4'). N. &W. R. R. 

(1890.) 

form an arch above any cavity and thus relieve an uncertain 
and probably variable proportion of the pressure that might 
otherwise exist. The higher the embankment the less the pro- 



§224. 



CULVERTS AND MINOR BRIDGES. 



257 



porfionate loading, until at some uncertain height an increase 
in height will not increase the load on the cover-stones. The 
'feffect of frost is likewise large, but uncertain and not computable. 
The usual practice is therefore to make the thickness such as 
experience has shown to be safe with a good quality of stone, 
i.e., about 10 or 12 inches for 2 feet span and up to 16 or 18 
inches for 4 feet span- The side waUs should be carried down 
deep enough to prevent their being undermined by scour or 
heaved by frost. The use of cement mortar is also an important 
feature of first-class work, especially when there is a rapid scour- 
ing current or a liability that the culvert will run under a head. 
In Figs. 102 and 103 are shown standard plans for single and 
double stone box culverts as used on the Norfolk & Western R.R. 
224. Old^rail culverts. It sometimes happens (although very 
rarely) that it is necessary to bring the grade line within 3 or 4 
feet of the bottom of a stream and yet allow an area of 10 or 12 
Bquare feet. A single large pipe of sufficient area could not be 
used in this case. The use of several smaller pipes side by side 
would be both expensive and inefficient. For similar reasons 
neither wooden nor stone box culverts could be used. In such 
Cases, as well as in many others where the head-room is not so 
limited, the plan illustrated in Fig. 104 is a very satisfactory 




BROKEN STONE BALLAST _ 

^TTTTTTTTTT»TTrTTTTTTTtTTTTTTTTTTTTTT| 




Fig. 104, — Standard Old-bail Culvert. N. & W. R.R. (1895.) 

solution of the problem. The old rails, having a length of 8 or 
9 feet, are laid close together across a 6-foot opening. Some- 
times the rails are held together by long bolts passing througl? 



258 RAILROAD CONSTRUCTION. § 225. I 

the webs of the rails. In the plan shown the rails are confined 
by low end walls on each abutment. This plan requires only 
15 inches between the base of the rail and the top of the culvert 
channel. It also gives a continuous ballasted roadbed. 

225. Reinforced Concrete Culverts. The development of 
reinforced concrete as a structural material is illustrated in its 
extensive adoption for arches and also for culverts. One of the 
special types which has been adopted is that of a box culvert 
which has a concrete bottom. Since this bottom can be made 
so that it will withstand an upward transverse stress, it furnishes I 
a broad foundation for the whole culvert, and thus entirely 1 
eliminates the necessity for extensive footing to the side walls of 
the culvert, such as are necessary in soft ground with an ordinary j 
stone culvert. Another advantage is that the inside of the cul- ^ 
vert may be made perfectly smooth and thus offer less resistance 
to the passage of water through it. As may be noticed from, 
Fig. 105, such a culvert is provided with flaring head walls, and 
sunken end walls, so that the water may not scour underneath 
the culvert, and other features common to other types. No 
attempt will here be made to discuss the design of reinforced 
concrete, except to say that all four sides of such a box culvert 
are designed to withstand a computed bursting pressure which 
tends to crush the flat sides inward. In Fig, 105 is shown one 
illustration of the many types of culverts which have been 
designed of reinforced concrete. 



ARCH CULVERTS. 

226. Influence of design on flow. The variations in the design 
of arch culverts have a very marked influence on the cost and 
efficiency. To combine the least cost with the greatest effi- 
ciency, due weight should be given to the following elements: 
(a) amount of masonry. (6) the simplicity of the constructive 
work, (c) the design of the wing walls, (d) the design of the 
junction of the wing walls with the barrel and faces of the arch, 
and (e) the safety and permanency of the construction. These 
elements are more or less antagonistic to each other, and the 
defects of most designs are due to a lack of proper proportion 
in the design of these opposing interests. The simplest con-- 
struction (satisfying elements h and e) is the straight barrel arch 



STANDARD ARCH CULVERT 

8 FEET SEAN 

I5ORFOLK & WESTERN R.R. 

(1891) 




note:- IN PLAGE OF BRICK ARCH, ^ 

RUBBLE STONE ARCH Of SAM£ THICK" ' \ 
MESS MAY BE USED ^ 



{To face 'page 2m.) 



t-:i. 



PLATE IV 




■RONG CEMENT MORTAR J, TO BE TARRIED OVER CROWNy 

w ai st nn . ttJ'nitAiir/ai .t aiKto 

' I ' I 'i ' I ' I 'i 



3 RINGS OF BRICK ( ' ^. | 

I ' — r — ' — T^ 




I 



§226. 



CULVERTS AND MINOR BRIDGES. 



259 





o 

I- 
o 

UJ 
CO 

CO 
CO 

O 
a: 
o 




260 



RAILROAD CONSTRUCTION. 



§227. 



between two parallel vertical head walls, as sketched in Fig. 
106, a. From a hydraulic standpoint the design is poor, as the 
water eddies around the corners, causing a great resistance 
which decreases the flow. Fig. 106; b, shows a mucji fetter 




Fig. 106. — Types of Culvekts. 

design in many respects, but much depends on the details of the 
design as indicated in elements (6) and (d). As a general thing 
a good hydraulic design requires complicated and expensive 
masonry construction, i.e., elements (b) and (d) are opposed. 
Design 106, c, is sometimes inapplicable because the water is 
liable to work in behind the masonry during floods and perhaps 
cause scour. This design uses less masonry than (a) or (b). 

227. Example of arch culvert design. In Plate IV is shown 
the design for an 8-foot arch culvert according to the standard 
of the Norfolk and Western R. R. Note that the plan uses the 
fl^,ring wing walls (Fig. 106, 6) on the up-stream side (thus 
protecting the abutments from scour) and straight wing walls 
(similar to Fig, 106, c) on the down-stream end. This econo- 
mizes masonry and also simplifies the constructive work. Note 
also the simplicity of the junction of the wing walls with the 
barrel of the arch, there being no re-entrant angles below the 
springing line of the arch. The design here shown is but one 
of a set of designs for arches varying in span from 6' to 30'. 



MINOR OPENINGS. 



228. Cattle-guards, (a) Pit guards. Cattle-guards will be 
considered under the head of minor openings, since the old- 
fashioned plan of pit guards, which are even now defended and 



§228. 



CULVERTS AND MINOR BRIDGES. 



261 



preferred by some railroad men, requires a break in the con- 
tinuity of the roadbed. A pit about three feet deep, five feet 




Fig. 107. — Cattle-suaed with Wooden Slats. 



long, and as wide as the width of the roadbed, is walled up with 
stone (sometimes with wood), and the rails are supported on 
heavy timbers laid longitudinally with the rails. The break in 
the continuity of the roadbed produces a disturbance in the 
elastic wave running through the rails, the effect of which is 
noticeable at high velocities. The greatest objection, however, 
lies in the dangerous consequences of a derailment or a failure 
of the timbers owing to unobserved decay or destruction by 
fire — caused perhaps by sparks and cinders from passing loco- 
motives. The very insignificance of the structure often leads 
to careless inspection. But if a single pair of wheels gets off the 
rails and drops into the pit, a costly wreck is inevitable. 

(b) Surface cattle-guards. These are fastened on top of the 
ties; the continuity of the roadbed is absolutely unbroken and 
thus is avoided much of the danger of a bad wrecK owing to a 
possible derailment. The device consists essentially of overlay 
ing the ties (both inside and outside the rails) with a surface on 



262 



RAILROAD CONSTRUCTION. ' 



§228. 



which cattle will not walk. The multitudinous designs for such 
a surface are variously effective in this respect. An objection, 




Fig. 108. — Sheffield CATTLE-atTABn. . 



CENTER 8EOTIOI4 







lUustratlng how 
.Sections should be 
placed on ties and 
manner of fastening 




\ho section 



FiQ. 109.— Climax Cattle-ouard (tilb). 

whicli is often urged indiscriminately against all such designs, is 
the liability that a brake-chain which may happen to be drag- 
ging may catch in the rough bars which are used. The bars 



§ 229. CULVERTS AND MINOR BRIDGES. 263 

are sometimes " home-made," of wood, as shown in Fig. 107. 
Steel guards may be made as shown in Fig. 108. The general 
construction is the same as for the wooden bars. The metal 
bars have far greater durability, and it is claimed that they are 
more effective in discouraging cattle from attempting to cross. 
229. Cattle-passes. Frequently when a railroad crosses a 
farm on an embankment, cutting the farm into two parts, the 
railroad company is obliged to agree to make a passageway 
through the embankment sufficient for the passage of cattle and 
perhaps even farm-wagons. If the embankment is high enough 
so that a stone arch is practicable, the initial cost is the only 
great objection to such a construction; but if an open wooden 
structure is necessary, all the objections against the old-fashioned 
cattle-guards apply with equal force here. The avoidance of a 
grade crossing which would otherwise be necessary is one of the 
great compensations for the expense of the construction and 
maintenance of these structures. The construction is some- 
times made by placing two pile trestle bents about 6 to 8 feet 
apart, supporting the rails by stringers in the usual way, the 
special feature of this construction being that the embankments 
are filled in behind the trestle bents, and the thrust of the em- 
bankments is mutually taken up through the stringers, which 
are notched at the ends or otherwise constructed so that they 
may take up such a thrust. The designs for old-rail culverts 
and arch culverts are also utilized for cattle-passes when suitable 
and convenient, as well as the designs illustrated in the following 
section, and the reinforced concrete design of § 225, 

230. Standard stringer and I-beam bridges. The advantages 
of standard designs apply even to the covering of short spans 
with wooden stringers or with I beams — especially since the 
methods do not require much vertical space between the rails 
and the upper side of the clear opening, a feature which is often 
of prime importance. These designs are chiefly used for cul- 
verts or cattle-passes and for crossing over highways — providing 
such a narrow opening would be tolerated. The plans all imply 
stone abutments, or at least abutments of sufficient stability to 
withstand all thrust of the embankments. Some of the designs 
are illustrated in Plate V. The preparation of these standard 
designs should be attacked by the same general methods as 
already illustrated in § 190. When computing the required 



264 



EAlLROAi) CONSTRUCTION. 



§230, 



transverse strength, due allowance should be made for lateral 
bracing, which should be amply provided for. Note particu- 
larly the methods of bracing illustrated in Plate V. The designs 
calhng for iron (or steel) stringers may be classed as permanent 
constructions, which are cheap, safe, easily inspected and main-^ 
tained, and therefore a desirable method of construction. 



PLATE V. 



M BOLT EVERY THIRD TIE, 





15"]l-beam (double,) so lbs. per foot. 
17 ft. long, u feet clear span. 

^IqI 



ii '^ 8 X 10 TIES, U LONG, 15 C.TO C. ;[ 



ANCHOR BOLT, 1 DIA 
, lo" LONG 




s=-r 



10'c-20*6VlONG. 



TT 



=P H BOLTS, 83i UNDER HEAD. 




STANDARD LBRIDGES-14-FT. SPAN. 

NORFOLK AND WESTERN R.R. 
C1891.) 




iJi 



V TYPE "Bj' 

FROM 40 TO 70 FEET 



I 



TYPES OF PLATE GIRDER BRIDGES. 

C. M. & St.P. RY. 
(Dec. 1895.) 




(Z'o face pays 2Q4:.) 



CHAPTER Vll. 

BALLAST. 
In. 

231. Purpose and requirements. " The object of the ballast 
is to transfer the applied lo?.d over a large surface; to hold the 
timber work in place horizontally; to carry off the rain-water 
from the superstructure and to prevent freezing up in winter; 
to afford means of keeping the ties truly up to the grade line; 
and to give elasticity to the roadbed." Thi^ extremely con- 
densed statement is a description of an ideally perfect ballast. 
The value of any given kind of ballast is proportional to the 
extent to which it fulfills these requirements. The ideally 
perfect ballast is not necessarily the most economical ballast 
for all roads. Light traffic generally justifies something cheaper, 
but a very common error is to use a very cheap ballast when a 
email additional expenditure would procure a much better 
ballast, which would be much more economical in the long run. 

232. Materials. The materials most commonly employed are 
gravel and broken stone. In many sections of the country 
other materials which more or less perfectly fulfill the require- 
ments as given above, are used. The various materials includ- 
ing some of these special types have been defined by the American 
Railway Engineering Association as follows: 

DEFINITIONS. 

Ballast. Selected material placed on the roadbed for the 
purpose of holding the track in line and surface. 

Sub-ballast. Any material of a character superior to that 
in the adjacent cuts, which is spread on the finished sub-grade 
of the roadbed and below the top ballast, to provide better 
drainage, prevent upheaval by frost, and better distribute the 
load over the roadbed. 

Top-ballast. Any material of a superior character spread 
over a sub-ballast to support the track structure, distribute the 
load to the sub-ballast, and provide good initial drainage. 

265 



266 RAILROAD CONSTRUCTION. § 232. 

Stone ballast. Stone broken by artificial means Into small 
fragments of specified sizes. 

Burnt clay. A clay or gumbo which has been burned into 
material for ballast. 

Chats. Tailings from mills in which zinc, lead, silver and 
other ores are separated from the rocks in which they occur. 

Chert. An impure flint or hornstone occurring in natural 
deposits. 

Cinders. The residue from the coal used in locomotives and 
other furnaces. 

Gravel. Worn fragments of rock, occurring in natural de- 
posits, that will pass through a 2|-inch ring and be retained 
upon a No. 10 screen. 

Gumbo. A term commonly used for a peculiarly tenacious 
clay, containing no sand. 

Sand. Any hard, granular, comminuted rock which will pass 
through a No. 10 screen and be retained upon a No. 50 screen. 

Slag. The waste product, in a more or less vitrified form, 
of furnaces for reduction of ore. Usually the product of a blast- 
furnace. 

There is still another classification which may or may not be 
considered as ballast. It is perhaps hardly correct to speak of 
the natural soils as ballast, yet many miles of cheap railways 
are " ballasted " with the natural soil, which is then called Mud 
ballast. 

Broken or crushed stone. Rock ballast is specified to be that 
which will all pass in any position through a 2|-inch ring, but 
which cannot pass through a |-inch mesh. It is most easily 
handled with forks. This method also has the advantage that 
when it is being rehandled the fine chips which would interfere 
w'/th effectual drainage will be screened out. Rock ballast is more 
expensive in first cost and is also more troublesome to handle, 
but in heavy traffic especially, the track will be kept in better 
surface and will require less work for maintenance after the ties 
have become thoroughly bedded. 

Burnt clay. This material has been used in many sections 
of the country where broken stone or gravel are unobtainable 
except at a prohibitive cost, and where a suitable quality of 
clay is readily obtained. This clay should be of " gumbo " 
variety and contain no gravel. It is sometimes burnt in a kiln, 
or it is sometimes burnt by piling the clay in long heaps over 



I 



§ 232. BALLAST. 267 

a mass of fuel, the pile being formed in such a way that a tem- 
porary but effectual kiln is made. It is necessary that a clear, 
clean fuel shall be used and that the firing shall be done by a 
man who is experienced in maintaining such a fire until the 
burning is completed. Such ballast may be burned very hard 
and it will last from four to six years. The cost of burning 
varies from 30 to 60 cents per cubic yard, according to the 
circumstances. 

Chats. This is a form of ballast which is peculiar to South- 
western Missouri and Southeastern Kansas. When this mate- 
rial was first used it was obtained from the refuse piles of the 
mills which treated the zinc and lead ores mined in those regions. 
With the processes then employed the material was obtained 
in lumps as large as broken stone, and they were considered to 
be as valuable as broken stone for ballast. Improvements in 
the processes of treating the ores have resulted in making this 
by-product very much smaller grained and of less value as bal- 
last, although it is still considered a desirable form of ballast 
where it may readily be obtained. It should be noted that it 
is classed with gravel and cinders in the forms of cross-section 
shown later. 

Chert. This is a form of flint or hornstone which occurs in 
nodules of a size that is suitable for ballast, and is a very good 
type of ballast wherever it is found, but its occurrence is com- 
paratively infrequent. It is classed with cemented gravel in 
the design of cross-sections of ballast. 

Cinders. This is one of the most universal forms of ballast, 
since it is a by-product of every road which uses coal as fuel. 
The advantages consist in the fairly good drainage, the ease of 
handling and the cheapness — after the road is in operation. 
One of the greatest disadvantages is the fact that the cinders 
are readily reduced to dust, which in dry weather becomes very 
objectionable. Cinders are usually considered preferable to 
gravel in yards. 

Gravel. This is one of the most common forms of good bal- 
last. There are comparatively few railroads which cannot find, 
at some place along their line, a gravel pit which will afford a 
suitable supply of gravel for ballast. Sometimes it is used just 
as found in the pit, but for Class A and even Class B roads it is 
usually necessary to screen it. See § 238a for specifications. 

Sand. Railroads which run along the coast are frequently 



268 RAILROAD CONSTRUCTION. § 233. 

ballasted merely with the sand obtained in the immediate neigh- 
borhood. One great advantage lies in the almost perfect drain- 
age which is obtained. 

Slag. When slag is readily obtainable it furnishes an excel- 
lent ballast which is free from diist and perfect in drainage 
qualities. Slag is classified with crushed rock in the cross- 
seetiona shown below, but it should be noted that this only 
applies to the best qualities of slag, since its quality is quite 
variable. 

Mud ballast. When the natural isoil is gravelly so that rain 
will drain through it quiekly, it will make a fair roadbed for 
light traffic, but for heavy traffic, and for the greater part of 
the length of most roads, the natural soil is a very poor material 
for ballast; for, no matter how suitable the sofl might be along 
limited sections of the road, it would practically ilever happen 
that the soil would be uniformly good throughout the whole 
length of the road. Considering that a heavy rain will in one 
day spoil the results of weeks of patient ** surfacing" with mud 
ballast, it is seldim (economical to use "mud" if there is a 
gravel-bed or other source of ballast anywhere on the line of 
the road. 

233. GrosS-sectionSi The reqiiirdd depth of the cross-section 
to the sub-soil depends largely on the Weight ef the rolling 
stock which is to pacs over the track. A eareful examination 
cf a roadbed to determine the charig-es which take place under 
the ties and also an examination of the track and ties during 
the passage of at heavy train shows that the heavy loads which 
are now common on railroad tracks force the tie into the bal- 
last with the passage of every wheel load. The effect on the 
ballast is a greater or less aniount of crushing of the ballast. 
Even thft very hardest grades of broken stone are more or less 
crushed by grinding against each other during the passage of a 
train. The softer and weaker forms of ballast are ground up 
much more quickly. One result is the formation of a fine dust 
which interferes with the proper drainage of Water through the 
[ballast. A second result is the compression of the ballast imme- 
diately under the tie into the sub-soil. In a comparatively 
short time a hole is formed under the tie which acts virtua,lly 
like a pump. With every rise and fall of the tie under each 
wheel load, the tie actually pumps the water from the surround- 
ing ballast and sub-soil into these various holes. When the 



§ 234. BALLAST. 269 

ballast is of such a character that the water does not drain 
through it easily, the water will settle in these holes long enough 
to seriously deteriorate the ties. When the track becomes so 
much out of line or level, or so loose that it needs to be tamped 
up, the process of tamping has practically the effect of deepen- 
itig the amount of ballast immediately under the tie, while the 
sub-soil is forced up between the ties. A longitudinal section 
of the sub-soil of a track which has been frequently tamped 
generally has a saw-tooth appearance, and the sub-soil, instead 
of being a uniform line, has a high spot between each tie, while 
the ballast is considerably below its normal level immediately 
under the tie. 

234. Classification of Railroads. The American Railway En- 
gineering Association has divided railroads into three classes 
with respect to the standards of construction which should be 
adopted for ballasting, as well as other details of construction. 
The three classes are as follows (quoted from the Association 
Manual) : 

** Class 'A' shall include all districts of a railway having more 
than one main track, or those districts of a railway having a 
single main track with a traffic that equals or exceeds the follow- 
ing: 

Freight-car mileage passing over districts per year per 

mile 150000 

or, 
Passenger-car mileage per annum per mile of district. . . 10000 

with maximum speed of passenger-trains of 50 miles per hour. 

''Class *B' shall include all districts of a railway having a 
single main track with a traffic that is less than the minimum 
prescribed for Class * A' and that equals or exceeds the following : 

Freight-car mileage passing over districts per year per 

mile 50000 

or, 
Passenger-car mileage per annum per mile of district. . . . 5000 

with maximum speed of passenger-trains of 40 miles per hour. ^ 
"Class 'C shall include all districts of a railway not meeting 

the traffic requirements of Classes 'A' or 'B.' " 

The classification was • adopted on the consideration that 

quality of traffic as well as mere tonnage should determine 



270 



KAILROAD CONSTRUCTION. 



§ 235. 



the classification of a railroad. For example, it is considered 
that a road which operates a train at a speed of 50 miles an 
hour should adopt the first class or Class ''A" standards, even 
though there is but one train per day on that railroad. It 
likewise means that any road whose traffic makes necessary the 
construction of a regular double track should adopt the first 
class specifications. 

235. Recommended sections for the several classifications. 
In Fig. 110 are shov/n a series of cross-sections which were 



hi'6^ 



-ii'ik'l 



-ll'lV," 



^ 5 IH- ->!< 4 0-^— >|< 4'0'— >t<:, 

--J.<gJr.~7 , ,, I , . ~7^ 






SINGLE TRACK ON TANGENT 
',1," 




-><-,-4|0— >i<- 




SINGLE TRACK ON CURVE, MAXIMUM ELEVATION 
Distances, A, B, C, D & E vary with elevation 

Fig. 110. — Cross-sections of Ballast for Class "A" Roads. 



recommended by the A. R. E. A. for Class " A " traffic. It 
should be noticed that in each case the cross-section of the 
roadbed from shoulder to shoulder of the roadbed is 22' 3'' 
plus the space between track centers for double track if any. 
The width of side ditches is merely added to that of the roadbed. 
The clear thickness of the ballast underneath the ties is made 
24 inches. The slope of | inch to the foot from the center of 
the track to the end of the tie, which is common to all the cross- 
sections, is designed with the idea of allowing a clear space of 
1 inch underneath the rail. The ballast is then rounded off 



§235. 



BALLAST. 



271 



on a curve of 4 feet radius and finally reaches the subsoil on a 
slope which is 2 : 1. 

In Fig. Ill are shown a series of cross-sections for various 
classes of ballast for railroads that belong to Class " B." It 




CRUSHED ROCK AND SLAG, 

J 




GRAVEL. CINDERS, CHATS, ETC. 




CEMENTING GRAVEL AND CHERT. 
Fig. 111. — Ckoss-sections of Ballast for Class "B" Roads. 

may be noted that the thickness of the ballast under the tie 
is 9 inches for this class. The width of roadbed between the 
shoulders, recommended for Class " B " is 16 feet. As before, 
the width of the ditches is supposed to be added to this width. 
It should be noted that when using cementing gravel and chert 
the slope of 3 : 1 is made to begin at the bottom of the tie in- 
stead of at a point about 2 inches below the top of the tie. 
This is done in order to prevent water from accumulating 
around the end of the tie in a material which is less permeable 
than the other forms of ballast, 



272 



RAILROAD CONSTRUCTION. 



§236. 



In Fig. 112 are shown two cross-sections for ballast for roads 
belonging to Class '' C." On roads of this class it is assumed 
that crushed rock will not be used for ballast. The width of 
roadbed between shoulders is 14 feet, while the depth of ballast 
underneath the tie is 6 inches. 

It should be noticed that the above sections issued by the 
association do not include any cross-section which is recom- 
mended when no special ballast is used other than the natural 
soil. In such a case a cross-section very similar to the sec- 
tions shown for cementing gravel and chert should be used. The 

V 

.g/g? J ^_]_SLOPE}6"tO THE FOOT 

JL y -LS-~^. , i 1, ^SLOPES TO 1 




GRAVEL, CINDERS. CHATS, ETC, 




CEMENTING GRAVEL AND CHERT. 
Fig. 112. — Ceoss-sections of Ballast fob Class "C" Roads. 



essential feature of such a section is that the soil, which is 
probably not readily permeable, should be kept away from 
the ends of the ties. Specifications for the placing of mud bal- 
last, as well as other forms of ballast, have frequently specified 
that the ballast should be crowned about 1 inch above the level 
of the tops of the ties in the center of the track. This feature 
of any cross-section, although proposed, was rejected by the 
association, in spite of the fact that when a tie is so imbedded 
it certainly will have a somewhat greater holding power in the 
ballast. 

236. Proper depth of ballast. The depth of ballast is officially 
defined by the A. R. E. A. as " the distance from the bottom of 



§ 237. BALLAST. 273 

the tie to the top of the subgrade." In the recommended sec- 
tions (Figs. 110 to 112) the depth shown varies from 6 inches 
to 24 inches. But the Ballast Committee reported in 1915 as 
a recommended conclusion that " From the data available, it 
is concluded that with ties 7 in. by 9 in. by 8^ ft., spaced approx- 
imately 24 in. to 25.5 ins., center to center, a depth of 24 inches 
of stone ballast is necessary to produce imiform pressure on the 
subgrade, and a combination of a lower layer of gravel or cinder 
ballast, 18 inches to 14 inches, and an upper layer of stone ballast, 
6 inches to 10 inches, approximately 24 inches deep in the aggre- 
gate, with the same spacing of the ties, will produce nearly the 
same results." New sections for Class " A " roads which would 
conform with the above were also recommended. The sections 
shown in Fig. 110, which are similar to those recommended 
in 1915, were adopted in 1921. The investigations of the 
Committee on Track Stresses (see Chap. XXV) have shown 
why deep ballast is necessary, but the economy of using a 
second-grade ballast as sub-ballast is possible. As previously 
stated, old track generally has a depth of ballast under the 
tie which is greater than the 2 feet recommended — ^often 3 or 4 
feet. 

237. Methods of laying ballast. The cheapest method of lay- 
ing ballast on new roads is to lay ties and rails directly on the 
prepared subgrade and run a construction train over the track 
to distribute the ballast. Then the track is lifted up until 
sufficient ballast is worked under the ties and the track is prop- 
erly surfaced. This method, although cheap, is apt to injure 
the rails by causing bends and kinks, due to the passage of 
loaded construction trains when the ties are very unevenly and 
roughly supported, and the method is therefore condemned and 
prohibited in some specifications. The best method is to draw 
in carts (or on a contractor's temporary track) the ballast that 
is required under the level of the bottom of the ties. Spread 
this ballast carefully to the required surface. Then lay the ties 
and rails, which will then have a very fair surface and uniform 
support, A construction train can then be run on the rails 
and distribute sufficient additional ballast to pack around and 
between the ties and make the required cross-section. 

The necessity for constructing some lines at an absolute 
minimum of cost and of opening them for traffic as soon as 
possible has often led to the policy of starting traffic when there 



274 RAILROAD CONSTRUCTION. § 238. 

is little or no ballast — perhaps nothing more than a mere 
tamping of the natural soil under the ties. When this is done 
ballast may subsequently be drawn where required by the train- 
load on flat cars and unloaded at a minimum of cost by means 
of a " plough." The plough has the same width as the cars 
and is guided either by a ridge along the center of each car or by 
short posts set up at the sides of the cars. It is drawn from one 
end of the train to the other by means of a cable. The cable is 
sometimes operated by means of a small hoisting-engine carried 
on a car at one end of the train. Sometimes the locomotive is 
detached temporarily from the train and is run ahead with the 
cable attached to it. 

238. Cost. The cost of ballast in the track is quite a variable 
item for different roads, since it depends (a) on the first cost of 
the material as it comes to the road, (6) on the distance from 
the source of supply to the place where it is used, and (c) on 
the method of handling. The first cost of cinder or slag is 
frequently insignificant. A gravel-pit may cost nothing except 
the price of a little additional land beyond the usual limits of 
the right of way. Broken stone will iisually cost $1 or more 
per cubic yard. If suitable stone is obtainable on the com- 
pany's land, the cost of blasting and breaking should be some- 
what less than this. The cost of hauling will depend on the 
distance hauled, and also, to a considerable extent, on the limi- 
tations on the operation of the train due to the necessity of keep- 
ing out of the way of regular trains. There is often a needless 
waste in this way. The " mud train " is considered a pariah 
and entitled to no rights whatever, regardless of the large daily 
cost of such a train and of the necessary gang of men, " The 
cost of broken-stone ballast in the track is estimated at $1.25 per 
cubic yard. The cost of gravel ballast is estimated at 60 c. 
per cubic yard in the track. The cost of placing and tamping 
gravel ballast is estimated at 20 c. to 24 c. per cubic yard, for 
cinders 12 c. to 15 c. per cubic yard. The cost of loading 
gravel on cars, using a steam-shovel, is estimated at 6 c. to 10 c. 
per cubic yard." — Report Roadmasters' Association, 1885. 

238a. Specifications. (Condensed from Am. Rwy. Eng. 
Assoc. Manual, 1915.) Broken stone ballast. To be selected 
on the basis of maximum (or minimum) figures for the following 
qualities: (a) weight per cubic foot, maximum; (6) water 
absorption in pounds per cubic foot, minimum; (c) per cent of 



§ 238a. BALLAST. 275 

wear, minimum; (d) hardness, maximum; (e) toughness, maxi- 
mum; (/) cementing value, minimum; (gr) compression test, 
maximum. Gravel ballast. For Class A railways: Bank 
gravel which contains more than two (2) per cent dust or forty 
(40) per cent sand should be washed or screened. Washed 
or screened gravel should contain not less than twenty-five 
(25) per cent nor more than thirty-five (35) per cent sand. 
For Class B railways: Bank gravel which contains more 
than three (3) per cent dust or sixty (60) per cent sand should 
be screened or washed. Washed or screened gravel should not 
contain less than twenty-five (25) per cent nor more than fifty 
(50) per cent sand. For Class C railways. Any material which 
makes better track than the natural roadbed may be econom- 
ically used. 

Testing gravel for ballast. Obtain five samples, each about 
one cubic foot, from various parts of the pit; mix thoroughly; 
make up a sample of about one cubic foot from the mixture. 
Sift through a screen, 10 meshes per linear inch, made of No. 24 
B. & S. wire; the residue is the " gravel," G. Sift the remainder 
through a screen, 50 meshes per linear inch, made of No. 31 
B. & S. wire; the residue is the " sand," S. That which passed 
through the screen is " dust," D. The percentage of sand, for 
example, equals S-7-{G-\-S-{-D). 



CHAPTER VIII. 

TIES, 
AND OTHER FORMS OF RAIL SUPPORT. 

239. Various methods of supporting rails. It is necessary 
that the rails shall be sufficiently supported and braced, so that 
the gauge shall be kept constant and that the rails shall not be 
subjected to excessive transverse stress. It is also preferable 
that the rail support shall be neither rigid (as if on solid rock) 
nor too yielding, but shall have a uniform elasticity throughout. 
These requirements are more or less fulfilled by the following 
methods. 

(a) Longitudinals. The fundamental idea is to have con- 
tinous support for the rail rather than to have it act as a con- 
tinuous girder with numerous supporting points — the ties. In 
§ 264 will be described a system of rails, used to some extent 
in Europe, having such broad bases that they are self-supporting 
on the ballast and are only connected by tie-rods to maintain 
the gauge. 

(b) Cast-iron "bowls" or "pots." These are castings resem- 
bling large inverted bowls or pots, having suitab.le chairs on 
top for holding and supporting the rails, and tied together with 
tie-rods. They will be described more fully later (§ 263). 

(c) Cross-ties of metal or wood. These will be discussed in 
the following sections. 

240. Economics of ties. The true cost of ties depends on the 
relative total cost of maintenance for long periods of time. The 
first cost of the ties delivered to the road is but one item in the 
economics of the question. Cheap ties require frequent renew- 
als, which cost for the labor of each renewal practically the 
same whether the tie is of oak or of hemlock. Cheap ties make 
a poor roadbed which will require more track labor to keep even 
in tolerable condition. The roadbed will require to be disturbed 
so frequently on account of renewals that the ties never get an 
opportunity to get settled and to form a smooth roadbed for any 
length of time. Irregularity in width, thickness, or length of 
ties is especially detrimental in causing the ballast to act and 

276 



I 



241. 



TIES. 



277 



wear unevenly. The life of ties has thus a more or less direct 
influence on the life of the rails, on the wear of rolling stock, and 
on the speed of trains. These last items are not so readily- 
reducible to dollars and cents, but when it can be shown that 
the total cost, for a long period of time, of several renewals of 
cheap ties, with all the extra track labor involved, is as great as 
or greater than that of a few renewals of durable ties, then there 
is no question as to the real economy. In the following dis- 
cussions of the merits of untreated ties (either cheap or costly), 
chemically treated ties, or metal ties, the true question is there- 
fore of the ultimate cost of maintaining any particular kind of 
ties for an indefinite period, the cost including the first cost of 
the ties, the labor of placing them and maintaining them to 
surface, and the somewhat uncertain (but not therefore non- 
existent) effect of frequent renewals on repairs of rolling stock, 
on possible speed, etc. 



WOODEN TIES. 

241. Choice of wood. This naturally depends, for any partic- 
ular section of country, on the supply of wood which is most 
readily available. Table XXII shows the relative use of the 
chief varieties in the U. S. Two-thirds of the entire list is white 

oak, red oak and southern 

TABLE XXII. NUMBER AND KINDS OF 

CROSS TIES USED BY 78% OF TOTAL 
MILEAGE OF STEAM RAILROADS IN 
UNITED STATES IN 1915. 

(Bull. 549, U. S. Dept. Agric). 



Kind of wood. 



White oak 

Red oak 

Southern pine 

Douglas fir 

Cypress 

Cedar . , 

Chestnut 

Eastern tamarack. . . 
Lodge pole pine. . . . , 

Western larch 

Western yellow pine 

Beech 

Maple 

Hemlock 

All other 

Total 



Number 
of ties. 



30,160,316 

15,989,605 

13,226,654 

6,308,685 

4,375,012 

4,121,570 

2,666,402 

2,520,475 

1,254,420 

1,196,415 

1,183,535 

1,139,457 

1,062,086 

839,924 

2,454,099 



88,498,655 



Per 

cent. 



34.1 

18.1 

15.0 

7.1 

4.9 



.7 

,8 
,4 
3 
3 
3 
1.2 
1.0 
2.8 



100.0 



pine. Douglas fir, which 
grows only in the west, is 
being transported to the 
east in increasingly large 
quantities and is displac- 
ing other woods. The 
use of eastern tamarack, 
lodge pole pine, western 
larch, western yellow pine, 
and hemlock is almost 
confined to the " western 
region" — ^west of the 
Mississippi river. Red- 
wood was formerly used 
quite extensively in the 
west, on account of cheap- 
ness and immunity from 
decay, but the wood is 



278 



EAILROAD CONSTRUCTION. 



242. 



too soft. The use of cypress is nearly confined to the west and 
south, and on the other hand the use of chestnut is nearly con- 
fined to the " eastern region " — north of the Ohio and Potomac 
and east of Chicago. 

On the basis of 88,498,655 ties for 78.46% of the mileage, the 
proportionate total is 112,974,615 ties. 100,000,000 to 125,000,000 
ties per year is elsewhere stated to be the normal demand. 
This means an annual average of about 290 ties for each mile of 
track, including sidings. 

242. Durability. The durability of ties depends on the cli- 
mate; the drainage of the ballast; the volume, weight, and 
speed of the. traffic; the curvature, if any; the use of tie-plates; 
the time of year of cutting the timber; the age of the timber 
and the degree of its seasoning before placing in the track ; the 
nature of the soil in which the timber is grown; and, chiefly, for 
untreated ties, on the species of wood employed. The vari- 
ability in these items will account for the discrepancies in the 
reports on the life of various woods used for ties. For example, 
six records of untreated white oak ties on six different roads 
gave figures varying from 3 to 14 years. Such a range of values 
is too wide for practical utilization. 

The variability in the actual life of a " group " of ties of nom- 
inally the same quality and placed in the track at the same time 



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Average Life-Percent 

Fia. 112a. — Relative Actual, Life of Ties of Nominally Uniform 

Quality, 

is shown in a study * made by the Forest Products Laboratory, 
U. S. Forest Service. Records show that there will be, in general. 



* "Relation between average life of ties and percentage of renewals," by 
Mabel E. Thorne, Statistician. 



§243. TIES. 279 

no replacements until after about 30% of the average life of the 
whole group. Then the replacements will commence and grow 
more frequent until at the time of the average life of the whole 
group, about 60% will have been replaced. After a time equal 
to 120% of the average life, about 90% of the ties will have been 
replaced, but a few of the remainder may stay in the track 
until nearly or quite 200% of the average life. The law is based 
on the records of 43 groups of ties comprising 42936 ties, or an 
average of about 1000 ties per group. The law is substantially 
true whether applied to short-lived untreated ties or to long- 
lived treated ties. The law may even be considered as suffici- 
ently established so that when 10% of a " group " of ties have 
been removed from a track, the time already elapsed may be 
considered as approximately 70% of the average life of the 
entire group, and the probable life of the remaining 90% of 
ties may be estimated accordingly. 

Some of the softer woods used for ties, such as cedar and 
redwood, resist decay very well, but are so soft that they are 
badly cut by the rail-flanges and do not hold the spikes very 
well, necessitating frequent respiking. Since the spikes must 
be driven within certain very limited areas on the face of each 
tie, it does not require many spike holes to " spike-kill "Jh.e tie. 
On sharp curves, especially with heavy traffic, the wheel-flange 
pressure produces a side pressure on the rail tending to over- 
turn it, which tendency is resisted by the spike, aided some- 
times by rail-braces. Whenever the pressure becomes too great 
the spike will yield somewhat and will be slightly withdrawn. 
The resistance is then somewhat less and the spike is soon so 
loose that it must be redriven in a new hole. If this occurs 
very often, the tie may need to be replaced long before any decay 
has set in. 

243. Dimensions. The usual dimensions for the best roads 
(standard gauge) are 8' to 9' long, 6" to 7" thick, and 8" to 
10" wide on top and bottom if they are sawed. Hewed ties 
(with rounded sides) shall have the faces not less than 6 inches 
wide, but the cross sectional area must not be less than a sawed 
tie of the same class. Narrow gauge and very-light-traffic 
roads will reduce these dimensions as much as twenty per 
cent. 

244. Spacing. The Penna. R. R. standard spacing (1921) 
called for 14, 16, 18 or 20 ties per 33-foot rail, according to the 



280 KAILROAD CONSTRUCTION. § 245. 

classification of track. The joints of the two lines of rails are 
placed " staggered " rather than " opposite " each other. The 
joints are " suspended " (see § 282) on two ties spaced 20" c. c. 
There are for each rail length two spaces 20" each and 12, 14, 16 
or 18 spaces of 29f ", 25f ", 22^", or 19|" each. 

245. Specifications. The specifications for ties are apt to 
include the items of size, kind of wood, and method of construc- 
tion, besides other minor directions about time of cutting, sea- 
soning, delivery, quality of timber, etc. 

(a) Size. The particular size or sizes required will be some- 
what as indicated in § 243. 

(b) Kind of wood. When the kind or kinds of wood are 
specified^ the most suitable kinds that are available in that 
section of country are usually required. 

(c) Method of construction. It is generally specified that the 
ties shall be hewed on two sides; that the two faces thus made 
shall be parallel planes and that the bark shall be removed. It 
is sometimes required that the ends shall be sawed off square; 
that the timber shall be cut in the winter (when the sap is down) ; 
and that the ties shall be seasoned for six months These last 
specifications are not required or lived up to as much as their 
importance deserves. It is sometimes required that the ties shall 
be delivered on the right of way, neatly piled in rows, the alter- 
nate rows at right angles, piled if possible on ground not lower 
than the rails and at least ten feet away from the nearest rail, 
the lower row of ties resting on two ties which are themselves 
supported so as to be clear of the ground. 

(d) Quality of timber. The usual specifications for sound 
timber are required, except that they are not so rigid as for a 
better class of timber work The ties must be sound, reason- 
ably straight-grained, and not very crooked — one test being that 
a line joining the center of one end with the center of the middle 
shall not pass outside of the other end. Splits or shakes, espe- 
cially if severe, should cause rejection. 

Specifications sometimes require that the ties shall be cut 

from small trees, making 
what is known as "pole 
ties" and definitely con- 
demning those which are 

POLE TIE. SLAB TIE. QUARTER TIE. '=' . 

^ cut or split from larger 

Fio. 113. — Methods of cutting Ties. ^ -, . . . /< i 1 

trunks, givmg two slab 





§ 246. TIES. 281 

ties " or four " quarter ties " for each cross-section, as is illus- 
trated in Fig. 113. Even if pole ties are better, their exclusive 
use means the rapid destruction of forests of young trees. 

246. Regulations for laying and renewing ties. — The regula- 
tions issued by railroad companies to their track foremen will 
generally include the following, in addition to directions regard- 
ing dimensions, spacing, and specifications given in §§ 242—245. 
When hewn ties of somewhat variable size are used, as is fre- 
quently the case, the largest and best are to be selected for use 
as joint ties. If the upper surface of a tie is found to be warped 
(contrary to the usual specifications) so that one or both rails do 
not got a full bearing across the whole width of the tie, it must 
be adzed to a true surface along its whole length and not mertis 
notched for a rail-seat. When respiking is necessary and spikes 
have been pulled out, the holes should be immediately plugged 
with ''wooden spikes," which are supplied to the foreman for 
that express purpose, so as to fill up the holes and prevent the 
decay w^hich would otherwise take place when the hole becomes 
filled with rain-water. Ties should always be laid at right angles 
to the rails and never obliquely Minute regulations to prevent 
premature rejection and renewal of ties are frequently made. It 
is generally required that the requisitions for renewals shall be 
made by the actual coimt of the individual ties to be renewed 
instead of by any wholesale estimates. It is unwise to have ties 
of wddely variable size, hardness, or durability adjacent to each 
other in the track, for the uniform elasticity, so necessary for 
smooth ridingj will be unobtainable under those circumstances. 

After a considerable discussion of the two policies of tie 
renewals over long continuous stretches of track or of single tie 
renewals where individually needed, the A. R. E. A, has decided 
in favor of single tie renewals, as being most economical and 
producing least track disturbance. 

247. Dating nails. These are made of iron or steel, galvanized 
with zinc. They should be 2^ inches long, j inch in diameter, 
with f -inch head, which has two figures Ye i^ch high, denoting 
the year, which are stamped, by depression, into the head. 
They should be driven into the upper side of all treated ties, 
lb inches inside the rail, on the line side of the track. The use 
of such dates gives definite knowledge of the life of the tie when 
it is renewed and a means of studving the effectiveness of the tie 
treatment. 



282 EAILROAD CONSTRUCTION. § 248. 

248. Cost of ties. When railroads can obtain ties cut by 
farmers from woodlands in the immediate neighorhood, they 
sometimes advertise a schedule of prices which they will pay, 
the prices being considerably lower than the prices demanded 
by dealers. Prices as low as 35 c. were formerly paid directly 
to tie cutters in tie growing sections, but increasing scarcity 
has raised the price. A great railway paid $610,713 for 453,000 
ties in 1920, an average of $1.31 each. These were of higher 
grade than the average. The following schedule shows pro- 

, portionate prices : white oak, $1.39; heart pine, $1.66; chestnut, 
$1.37; red oak, $1.34; sap pine, $1.19; maple, beech and birch, 
$1.27. 

PKESERYATIVE PROCESSES FOR WOODEN TIES. 

249. General principles. Wood has a fibrous cellular struc- 
ture, the cells being filled with sap or air. The woody fiber is 
but little subject to decay unless the sap undergoes fermentation. 
Preservative processes generally aim at removing as much of the 
water and sap as possible and filling up the pores of the wood 
with an antiseptic compound. The most common methods 
all agree in this general process and only differ in the method 
employed to get rid of the sap and in the antiseptic chemical 
with which the fibers are filled. One valuable feature of these 
processes lies in the fact that the softer cheaper woods are more 
readily treated than are the harder woods and from them a tie 
can be made which will be as durable as the best (from the stand- 
point of decay), and, if protected from mechanical wear by tie- 
plates, will have a very long life. The following woods may be 
used without preservative treatment: White oak family, long- 
leaf strict heart yellow pine, cypress, excepting the white cypress, 
redwood, white cedar, chestnut, catalpa, locust, except the 
honey locust, walnut and black cherry. The following woods 
should preferably not be used without preservative treatment: 
Red oak family, beech, elm, maple, gum, loblolly, short-leaf. 
Western yellow pine, Norway, North Carolina pine and other 
sap pines, red fir, spruce, hemlock, and tamarack. It is better 
to use an excess of chemical rather than not enough. Ties 
should be grouped before treatment; for example, green ties 
should not be mixed with seasoned ties, since the treatment 
should be different. Ties should be air-seasoned before being 



§250. TIES. 283 

treated. When there is time to air-season them at the plant 
before treatment, tbey should be piled in groups having the same 
degree of seasoning, so that they rest on seasoned stringers, the 
lowest ties at least 6 inches from the ground, which should be 
thoroughly drained and cleared from weeds, high grass and 
decaying matter. The ties should not be allowed to over- 
season or deteriorate. Ties which show signs of checking should 
be secured with S-irons or bolts to prevent further checking. 
When ties are to be adzed or bored for the use of tie-plates or 
screw spikes, the adzing or boring should be done before chemical 
treatment. Steam seasoning, if excessive, weakens the wood. 
It should therefore be limited, unless it is imperative to treat 
green ties because air-seasoned ties are not obtainable. 

To do the work, long cylinders, which may be opened at the 
ends, are necessary. Usually the timbers are run in and out on 
iron carriages running on rails fastened to braces on the inside of 
the cylinder. When the load has been run in, the ends of the 
cylinder are fastened on. The water and air in the pores of the 
wood are drawn out by subjecting the wood alternately to steam- 
pressure and to the action of a vacuum-pump. Live steam 
should be admitted so that a pressure of 20 lbs. is produced 
within 30 to 50 minutes. This pressure may be maintained from 
1 to 5 hours, depending on the condition of the wood, but the 
pressure should never exceed 20 lbs. A vent should be provided 
to allow the escape of air and condensed water. After steaming, 
a vacuum of not less than 24 inches of mercury at sea-level (or 
correspondingly less for higher altitudes), shall be produced 
and maintained for half an hour. Then, without breaking the 
vacuum, the chemical shall be admitted. 

250. Creosoting. This process consists in impregnating the 
wood with creosote oil, a product obtained from coal-gas tar 
or coke oven tar which shall be free from any tar, including coal- 
gas tar, oil or residue obtained from petroleum or any other 
source. The pure creosote oil is strongly recommended by the 
A.R. E. A., but they recognize that the practice of using other coal 
tar distillates, v^^hen the available supply of creosote is inadequate, 
is firmly established, and have made specifications accordingly. 

It would require about 35 to 50 lbs. of creosote to completely 
fill the pores of a cubic foot of wood. But it would be impossible 
to force such an amount into the wood, nor is it necessary or 
desirable. After one of the vacuum periods, the cylinder is 



284 RAILROAD CONSTRUCTION. § 251. 

filled with creosote oil having a temperature of not less than 
160° F. The cylinders should be provided with steam coils in 
order to maintain that temperature during injection. The 
pressure should immediately be raised to 75 lbs. per square 
inch, and then by a gradual increase to a maximum of 175 or 
200 lbs. or until about 6 to 10 lbs. per cubic foot, or about 21 
to 35 lbs. per tie, is absorbed, this amount being indicated by 
calculations based on gauge readings of the oil in the oil reservoir, 
taken before and after the introduction and withdrawal of the 
oil from the cylinder. Owing to variations in the volume of 
the creosote due to change of temperature during treatment, 
also to variations in the capacity volume of the cylinder due 
to change in temperature of the metal, and several other 
causes, the determination of the volume of the oil actually 
absorbed by the ties is not simple. Each cylinder must be 
calibrated by a series of tests, since these causes may easily 
produce an error of 25% in the nominal results. As a check, 
the ties on a cylinder tram-car should occasionally be weighed 
before and after treatment. Even this check will not be con- 
clusive if the ties have been steam seasoned, since steam 
seasoning usually increases the weight and this increase would 
be credited as absorption of chemical. 

251. Burnettizing (chloride-of-zinc process).. This process is 
very similar to the creosoting process except that the chemical is 
chloride of zinc. The chemical is heated to 140° F, before using. 
The preliminary treatment of the wood to alternate vacuum and 
pressure is not continued for quite so long a period as in the 
creosoting process. Care must be taken, in using this process, 
that the ties are of as uniform quality as possible, for seasoned 
ties will absorb much more zinc-chloride than unseasoned (in the 
same time), and the product will lack uniformity unless the sea- 
soning is uniform. The amount of solution injected shall be 
equivalent to | lb. of dry soluble zinc-chloride per cubic foot of 
timber. The solution shall be as weak as can be used and still 
obtain the desired absorption of zinc-chloride, and shall not be 
stronger than 5%. If the cylinders are provided with steam 
coils, steam pressure shall be maintained in these coils during 
treatment. One great objection to burnettized ties is the fact 
that the chemical is somewhat easily washed out, when the wood 
again becomes subject to decay. Another objection is the fact 
that when the solution of zinc-chloride is made strong (over 3%) 



§252. TIES. 285 

the timber is made very brittle and its strength is reduced. 
The reduction in strength has been shown by tests to amount 
to -|- to yV *^^ ^^® ultimate strength, and that the elastic limit 
has been reduced by about ^. 

252. Kyanizing (bichloride-of-mercury or corrosive-sublimate 
process). This process has been much used, but it is so objec- 
tionable, on account of the chemical being such a virulent poison 
that workmen are sickened by fumes arising from the tanks, that 
it is no longer included as one of the standard methods, 

253. Zinc-tannin process. The last two methods described 
(as well as some others employing similar chemicals) are open 
to the objection that since the wood is impregnated with an 
aqueous solution, it is liable to be washed out very rapidly if the 
v/ood is placed under water, and will even disappear, although 
more slowly, under the action of moisture and rain. Several 
processes have been proposed or patented to prevent this. By 
one of these processes the timber is successively subjected to the 
action of chemicals, each individually soluble in water, and hence 
readily impregnating the timber, but the chemicals when brought 
in contact form insoluble compounds which cannot be washed 
out of the wood-cells. After • injecting the zinc-chloride, as 
before described, the solution is run off and the ties drained for 
15 minutes. Then a 2% solution of tannic acid, made from 
6 1 lbs. of 30% extract of tannin and 100 lbs. of water is run in 
and maintained at 100 lbs. pressure for one-half hour. Then 
a solution of glue made by dissolving 2.1 lbs. of glue containing 
50% gelatine in 100 lbs. of water is rim in and maintained at 
100 lbs. pressure for one-half hour. The glue and tannin com- 
bine to form an insoluble leathery compound in the cells, which 
will prevent the zinc-chloride from being washed out. 

254. Zinc-creosote emulsion process. The chemical is an emul- 
sion which will leave in the wood an equivalent of 0.4 lb. of dry, 
soluble zinc-chloride and from 1.25 to 1.5 lbs. of creosote per cubic 
foot. The zinc-chloride must not be stronger than 3.5%. The 
emulsion must be effectively mixed in a storage tank and heated 
to at least 140° F. before it enters the cylinder, where the pressure 
is raised to 100 lbs. per square inch and maintained there until 
the required amount of chemical has been absorbed by the wood. 

255. Two-injection zinc-creosote process. The zinc-chloride 
and creosote are injected separately. The zinc-chloride must be 
as weak as possible (ngt more than 5%), and yet strong enough 



286 



RAILROAD CONSTRUCTION. 



§256. 



so that the equivalent of 0.3 lb. can be injected per cubic foot. 
After impregnation, the remaining zinc-chloride is run out and 
the creosote is forced in and maintained at 100 lbs. pressure 
until the wood has absorbed about 3 lbs. of oil per cubic foot. 

256. Cost of treating. The cost of treating ties by the vari- 
ous methods has been estimated as follows.* The total cost 
is divided into (1) seasoning; (2) labor; (3) fuel; (4) main- 
tenance and (5) chemicals. 

Seasoning. The labor required for air-seasoning, the usual 
practice, is estimated at from 0.75 c. to 1.5 c. per tie, or is aver- 
aged at 1.0 c. per tie. Labor. The labor involved in all other 
handling of the ties is averaged at 6.0 c. per tie. Fuel may 
cost 0.5 c. per tie when natural gas or oil is obtainable and up 
to 2.0 c. per tie for other scarcer fuels; it is averaged at 1.0 c. 
per tie. Maintenance of the plant is estimated at 1.25 c. to 
2.0 c. per tie; as an average it is placed at 1.5 c. for creosoting 
plants and 1.6 c. for plants using zinc-chloride, since it is more 
corrosive. Chemicals. On the basis of a 7" X9" X8' tie, having 
a volume of 3.5 cubic feet, and ^ lb. of zinc-chloride per cubic 
foot, the amount of ZnCl2 is 1.75 lbs. per tie; at 4c. per pound 
this would cost 7 c. per tie. Using 10 lbs. of creosote per cubic 
foot or 35 lbs. per tie, 4.08 gallons (8.58 lbs. per gallon) of creosote 
would be used per tie, A price of 6 to 10 cents per gallon is 
quoted for large quantities of creosote. Apparently 6.84 c. 
per gallon was used in the calculation, since the cost of the creo- 
sote was put at 27.9 c. per tie. Summarizing, the cost by the 
several methods was as given below. 



Chemical used. 


Quantity 

per cubic 

foot. 


Seasoning, 

labor, 

fuel. 


Mainte- 
nance of 
plant. 


Chemical 

cost. 


Total. 


Creosote 


10 lbs. 
6 " 
3 " 
i " 
i " 


8.0c. 
8.0 " 

} 8.0" 

8.0" 


1.5 c. 
1.5" . 

1.6 " 
1.6 " 


27.9 c. 
16.8 " 

15.4 " 

7.0" 


37.5 c. 
26 . 3 " 


/ Creosote 

1 ZnCl2 

Zinc chloride 


25.0" 
16.6 " 



Of course the above figures are merely illustrative. Variations 
in the cost of labor and materials will probably change all these 
figures. ■ Nothing is included for interest, depreciation, super- 
intendence or profit. 



*Bull. No. 118, U. S. Dept. of Agric, Div. of Forestry. Nov., 1912. 



§257. TIES. 287 

257. Economics of treated ties. The fact that treated ties are 
not universally adopted is due to the argument that the added 
life of the tie is not worth the extra cost. If ties can be bought 
for 25 c, and cost 25 c. for treatment, and the treatment only 
doubles their life, there is apparently but little gained except 
the work of placing the extra tie in the track, which is more 
or less offset by the interest on 25 c. for the life of the untreated 
tie, and the larger initial outlay makes a stronger impression on 
the mind than the computed ultimate economy. But when 
(utilizing some statistics from the Pittsburg, Ft. Wayne & 
Chicago Railroad) it is found that white oak ties laid in rock 
ballast had a life of 10.17 years, and that hemlock ties treated 
with the zinc-tannin process and laid in the same kind of ballast 
lasted 10.71 years, then the economy is far more apparent. 
Unfortunately no figures were given for the cost of these ties 
nor for the cost of the treatment; but if we assume that the 
white oak ties cost 75 c. and the hemlock ties 35 c. plus 20 c. 
for treatment, there is not only a saving of 20 c. on each tie, 
but also the advantage of the slightly longer life of the treated 
tie. In the above case the total life of the two kinds of ties 
is so nearly the same that we may make an approximation of 
their relative worth by merely comparing the initial cost; but 
usually it is necessary to compare the value of two ties one 
of which may cost more than the other, but will last considerably 
longer. The mathematical comparison of the real value of 
two ties under such conditions may be developed as follows: 
The real cost of a tie, or any other similar item of constructive 
work, is measured by the cost of perpetually maintaining that 
item in proper condition in the structure. It will be here 
assumed that the annual cost of the trackwork, which is assign- 
able to the tie, is the same for all kinds of ties, although the 
di 'Terence probably lies in favor of the more expensive and 
most durable ties. By assuming this expense as constant, the 
remaining expense may be considered as that due to the cost 
of the new ties whenever necessary, plus the cost of placing 
thrin in the track. We also may combine these two items 
in one, and consider that the cost of placing a tie in the track, 
which we will assume at the constant value of 20 c. per tie, 
regardless of the kind of tie, is merely an item of 20 c. in the 
total cost of the tie. We will assume that Tj is the present 
cost of a tie, the cost including the preservative treatment if 



288 RAILROAD CONSTRUCTION. § 257. 

any, and the cost of placing in the track. The tie is assumed 

to last n years. At the end of n years another tie is placed 

in the track, and, for lack of more precise knowledge, we will 

assume that this cost Tj equals T^. The "present worth" 

of T2 is the sum which, placed at compound interest, would 

equal Tj ^t the end of n years, and is expressed by the quantity 

T 
(1+ \n f ^^ which r equals the rate of interest. Similarly at 

the end of 2n years we must expend a sum Tg to put in the third 
tie, and the present worth of the cost of that third tie is ex- 

T 
pressed by the fraction , i^xo "' ^® ^^y similarly express 

the present worths of the cost of ties for that particular spot 

for an indefinite period. The sum of all these present worths 

is given by the sum of a converging series and equals (assuming 

TX (1+^)^ 
that all the T's are equal) .^ , \ — — —. But instead of laying 

^ {l+r)n—l '' ° 

aside a sum of money which will maintain a tie in that par- 
ticular place in perpetuity, we may compute the annual sum 
which must be paid at the end of each year, which would be 
the equivalent. We will call that annual payment A, and 
then the present worths of all these items are as follows: 

A 
For the first payment .*...,. 7i"t~t» 

For the second payment TfX^* 



For the third payment ^^ , „\z* 



A 

(i+ry 

A 



For the nth payment ,-^, ,— , 

^ -^ (l+r)n 

After the next tie is put in place we have the present worths 
of the annual payments on the second tie, of which the first 
one would be 

A 



For the (n + 1) payment 



(l+^)(n+l)* 



Similarly after x ties have been put in place the last pay- 

A 

ment for the x tie would have a present worth 7- The 

■^ (l+r)na:* 



§ 257. TIES. 289 

sum of all these present worths is represented by the sum of 

A 

a converging series and equals the very simple expression — • 

But since the sum of the present worths of these annual pay- 
ments must equal the sum of the present worths of the payments 
made at intervals of n years, we may place these two summa^ 
tions equal to each other, and say that 

^^ rxrx(l+r)^ / 
(l+rr-l 

Values of ^ for various costs of a tie T on the basis that r 
equals 5% have been computed and placed in Table XVIII. 
To illustrate the use of this table, assume that we are comparing 
the relative values of two ties, both untreated, one of them 
a white oak tie which will cost, say 75 c, and will last twelve 
years, the other a yellow pine tie which will cost, say 35 c, 
and will last six years. Assuming a charge for each case of 
20 c. for placing the tie in the track, we have as the annual 
charge against the white oak tie, which costs 95 c. in the track, 
10.72 c. The pine tie, costing 55 c. in the track and lasting 
six years, will be charged with .an annual cost of 10.48 c, which 
shows that the costs are practically equal. It is probably 
true that the track work for maintaining the white oak would 
be less than that for the pine tie, but since the initial cost of 
the pine tie is less than that of the oak tie, it would probably 
be preferred in this case, especially if money was difficult to 
obtain. It may be interesting to note that if a comparison is 
made from a similar table which is computed on the basis of 
compounding the money at 4% instead of 5%, the annual 
charges would be 10.13 and 10.49 c. for the oak and pine ties 
respectively, thus showing that when money is "easier" the 
higher priced tie has tlie greater advantage. 

Example 2. Considering again the comparison previously 
made of a white oak untreated tie which was assumed to cost 
75 c, and a hemlock treated tie, which cost 35 c. for the tie 
and 20 c. for the treatment, the total costs of these ties laid 
in the track would therefore be 95 c. and 75 c. respectively. 
These ties had practically the same life (10.17 and 10.71 years), 
but in order to use the table, we will call it ten years for each 
tie. The annual charge against ihe oak tie would therefore 



290 EAILROAD CONSTRUCTION. §258. 

be 12.30 c, while that against the hemlock tie would be 9.72 c. 
This gives an advantage in the use of the treated tie of 2.58 c. 
per year, which capitalized at 5% would have a capitalized 
value of 51.6 c. 

The Atchison, Topeka and Santa Fe R. R. has compiled a 
record of treated pine ties removed in 1897, '98, '99, and 1900, 
showing that the average life of the ties removed had been about 
11 years. On the Chicago, Rock Island and Pacific R. R., the 
average life of a very large number of treated hemlocK and 
tamarack ties was found to be 10.57 years. Of one lot of 21,850 
ties, 12% still remained in the track after 15 years' exposure. 

It has been demonstrated that much depends on the minoir 
details of the process — ^whatever it may be. As an illustra- 
tion, an examination of a batch of ties, treated by the zinc- 
creosote process, showed 84% in service after 13 years' expo- 
sure; another batch, treated by another contractor by the same 
process (nominally), showed 50% worthless after a service of six 
years. 

METAL TIES. 

258. Extent of use. In 1894 * there were nearly 35000 miles 
of " metal track " in various parts of the world. Of this total, 
there were 3645 miles of " longitudinals " (see § 264), found 
exclusively in Europe, nearly all of it being in Germany. There 
were over 12000 miles of " bowls and plates " (see § 263), foimd 
almost entirely in British India and in the Argentine Republic. 
The remainder, over 18000 miles, was laid with metal cross-ties 
of various designs. There were over 8000 miles of metal cross- 
ties in Germany alone, about 1500 miles in the rest of Europe, 
over 6000 miles in British India, nearly 1000 miles in the rest 
of Asia, and about 1500 miles more in various other parts of the 
world. Several railroads in this country have tried various 
designs of these ties, but their use has never passed the experi- 
mental stage. These 35000 miles represent about 9% of the 
total railroad mileage of the world — nearly 400000 miles. They 
represent about 17.6% of the total railroad mileage, exclusive of 
the United States and Canada, where they are used but little, 
except experimentally. In the four years from 1890 to 1894 the 
use of metal track increased from less than 25000 miles to nearly 

* Bulletin No, 9, U, S. Dept. of Agriculture, Div. of Forestry. 



i§259. TIES. 291 

35000 miles. This increase was practically equal to the total 
increase in railroad mileage during that time, exclusive of the 
increase in the United States and Canada. This indicates a 
large growth in the percentage of metal track to total mileage, 
and therefore- an increased appreciation of the advantages to be 
derived from their use. 

The above figures were true in 1894. Since then there has 
been considerable development. In 1915, over one million of 
the " Carnegie " steel ties, M21 section, had been laid on the 
Bessemer and Lake Erie R. R. It is now the standard on that 
road. On several other roads these ties are used extensively 
and " are not in the nature of test installations." The National 
Railways of Mexico have adopted as standard a pressed steel 
tie. The scarcity of tie timber in Mexico, the comparatively 
light weight of rolling stock and comparatively low speed, com- 
bine to favor this form of tie, which is very similar to a tie tried 
as an experiment by the N. Y. C. & H. R. R. R. in 1892, but 
which was found unsuitable for their requirements. 

259. Forms and dimensions of some metal ties. As shown on 
Plate VI, the tics have approximately the same external dimen- 
sions as wooden ties. Stability in the ballast requires that they 
shall be heavy, at least as heavy as a wooden tie, and that the 
shape shall be such that, when surrounded by ballast, they shall 
be anchored against horizontal or vertical motion. The broad 
lower flange of the Carnegie tie apparently fulfils the latter 
requirement. The " Champion " tie, shown on Plate VI, is 
essentially an inverted T, of j^" metal, with a base 10" wide, 
and a flange 5" high. Two pairs of white oak blocks, easily 
renewable, and into which cut spikes or screw spikes may be 
driven, are higher than the flange and there is therefore no trouble 
about the insulation of track circuits. The " System Couillet," 
used in Europe, has some of the same principles, but is much 
lighter and only serviceable for lighter rolling stock. 

260. Durability. Many metal ties have failed because of 
breakage, which generally begins at some opening, perhaps a 
bolt hole, or a place where the metal has been sheared on three 
sides and bent down on the fourth side to form a lug; the break 
invariably begins at some corner, if the opening has sharp cor- 
ners. Some metal ties have crushed down immediately under 
the rail, showing that the design was too light and that there 
was too little metal there for the traffic it had to carry. 



292 EAILKOAD CONSTRUCTION. § 260 

Metal ties are subject to rust, especially when in damp local- 
ities, such as tunnels; but on the other hand it is in such con- 
fined localities, where renewals are troublesome, that it is especi- 
ally desirable to employ the best and longest-lived ties. Paint, 
tar, etc., have been tried as a protection against rust, but such 
protection is quickly scraped off and the conditions prevent any 
renewal of the protection, such as may be done by^repainting 
a bridge, for example. Thirty Carnegie ties, which weighed 
originally 5213 pounds, were taken from the track after six 
years' service; after the dirt and rust had been removed, they 
were found to weigh 4912 pounds, a loss of 301 pounds, or an 
average loss of less than 1% per year. A metal tie could per- 
haps lose 35% of its weight by rusting before this cause alone 
would require its removal. 

Virtual failures, necessitating remioval, are frequently due to 
defects in the device for fastening the rails to the ties. Some 
of the designs include a lug, fitting over the base of the rail and 
held in place by a bolt and nut. These are often jarred loose, 
unless the nuts are held by nutlocks. 

Many ties, both steel and concrete, which have abundant 
strength to support the mere weight of the traffic, are immedi- 
ately broken when a derailment causes car wheels or engine 
drivers to strike them directly. They do not have the tough- 
ness and resiliency of wooden ties to withstand such shocks. 

The Carnegie tie is the only steel tie which has been used in 
sufficient quantities and for such a length of time that any 
rational estimate of its life may be made — except those experi- 
mental types of ties whose life has been so short that they are 
evidently failures. 22400 Carnegie ties were laid on the Duluth, 
Missabe & Northern Rwy. in 1908. In 1916 one tie was removed 
under special circumstances. In 1919, 30 had failed by crush- 
ing under the rail seat. By 1920, a total of about 100 had failed. 
This is a little over 0.4% after a period of twelve years. This, 
ratio is too small to apply to the curve shown in Fig. 112a, § 242, 
but it indicates a very long average life. Another far less favor- 
able case is that of 384 ties placed in the Erie R. R. in 1909. 
Ten were removed in 1916, eighteen more in 1918, and fourteen 
more in June, 1919. A later report states that the last of them 
were removed by August, 1919, after an average of over nine 
years' service. " The majority of them were crushed under 
the rail seat," which indicates that they were too light for the 



^M^ 



s:tO^ 



-p — rr 

i> II 
u u 



=2^'= 



— " — - yp p TPT sf 

SOCKETS FOR SCREy/ SpiKES II If^ tr 



r"- 



i^'GVa^H 2 2-- ,-,;- 



-^6}t>K- 



H2F 






PERCIVAL REINFORCED CONCRETE TIE (1910) 




^ 




BATES REINFORCED CQNCR.ETE TIE (1912} 





"system griffin 

C.BOWL) 






LlVESEY BOWL, {jse4) 



T«-»! 




CHAMPION STEEL (1920) 



t£r 




i^^^^'s^^^^^^^^^^^^^^^^^^??^^^^^ y. 



-^ 



CARNEGIE STEEL TIE (1916> 




Plate VI. — Some Forms op Metal Ties. 
(Between pp. 292 and 293.) 



§261. TIES. 293 

work they had to do. Several other roads have made similar 
reports — a few experimental ties have crushed under the head 
after a few years' service, evidently because the type chosen 
(there are five weights) was too light for the weight of the roll- 
ing stock. 

261. Economics of steel ties. Perhaps the most potent 
reason for the slow adoption of a substitute for the wooden tie 
is the plain matter of cost. In spite of the fact that the available 
supplies of tie timber are being used up at a rate which is several 
times the rate of renewal of such supplies by growth, the relative 
cost of steel and wooden ties is such that the steel tie must show 
a great superiority in order to justify its extra cost. Present 
prices (1921) are abonormal but are perhaps relatively nearly 
the same. Assume that a white oak tie costs $1.40 and that 
it costs $1.12 more for spikes and tie plates, and 30 c. more to 
place it in the track; assume that this tie will last 8 years under 
a certain class of traffic. Then, by Table XVIII, the annual 
charge for an initial cost of $2.82 is 2.82X15.47 = 43.63 c. The 
present quoted price for a Carnegie M21 tie, including fastenings, 
is $5.00; adding 30 c. for placing in track, we have a total of 
$5.30. 43.63-4-5.30 = 8.23, the annual charge in cents for each 
dollar of initial expenditure. By interpolation in Table XVIII 
between 8.27 for 19 years and 8.02 for 20 years, it is seen that 
the metal tie must have an average life of 19 yrs. 2 mo. to equal 
the economy of the oak tie. The above comparison assumes 
the substantial equality of cost of track labor and the main- 
tenance of the track fastenings with the two kinds of ties. 



263. Bowls or plates. As mentioned before, over 12000 miles 
of railway, chiefly in British India and in the Argentine Repub- 
lic, are laid with this form of track. It consists essentially of 
large cast-iron inverted " bowls " laid at intervals under each 
rail and opposite each other, the opposite bowls being tied 
together with tie-rods. A suitable chair is riveted or bolted on 
to the top of each bowl so as to properly hold the rail. Being 
made of cast iron, they are not so subject to corrosion as steel 
or wrought iron. They have the advantage that when old and 
worn out their scrap value is from 60% to 80% of their initial 
cfost, while the scrap value of a steel or wrought-iron tie is prac- 
tically nothing. Failure generally occurs from breakage, the 



294 KAILROAD CONSTRUCTION. § 264. 

failures from this cause in India being about 0.4% per annum. 
They weigh about 250 lbs. apiece and are therefore quite expen- 
sive in first cost and transportation charges. There are miles 
of them in India which have already lasted 25 years and are 
still in a serviceable condition. Some illustrations of this form 
of tie are shown in Plate VI. 

264. Longitudinals. Although the discussion of longitudinals 
might be considered to belong more properly to the subject of 
Rails, yet the essential idea of all designs mubt necessarily be 
the support of a rail-head on which the rolling stock may run, 
and therefore this form, unused in this country, will be briefly 
described here. This form, the use of which is confined almost 

exclusively to Germany, is being gradually 
replaced on many lines by metal cross-ties. 
The system generally consists of a compound 
rail of several parts, the upper bearing rail 
\zzzzzzzzz2z^ being very light and supported throughout its 
jPjQ jj^ length by other rails, which are suitably tied 

together with tie-rods so as to maintain the 
proper gauge, and which have a sufficiently broad base to be 
properly supported in the ballast. One great objection to this 
method of construction is the difficulty of obtaining proper 
drainage especially on grades, the drainage having a tendency to 
follow along the lines of the rails. The construction is much 
more complicated on sharp curves and at frogs and switches. 
Another fundamentally different form of longitudinal is the 
Haarman compound '' self -bearing rail," having a base 12" 
wide and a height of 8", the alternate sections breaking joints 
so as to form a practically continuous rail. 

Some of the other forms of longitudinals are illustrated in 
Plate VI. 

For a very complete discussion of the subject of metal ties, 
see the " Report on the Substitution of Metal for Wood in 
Railroad Ties " by E. E. Russell Tratman, it being Bulletin 
No. 4, Forestry Division of the U. S. Dept. of Agriculture. 

265. Reinforced concrete ties. The wide application of rein- 
forced concrete to various structural purposes, combined with 
its freedom from decay, has led to its attempted adoption for 
ties. For several years a standing committee of the Amer. 
Rwy.. Eng. Assoc, has systematically followed the experimental 
tests on several railroads of nmnerous substitutes for wooden 



§ 265. TIES. 295 

ties. Many of these ties are made of metal and have been pre- 
viously referred to. Others are made of concrete, reinforced 
with steel. The concrete is not subject to decay but it is so 
brittle that, when struck by a derailed car or locomotive, it 
will almost inevitably crack, and after that, its disintegration 
is a matter of a very short time. The Percival tie, shown on 
Plate VI, has been tested for several years on some roads having 
comparatively light traffic. The reports from these roads are 
encouraging, if not conclusive. The " Bates " tie consists of 
two concrete blocks, one under each rail, which are connected 
by a pair of trussed structures of steel. In the center space of 
of about two feet between the two blocks, the steel is exposed 
to rust. A report on these ties said that, after seven years 
service, the exposed trusses were " rusted to a maximum depth 
of possibly ys" but not to such an extent as to seriously weaken 
the trusses." It is a common belief that it is essentially impos- 
sible to design a concrete tie, even when reinforced with steel, 
which will have sufficient resiliency to withstand the shocks of 
rail traffic. Innumerable concrete ties have ignominiously failed 
after a very short service. 



4 



CHAPTER IX. 

RAILS. 

266. Early forms. The first rails ever laid were wooden 
stringers which were used on very short tram-roads around coal- 
mines. As the necessity for a more durable rail increased, 
owing chiefly to the invention of the locomotive as a motive 
power, there were invented successively the cast-iron "fish- 
belly" rail and various forms of wrought-iron strap rails which 
finally developed into the T rail used in this country and the 
double-headed rail, supported by chairs, used so extensively in 
England. The cast-iron rails were cast in lengths of about 3 
feet and were supported in iron chairs which were sometimes 
set upon stone piers, A great deal of the first railroad track 
of this country was laid with -longitudinal stringers of v/ood 
placed upon cross-ties, the inner edge of the stringers being 
protected by wrought- iron straps. The "bridge" rails were 
first rolled in this country in 18-14. The "pear" section was 
an approach to the present form, but was very defective on 
account of the difficulty of designing a good form of joint. The 
"Stevens" section was designed in 1830 by Col. Robert L. 
Stevens, Chief Engineer of the Camden and Amboy Railroad; 
although quite defective in its proportions, according to the 
present knowledge of the requirements, it is essentially the pres- 
ent form. In 1836, Charles Vignoles invented essentially the 
same form in England ; this form is therefore known throughout 
England and Europe as the Vignoles rail. 

267. Present standard forms. The larger part of modern 
railroad track is laid with rails v/hich are either "T" rails or 
the double-headed or " bull-headed " railc which are carried in 
chairs. The double-headed rail v/as designed with r, symmetri- 
cal form with the idea that after one head had been worn out 
by traffic the rail could be reversed, and that its life would be 
practically doubled. Experience has shown that the wear of the 

296 



§267. 



KAILS. 



297 



rail in the chairs is very great; so much so that when one head 
has been worn out by traffic the whole rail is generally useless, 




I 

STEVENS. 1830 




PEAR." 



'! BRIDGE" 1843. 
BALT. & OHIO R.R. 



VIGNOLES. 1886. 




STEPHENSON 
(ENGLISH) 1838 




"BULL-HEAD." 




"FISH-BELLY" — CAST IRON. 



^i: 



3 



CAST IRON. 




Fig. 115. 



reynolds— 1767. 
-Early Forms of Rails. 



If the rail is turned over, the worn places, caused by the chairs, 
make a rough track and the rail appears to be more brittle and 
subject to fracture, possibly due to the crystallization that may 
have occurred during the previous usage and to the reversal of 
stresses in the fibers. Whatever the explanation, experience has 
demonstrated the ]act. The ''bull-headed" 
rail has the lower head only large enough to 
properly hold the wooden keys with which 
the rail is secured to the chairs (see Fig. 116) 
and furnish the necessary strength. The use 
of these rails requires the use of two cast- 
iron chairs for each tie. It is claimed that 
such track is better for heavy and fast traffic, but it is more 




Fig. 116. — Bull- 
headed Rail and 
Chair. 



29§ 



RAILROAD CONSTRUCliON. 



§267. 



expensive to build and maintain. It is the standard form of 
track in England and some parts of Europe. 

Until after 1893 there was a very great multiplicity in the 
designs of " T " rails as used in this country, nearly every 
prominent railroad having its own special design, which perhaps 
differed from that of some other road by only a very minute and 
insignificant detail, but which nevertheless would require a 
complete new set of j oils for rolling. This had a very appreciable 
effect on the cost of rails. In 1893, the American Society of 
Civil Engineers, after a very exhaustive investigation of the 




Fig. 117. — Standakd Rail Sections. 



subject, extending over several years, having obtained the opin- 
ions of the best experts of the country, adopted a series of sec- 
tions which have been very extensively adopted by the railroads 
of this country. 

In 1909 the American Railway Association and the American 
Railway Engineering Association, by combined action, developed 
a series of sections. Fig. 117 shows diagrammatically all of 
these sections and their variations with different weights and 
systems are shown by the tabular values for the lettered dimen- 
sions. It may be noted that the radii of the upper and lower 
corners of the flanges and of the lower corners of the head are 
constant (r^") for all weights of rail and for all systems. 



§267. 



RAILS. 



299 







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300 



RAILROAD CONSTRUCTION. 



§268. 




Fig. 118. — Relation 
OF Rail to Wheel- 
tread. 



The chief features of disagreement among railroad men relate 
to the radius of the upper corner of the head and the slope of the 
side of the head. The radius (ys") adopted by the A. S. C. E. 
for the upper corner (constant for all weights) is a little more 
than is advocated by those in favor of " sharp corners " who 
prefer a radius of I". On the other hand it is much less than 

is advocated by those who consider that it 
should be nearly equal to (or even greater 
than) the larger radius universally adopted 
for the. corner of the wheel-flange. The 
discussion turns on the relative rapidity of 
rail wear and the wear of the wheel-flanges 
as affected by the relation of the form of the 
wheel- tread to that of the rail. It is argued 
that sharp rail corners wear the wheel- 
flanges so as to produce sharp flanges, 
which are liable to cause derailment at 
switches and also to require that the tires of 
engine-drivers must be more frequently 
turned down to their true form. On the 
other hand it is generally believed that rail wear is much less 
rapid when the area of contact between the rail and wheel- 
flange is small, and that when the rail has worn down, as k, inva- 
riably does, to nearly the same form as the wheel-flange, the rail 
wears away very quickly. The A. R. E. A. system uses f " radius 
i'or all rail weights. The " B " sections were proposed to satisfy 
*hose that desired that the head should be narrower and deeper 
than as found in the " A " sections. The A. R. E. A. Manual 
(1915), suggests that if a section is found to be inadequate because 
of lack of depth of head, the next heavier section will be found 
more desirable and economical. 

268. Weight for various kinds of traffic. The heaviest rails 
in use weigh 120 to 140 lbs. per yard, and even these are only 
used on some of the heaviest traffic sections of such roads as the 
N. Y. Central, the Pennsylvania, the N. Y., N. H. & H., and 
a few others. Probably the larger part of the mileage of the 
country is laid with 80- to 90-lb. rails — considering the fact that 
" the larger part of the mileage " consists of comparatively light- 
traffic roads and may exclude all the heavy trunk lines. Very 
light-traffic roads are sometimes laid with 70-lb. rails. Roads 
with fairly heavy traffic generally use 90- to 100-lb. rails, espe- 
cially when grades are heaVy and there is much and sharp curv- 



§268. RAILS. 301 

ature. The tencjency on all roads is toward an increase in the 
weight, rendered necessary on account of the increase in the 
weight and capacity of rolling stock, and due also to the fact that 
accumulated operating experience has shown that it is both 
better and cheaper to obtaiij a more solid and durable track 
by increasing the weight of the rail rather than by attempting 
to support a weak rail by an excessive number of ties or by exces- 
sive track labor in tamping. It should be remepjbered that in 
buying rails the mere weight is, in one sense, of no importance. 
The important thing to consider is the strength and the stiff- 
ness. If we assume that all weights of rails have similar cross- 
sections (which is nearly although not exactly true), then, since 
for beams of similar cross-sections the strength varies as the cube 
of the homologous dirnensions and the stiffness as the fourth power, 
while the area (and therefore the weight per unit of length) only 
varies as the square, it follows that the stiffness varies as the 
square of the weight, and the strength as the f power of the 
weight. Since for ordinary variations of weight the price per 
ton is the same, adding (say) 10% to the weight (and cost) adds 
21% to the stiffness and over 15% to the strength. As another 
illustration, using an 80-lb. rail instead of a 75-lb. rail adds only 
6f % to the cost, but adds about 14% to the stiffness and nearly 
11% to the strength. This shows why heavier rails are more 
economical and are being adopted even when they are not abso- 
lutely needed on account of heavier rolling stqck. The stiffness, 
strength, and consequent durability are increased in a much 
greater ratio than the cost. 

The relation between weight of rail and the weight on the 
drivers of the locomotives which are to run on it has been briefly 
expressed by the Baldwin Locomotive Works as " 300 pounds 
of wheel per pound of rail per yard." * This rule may be utilized 
by making a diagram as shown in Fig. 119. For example, if it 
is desired to use a type of locomotive with 170,000 lbs. on the 
drivers and also 75-lb. rails, four pairs of drivers will be needed 
and such a type of locomotive should be used. By usmg 95-lb. 
rails the same weight on the drivers could be placed on three axles. 
As another example, a Pacific-type locomotive, with 150,000 lbs. 
on its six drivers, should have a rail with a minimum weight of 83 
lbs., or say an 85-lb. rail. Whatever elements are given, the cor- 
responding proper value for the other element may be derived. 



* See § 447 (c) for expansion of this rule. 



302 



EAILROAD CONSTRUCTION. 



§269. 



269. Effect of stiffness on traction. A very important but 
generally unconsidered feature of a stiff rail is its effect on trac- 
tive force. An extreme illustration of this principle is seen 
when a vehicle is drawn over a soft sandy road. The constant 
compression of the sand in front of the wheel has virtually the 
same effect on traction as drawing the wheel up a grade whose 



600,000 



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"S 350,000 



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tut 
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50 



60 70 80 90 

Weight of Rail - pounds per yard 



100 



Pig. 113 — Curves for Finding the Number of Drivers Needed for 
Given Weight on Driving Wheels and Weight of Rails. 

steepness depends on the radius of the w^heel and the depth of 
the rut. On the other hand, if a wheel, made of perfectly 
elastic material, is rolled over a surface which, while supported 
with absolute rigidity, is also perfectly elastic, there would be a 
forward component, caused by the expanding of the compressed 
metal just behind the center of contact, which would just bal- 
ance the backward component. If the rail was supported 
throughout its length by an absolutely rigid support, the high 
elasticity of the wheel-tires and rails would reduce this form of 



§270. KAILS. 303 

resistance to an insignificant quantity, but the ballast and even 
the ties are comparatively inelastic. When a weak rail yields, 
the ballast is more or less compressed or displaced, and even 
though the elasticity of the rail brings it back to nearly its 
former place, the work done in compressing an inelastic material 
is wholly lost. The effect of this on the fuel account is certainly 
very considerable and yet is frequently entirely overlooked. It 
is practically impossible to compute the saving in tractive power, 
and therefore in cost of fuel, resulting from a given increase in 
the weight and stiffness of the rail, since the yielding of the rail 
is so dependent on the spacing of the ties, the tamping, etc. But 
it LS not difficult to perceive in a general way that such an econ- 
omy is possible and that it should not be neglected in considering 
the value of stiffness in rails. 

270. Length of rails. The recommended standard length of 
rails is 33 feet. Several years ago, many roads experimented with 
45-foot and even 60-foot rails. The argument in favor of longer 
rails is chiefly that of the reduction in track-joints, which are 
costly to construct and to maintain and are a fruitful source of 
accidents. Mr. Morrison of the Lehigh Valley R. R.* declared 
that, as a result of extensive experience with 45-foot rails on that 
road, he found that they are much less expensive to handle, 
and that, being so long, they can be laid around sharp curves 
without being curved in a machine, as is necessary with the 
shorter rails. The great objection to longer rails lies in the 
difficulty in allowing for the expansion, which will require, in 
the coldest weather, an opening at the joint of .nearly f" for a 
60-foot rail. The Pennsylvania R. R. and the Norfolk and 
Western R. R. each laid a considerable mileage with 60-foot 
rails. The net result is the fixed standard of 33 feet. 

271. Expansion of rails. Steel expands at the rate of .0000065 
of its length per degree Fahrenheit. The extreme range of tem- 
perature to which any rail will be subjected will be about 160°, 
or say from -20° F. to +140° F. With the above coefficient 
and a rail length of 60 feet the expansion would be 0.0624 foot, 
or about f inch. But it is doubtful whether there would ever 
be such a range of motion even if th^e were such a range of 
temperature. Mr. A. Torrey, chief engineer of the Mich. Cent. 
1$.. R., experimented with a section over 500 feet long, which, 

* Report, Roadmasters Association, 1895. 



304 EAILBOAD CONSTRUCTION. § 272. 

although not a single rail, was made ''continuous" by rigid 
splicing, and he found that there was no appreciable additional 
contraction of the rail at any temperature below +20° F. The 
reason is not clear, but the fact is undeniable. 

The heavy girder rails, used by the street railroads of the 
country, are bonded together with perfectly tight rigid joints 
which do not permit expansion. If the rails are laid at a tem- 
perature of 60° F. and the temperature sinks to 0°, the rails 
have a tendency to contract .00039 of their length. If this 
tendency is resisted by the friction of the pavement in which the 
rails are buried, it only results in a tension amounting to .00039 
of the modulus of elasticity, or say 10920 pounds per square 
inch, assuming 28 000000 as the modulus of elasticity. This 
stress is not dangerous and may be permitted. If the tempera- 
ture rises to 120° F., a tendency to expansion and buckling will 
take place, which will be resisted as before by the pavement, 
and a compression of 10920 pounds per square inch will be in- 
duced, which will likewise be harmless. The range of tempera- 
ture of rails which are buried in pavement is much less than 
when they are entirely above the ground and will probably 
never reach the above extremes. Rails supported on ties which 
are only held in place by ballast must be allowed to expand and 
contract almost freely, as the ballast cannot be depended on to 
resist the distortion induced by any considerable range of tem- 
perature, especially on curves. 

272. Rules for allowing for temperature. Track regulations 
generally require that the track foremen shall use iron • {not 
wooden) shims for placing between the ends of the rails while 
splicing them. The thickness of these shims should vary with 
the temperature. Some roads use such approximate rules as the 
following : '' The proper thickness for coldest weather is y^ of an 
inch; during spring and fall use ^ of an inch, and in the very 
hottest weather Te of an inch should be allowed.'" This is on 
the basis of a 30-foot rail. When a more accurate adjustment 
than this is desired, it may be done by assuming some very high 
temperature (100° to 125° F.) as a maximum, when the joints 
should be tight; then compute in tabular form the spacing for 
each temperature, varying by 25°, allowing 0".0643 (very 
nearly re") for each 25° change. Such a tabular form would 
be about as follows (rail length 33 feet): 



§273. 



RAILS. 



305 



Temperature. . 


Over l00° 


100°-75° 


75°-50° 


50°-2.5° 


25°-0° 


Below 0° 


Rail opening. . 


Close 


1 // 

To 


1// 

8 


^" 


in 


5 fr 



One practical difficulty in the way of great refinement in this 
work is the determination of the ifGal tetnperature of the rail 
when it is laid. A rail lying in the hot sUn has a very much 
higher temperature than the air. The temperature of the rail 
cannot be obtained even by exposilig a thermometer directly to 
the sun, although such a result might be the best that is easily 
obtainable. On a cloudy or rainy day the rail has practically 
the same temperature as the air; therefore on such days there 
need be no such trouble. 

273. Standard specifications. Specifications are 'coiistantly 
Varying. "They are always a conipromise between the wishes 
of railroad engineers and the interests of rail manufacturers. 
At present (1921) rail prices are high, the railroads are relatively 
in a low financial condition, and the specifications are much less 
rigid than those mutually accepted in 1910. Therefore, instead 
of quoting verbatim, in this edition, the specifications now cur- 
rent, the general features have been discussed, many of which 
\vill probably be modified in future specifications. When buy- 
ing rails for any road, the latest issue of standard Am^ Rwy. Eng. 
Assoc, specifications should be obtained for reference. 

2173a. Chemical composition. More than 98% of the com- 
position of steel rails is iron, but the value of tjie rail, as a rail, 
is almost wholly dependent upon the large number of other 
chemical elements which are, or may be, present in very small 
amounts. 

Carbon. Many years ago, when rails were comparatively 
light and the maximum wheel loads were correspondingly light, 
the carbon in rails ranged from 0.20% to 0.50%. But the great 
increase in wheel loads produces a concentrated pressure on the 
rails which causes the steel to " flow " if the steel is comparatively 
soft. An increase of a few hundredths of a percent of carbon 
makes the steel harder but an excess of carbon makes it too 
brittle. Since heavier wheel loads require heavier rails, more 
carbon is used in the heavier sections. Since it is safer to use 
more carbon in open-hearth rails than in Bessemer rails, a higher 
percentage is so used. The hmits at present (1921) are as 
follows : 



306 



RAILROAD CONSTRUCTION. 



§ 2736. 





Bessemer process. 


Open-hearth process. 


Chemical 
elements. 


Weight, pounds per 
yard. 


Weight, pounds per yard. 




70 to 84 


85 and over 


70 to 84 


85 to 110 


111 and over 


Carbon 

Phosphorus, 

not to exceed 
Manganese . . . 
Silicon, not less 

than 


0.40 to 0.50 

0.10 
0.80 to 1.10 

0.10 


0.45 to 0.55 

0.10 
0.80 to 1.10 

0.10 


0.53 to 0.68 

0.04 
0.60 to 0.90 

0.10 


0.62 to 0.77 

0.04 
0.60 to 0.90 

0.10 


0.67 to 0.82 

0.04 
0.60 to 0.90 

0.10 



Sulphur. Former specifications required that sulphur should 
not exceed 0.075% in Bessemer rails and 0.06% in open-hearth 
rails. Manufacturers now demand an excess price if a definite 
limitation is made but say that it is to their own interests, for 
other reasons, to have the sulphur within safe limits. As a 
compromise, no definite limitation is now made. This con- 
cession, now allowed by the railroads, illustrates forcibly how 
the railroads are compelled by financial considerations to relax 
from the former rigidity of specifications. 

When a railroad buys a large order of rails directly from a 
rail mill, the railroad usually sends an inspector, who is furnished 
by the manufacturer with chemical analyses of the steel, one 
for each day and night turn for Bessemer rails or one for each 
heat of open-hearth rails. Sometimes samples are furnished 
the inspector, if he is a chemist, and he is given facilities at the 
mill to make his own check analyses. 

273b. Physical requirements. These are increasingly depended 
on to determine (a) ductility or toughness as opposed to brittle- 
ness and (6) soundness, or its homogeneity and freedom from 
seams, laminations, cavities, or interposed foreign matter. The 
ductility is tested by dropping a tup weighing 2000 pounds, 
which has a striking face with a radius of 5 inches, on a test 
rail about 5 to 6 feet long, which is supported on two pedestals, 
also having bearing surfaces with radii of 5 mches, the pedestals 
being adjustable to spans varying from 3 feet to 4 feet 6 inches. 
The pedestals are spaced 3 feet for rails weighing 110 pounds 
per yard or less, and are spaced 4 feet for rails weighing 111 to 
140 pounds per yard. The pedestals are firmly secured to an 
anvil weighing 20,000 pounds which is supported on 20 very 
heavy springs. Gauge marks, one inch apart for three inches 



§273c.' RAILS. 307 

each side of the center, are marked in the center of the top of 
the rail. The rails are usually tested with the head in tension, 
or with the rail inverted. The tup falls from a height of 16 feet 
on 70- to 79-pound rails, 17 feet on 80- to 90-pound rails, 18 feet 
on 91- to 110-pound rails and 20 feet on 111- to 140-pound rails. 
Under such impacts the elongation on one inch of the six-inch 
scale, marked as above, shall be at least 8%. The permanent 
set, on a 3-foot chord, is noted for each blow. The test pieces, 
which do not break under ordinary blows, are nicked and broken 
so that the interior may be examined for " soundness," or for 
such flaws as fissures, laminations, cavities, etc. Fissures which 
are really indicative of structural defects in a rail are sometimes 
microscopic, even when a specimen is carefully cut from the 
rail and the surface polished. The defects may be deepened 
and accentuated by etching the surface with hot concentrated 
hydrochloric acid. 

By agreement between a railroad and a rail manufacturer, 
the physical test may be made by a quick-bend machine instead 
of a falling weight. Such a machine is essentially a hydraulic 
press of not less than 350 tons capacity. The bearing supports 
of the tested rail are flat surfaces, with vertical faces 48 inches 
apart, of which the inner edges are rounded to a |-inch radius. 
The head of the ram has a bearing surface with a radius of five 
inches. The percentage of elongation before failure may be 
observed as before. 

273c. Classification. Rails are classified as No. 1 and No. 2. 
No. 1 rails are those with no injurious defects or flaws. No. 2 
rails are those which arrive at the straightening presses more 
crooked than is allowed for No. 1 rails but which, in the judg- 
ment of the inspector, may be accepted in spite of this or other 
minor defects which do not impair their soundness and strength. 
No. 2 rails must not exceed 5 per cent of the whole order. They 
must have their ends painted white and have two prick-punch 
marks on the side of the web near the heat number, near the 
end of the rail, so placed as not to be covered by the joint bars. 

273d. Branding. The name of the manufacturer, the month 
and year of manufacture, and the weight and type of section of 
rail shall be rolled in raised letters and figures on one side of the 
web, where it will not be covered by joint bars. The markings 
shall be done so effectively that the marks may be read as long 
as the rails are in service. The type of section is indicated by 



30§ RAILROAD COJ^STRUCTION. § 273e. 

A. S. C. E., R. A.-A., R. A -B., or R. E,, to indicate one of 
the various types elaborated in Table XXIII. Open-hearth 
rails are branded or stamped O. H., in addition to the other 
marks. 

273e. Dimensions and drilling. The standard length is 33 
feet at a temperature of 60° F. Ten per cent of the entire order 
will be accepted in shorter lengths, varying by one foot from 
32 to 25 feet. A variation of I" from specified lengths is allowed, 
except that 15% of the order may vary |" from specified lengths. 
Drill holes may vary -^^" in size and location from the drawings 
furnished by the railroad company. The recommended position 
(vertically) in the web is " midway between the intersections 
of the vertical center line of the rail with the planes of the fishing 
surfaces of the head and base." The hole centers should he 
5^" apart, the first hole center being 2x^" from the end of the 
rail, which allows |-inch clearance when the rails are bolted 
together in normal position. 

273f. Finishing. Rails must be srnooth at the heads, straight 
in line and surface, and without twists, waves or kinks. The 
limiting allowable camber in a 33-foot rail is " 4 inches for thick 
base sections and 5 inches for thin bas6 sections." They shall 
be sawed square at the ends, a variation of not more than -^'^ 
being allowed. Burrs must be carefully removed. When a 
finished rail shows defects at either end or in any drilled hole, 
the entire rail shall be rejected. 

274. Life of rails. There has been a great development since 
1900 in the science of manufacturing rails. This is indicated by 
the decrease in rail '' failures." If there is a defect in a rail it 
will usually break or " fail '■ before it is worn down. If the defect 
is serious it will break in a few weeks or months. Minor defects 
require much longer time to develop. The accompanying 
tabular form shows the number of rail failures per 100 track 
miles, after one to five years' service, reported by several rail- 
roads of the United States. To appreciate the figures, note that 
there are 32000 rails 33 feet long in 100 miles of track. The 
record for rails rolled in 1913 showed that after five years' servipe, 
a total of 246.5 per 100 miles, or an average of 0.77%, had failed. 
Note that the increase of failures per year, after the first year, 
is regular, as it slyould be. Note also that there has been a steady 
improvement in the figures for 3, 4 and 5 years' service, but 
that since 1914 or 1915 the failures have increased somewhat, 



§ 275. HAILS. S09 

AVERAGE RAIL FAILURES PER lOO TRACK MILES 



Year 


Years of service. 


rolled. 





1 


2 


3 


4 


5 


1908 
1909 
1910 
1911 
1912 
1913 
1914 
1915 
1916 
1917 
1918 
1919 
1920 


'2!o' 

1.2 
0.7 
1.6 
5.3 
1.6 
2.0 
3.9 


2 8. 9 
12.5 

8.2 

8.9 
11.8 
21.6 " 

8.9 
14.8 


77.0 
32.1 
25.8 
19.8 
19.0 
29.2 
38.9 
27.6 


124 '. 6 
104.4 
49.3 
44.8 
32.9 
34.2 
47.7 
66.0 


224 !i 
152.7 
133.3 
78.9 
69.5 
50.9 
53.0 
70.6 


398.1 

277.8 

198.5 

176.3 

107.1 

91.9 

74.0 

82.4 



indicating perhaps that war conditibns had lowered thfe c(tiality 
of the tails. The reports also ishbw that failures ar§ liidre colii- 
mon using Bessemer than open-hearth rails, ahd, cdhsidering 
that Bessemer are used in general fdr lighter service, the ratib 
against Bessemer would probably be greater fofr the sanie ser- 
vifce. Bessemer rails cost less than open hearth, and this fsifcrt 
is perhaps the only reason fbr their use; The percentage of 
Bessemer rails to the total in 1913 was about 9.1%; in 1918 
the percentage was reduced to 2.7%. These figures are based 
on reports made to the A. R. E. A. Presumably cottiplete std,tis- 
tics (unobtainable) wdhld show a somewhat larger petcehtage 
of Bessemer rails, used oh small roads whitih did not make 
reports, but the above figttres show that Open-heatth rails ar^ 
Considered to be superior in spite of the higher price. 

275. liitensity of pressure oh rails. A Specigil committee of 
the. A. R. E. A. made an ih\^estigation to determine the inteh- 
sities of pressure prbduced by varying wheel loads oh the head 
of a rail and also the amount of permanent defot-mation of 
" flow " of the metal. The testing mechanishl made it possible 
to increase the " wheel load " up to 580,000 lbs., a figure about 
30 times as great as the greatest wotkin^ wheel loS-d. The hriit 
intensity of pressure incteiised with an increase of the wheel load 
from zero up to about 30,000 lbs. At that figure, which corrfe- 
fepiOnds to ah axle load of 60,000 lbs,, or nearly the iliaxiinuih 
of present practice, the unit intensity of presshre reached its 
maximum arid remained substantially constant while the Wheel 
load was increased from 30,000 to 580,000 lbs. In other wOtds, 



310 RAILED AD CONSTRUCTION. § 275a. 

the area of contact increased as fast as the pressure increased. 
This maximum average unit pressure varied considerably with 
the different rails tested, although it was nearly constant for any 
one rail. The unit values varied from 105,000 to 160,000 lbs, per 
square inch. 

275a. Flow of metal. The permanent deformation of the 
metal was measured by noting the reduction in the horizontal 
and vertical diameters of very small tapering holes drilled into 
the head of the tested rail slightly below the bearing surface. The 
testing wheel was caused to roll over the tested rail several 
hundred thousand times. In one test the initial load was 15,000 
lbs., increasing by steps up to 30,000 lbs. Up to a load of 
20,000 lbs. no permanent deformation of the holes was observ- 
able. With a load of 25,000 lbs. a slight set was observable 
which grew more rapid when the load was increased to 30,000 
lbs. But even then the deformation was not as great as it was 
in another test when the initial loading was 30,000 lbs. This 
indicates that the effect on a rail of continued rolling pressure 
of less than 20,000 lbs., or 40,000 lbs. per axle, is to harden the 
surface metal and make it better able to withstand wear. This 
seems to be corroborated by, and also explains, the remarkable 
wearing qualities of many old rails which were surfaced-hardened 
by comparatively light loads and which subsequently carried 
much heavier loads with less wear than new rails. 

276. Rail wear on tangents. The weight carried by a single 
engine driver is often from 24,000 to 30,000 lbs. Each of the 
eight wheels of a J.40,000-lb. capacity coal car, when loaded, 
carries nearly 25,000 lbs. Such loads will certainly cause a 
flow of metal, as shown by the laboratory test above described, 
but a four-weeks' service test on the same test rails (referred to 
above) showed a flow of metal as indicated in Fig. 120, the flow 
being considerably greater than that produced in the laboratory 
tests with the same weight and number of rollings. The aver- 
age wheel load in the service test was much less than the maxi- 
mum in the laboratory tests, but the greater effect was probably 
due to the great variety of wheel treads making different forms 
of contact between rail and tread, with occasional great con- 
centration of pressure. But Fig. 120 shows the typical normal 
wear of a rail on a tangent, the wear being all on the top. The 
center of pressure is usually about one-half inch inside from the 
center of the rail and is inclined outward. The wear is approxi- 



§ 276a. KAILS. 311 

mately symmetrical with this axis. Fig. 120 also shows the 
flow of metal outside of the original -contour, which all occurred 
on the gauge side. Very soft badly worn rails may show a fin 
on the outside. 





Fig. 120. — Rail Wear on a Fig. 121. — Rail Wear on Curves. 

Tangent. 

276a. Rail wear on curves. The pressure and grinding action 
of the wheel flanges against the rails wears away the inner side 
of the head of the outer rail on curves. If the rail is left in the 
track and the wear is permitted to continue, the head may be 
worn to approximately the form shown in Fig. 121. On the 
other hand, the inner rail is not subjected to any such lateral 
grinding action, and the rail is worn to substantially the same 
form as on a tangent, but the wear is more rapid due to the 
longitudinal slipping — see § 395, If the rails are soft or the 
traffic very heavy, thefe will be a flow of metal and a fin will 
form on the inner edge and perhaps also on the outer edge. 

277. Experimental determination of rail wear. Several years 
ago a series of tests for rail wear were made on the Northern 
Pacific R. R. by taking up, weighing, and replacing, each year, 
the several groups of rails under test. Some of these rails were 
on tangents, the others on curves of various curvature. Some 
of the rails of each group were made of Bessemer steel, the others 
of open-hearth steel. No tests were made to determine the loss 
of weight through mere oxidation. All of the rails were in service 
for five years and some lasted for six years or more, but the loss 
in weight during the sixth year was nearly always equal to, and 
in some cases twice as much as, the loss during the preceding 
five years. Some of the rails lost over 10% of their weight, or 
about one-fourth the weight of the head, before being removed. 
Although the tests were too few to establish any positive laws, 
some tendencies which may be observed will give at least an 
approximate idea of the laws of rail wear. 

1. The average loss of weight during the first five years on 



312 RAILKOAD CONSTRUCTION. § 278. 

20 rails on tangents was 0.412 lb. per yard per 10,000,000 tons 
of traffic. 

2. Ten of these same rails were kept in place at least one year 
longer and during the sixth year lost almost twice as much metal 
as during the previous five years; in other words, about two- 
thirds of the entire loss occurred during the sixth year. 

3. The average loss of weight during the first five years from 
20 rails on a tangent was 0.463 lb. per yard per 10,000 trains. 
The relation between mere tonnage and number of trains could 
not be even indicated by so few tests. There is reason to believe 
that engine drivers are more responsible for rail wear than mere 
car-wheel tonnage. This practically means that one effect of 
grade is to increase rail wear, since more (or heavier) engines 
are needed to haul a given car tonnage. 

4. The wear of the outer rail of curves is, of course, far greater 
than that of the inner rail, but the figures obtained did not seem 
to follow any rational law, the ratio of outer to inner rail wear 
varying from 144 to 244%, with an average of 182%. 

5. The average rail wear on curves, averaging inner and outer 
rails, per yard, per degree of curve, per 10,000,000 tons traffic, 
varied from 0.145 lb. for a 4° 04' curve down to 0.102 lb. per 
degree for a 10° 13' curve. Based on the four curves tested, the 
results seemed to point to the law that raiil wear on curves does 
not increase as fast as the degree of the curve. 

6. Although the tests were too few to establish any law, the 
increase of the mean rail wear on curves with increase in degree 
of curve was very regular and indicated that the average rail 
wear on a curve of about 6° 40' is about twice as great as that 
on a tangent. 

7. The wear on open-hearth rails was almost invariably less 
than that on Bessemer rails, under identical conditions. 

278. Cost of rails. In 1873 the cost of steel rails was about 
$120 per ton, and the cost of iron rails about $70 per ton. 
Although the steel rails were at once recognized as superior to 
iron rails on account of more uniform wear, they were an expen- 
sive luxury. The manufacture of steel rails by the Bessemer 
process created a revolution in prices, and they steadily dropped 
in price until, many years ago, steel rails were manufactured 
and sold for $22 per ton. For several years since then the price 
was very uniform at $28 per' ton at the mill. But now (1921) 
the advantages of open-hearth steel are better appreciated and 



§278. . RAILS. 313 

a large proportion of rails are being rolled from open-hearth 
steel, which commands about S2 per ton more. At present 
(1921) the current prices at Pittsburgh mills run at about $38 
per ton for Bessemer and $40 for open-hearth. 

There is no longer any demand for iron rails, since the cost 
of manufacturing them is substantially the same as that of steel 
rails, while their durability is unquestionably inferior to that 
of steel rails. Rail quotations are generally on the basis of 
" long tons " of 2240 lbs. 

The freight charge for transporting rails from the mill to the 
place where used is usually so large that it adds a very appreci- 
able amount to the cost per ton. As an approximation, the 
freight may be estimated as 0.6 cent per ton-mile, or $3.00 per 
ton for a haul of 500 miles. 



CHAPTER X. 

RAIL-FASTENINGS. 

RAIL-JOINTS 

279, Theoretical requirements for a perfect joint A perfect 
rail-joint is one that has the same strength and stiffness — no 
more and no less — as the rails which it joins, and which will 
not interfere with the regular and uniform spacing of ties. It 
should also be reasonably cheap both in first cost and in cost of 
maintenance. Since the action of heavy loads on an elastic rail 
is to cause a wave of translation in front of each wheel, any 
change in the stiffness or elasticity of the rail structure will 
cause more or less of a shock, which must be taken up and 
resisted by the joint. The greater the change in stiffness the 
greater the shock, and the greater the destructive action of the 
shock. The perfect rail-joint must keep both rail-ends truly in 
line both laterally and vertically, so that the flange or tread of 
the wheel need not jump or change its direction of motion sud- 
denly in passing from one rail to the other. A consideration of 
all the above requirements will show that only a perfect welding 
of rail-ends would produce a joint of uniform strength and stiff- 
ness which would give a uniform elastic wave ahead of each 
wheel. As welding is impracticable for ordinary railroad work 
(see § 271), some other contrivance is necessary which will 
approach this ideal as closely as may be. 

280. Efficiency of any type of rail-joint. Throughout the 
middle portion of a rail the rail acts as a continuous girder. If 
we consider for simplicity that the ties are unyielding, the deflec- 
tion of such a continuous girder between the ties will be but 
one-fourth of the deflection that would be found if the rail were 
cut half-way between the ties and an equal concentrated load 
were divided equally between the two unconnected ends. The 
maximum stress for the continuous girder would be but one-half 
of that in the cantilevers. Joining these ends with rail-joints 
will give the ordinary "suspended" joint. In order to main- 

314 



§ 281. RAIL-FASTENINGS. 315 

tain imiform strength and stiffness the rail-joint must supply 
the deficiency. These theoretical relations are modified to an 
unkown extent by the uncertain and variable yielding of the 
ties. Since a theoretically perfect joint is unattainable, on 
account of the necessity for allowing for expansion, the nearest 
approach appears to be a joint which, when tested in comparison 
with a solid rail on an equal span (20 inches), will withstand 
an equal load before permanent set takes place. Some very 
thorough tests of several types of joints were made on this basis 
by the Pennsylvania R. R. at Altoona in 1915. The types 
tested were plain angle-bars, the " Continuous," the " Bonzano," 
the " 100-per-cent," (see Plate VII) and also the " Duquesne," 
which is similar to the Bonzano and the 100-per-cent, except 
that the fin which projects below the rail between the ties has 
a different form. The " efficiency " of these joints was com- 
puted as the ratio of the load carried by the joint when it began 
to fail (or when permanent set commenced) to the load carried 
by the solid rail when it began to fail. The efficiencies for 
these joints, as ordinarily used, tested from 29% to 64%. 
Tests were also made of " heat-treated " joints (see § 285) 
which showed efficiencies from 60 to 150% higher than the 
untreated joints, the efficiencies being, in nearly every case, 
over 100%. The heat treatment costs about 0.2 c. per pound 
or say 16 c. for an 80-lb. pair. The added efficiency is so 
well worth the added cost that the use of heat treated 
splice bars is becoming more common and may soon become 
standard. 

281. Effect of rail gap at joints. It has been found that the 
jar at a joint is due almost entirely to the deflection of the joint 
and scarcely at all to the small gap required for expansion. 
This gap causes a drop equal to the versed sine of the arc having 
a chord equal to the gap and a radius equal to the radius of 
the wheel. Taking the extreme case (for a 30-foot rail) of a 
f " gap and a 33" freight-car wheel, the drop is about TTfTTTJ""- 
In order to test how much the jarring at a joint is due to a gap 
between the rails, the experiment was tried of cutting ^hallow 
notches in the top of an otherwise solid rail and running a loco- 
motive and an inspection car over them. The resulting jarring 
was practically imperceptible and not comparable to the jar pro- 
duced at joints. Notwithstanding this fact, many plans have 
been tried for avoiding this gap. The most of these plans con- 



316 



RAILEOAt) COJ^StRUGTlON. 



§282. 



sist essentially of some fotm of cdmpotind rail, the sections 
breaking joints. (Of course the design of the compound rail 
has also several other objects in view.) In Fig. 122 are shown a 
few cff the very tnaily desigrts which have beeri propotsed. These 
designs have invariably been abanddned after trial. Another 
plan, which has been extensively tried dn the Lehigh Valley 
R. R., is the use of mitered joints. The advantages gained by 
their use are as yet doubtful, while the added expense is Uiiques- 
tionable. The " Roadtnasters Association bf America '^ in 1895 





Fig. 122. — Compottnd HAil Sections. 

adopted a resolution recommending mitered joints for double 
track, but their use has been abandoned. 

282. " Supported," " suspended," and " bridge " joints. A 
joint is " supported " when a tie is placed immediately under 
the middle of the joint. The localized traffic stress at the joint 
must be carried almost exclusively by that one tie and compara- 
tively little is carried by the adjacent ties. A " suspended " 
joint is located symmetrically between two ties, which share 
equally the localized stress. Formerly there was a considerable 
proportion of railroad engineers who favored supported joints, 
but now the suspended joint is almost universally the stand- 
ard. 

" Bridge "-joints are similar to suspended joints in that the 
joint is supported on two ties, but there is the important differ- 
ence that the bridge joint supports the rail from underneath and 
there is no transverse stress in the rail, whereas the suspended 
joint requires the combined transverse strength of both angle- 
bars and rail. The " Fisher " bridge joint, now seldom seen, 
is purely of this type, only two bolts being used to hold the rail 
ends together. But the princijfle of supporting the base of the 
rail is seen in the Wolhaupter, the Weber, the Continuous and 
the Atlas. See Plate VII. Although some of these forms are 
in extensive use, the angle-bar (see § 284) is the standard on a 
large proportion of the mileage of the country. 

283. Failures of rail-joints. An instructive report was made 
in 1915 by an Engineer of Tests of the Pennsylvania R. R. on 



i 284. RAIL-t^ASTENINGS. 317 

^n examination of 960 angle-bars, foUnd in a scrap pile, to deter- 
mine the various causes for their removal from the track. The 
various causes were classified under five headings, the typical 
failures being illustrated in Fig. 123. (1) Abrasion on the 
top fishing surface, the depth of wear varying from i^" to ^" 
and extending perhaps 8 inches each way from the center. On 
short 4-hole bars the wear is almost wholly in the center; oh 
the longer 6-hole bars, wear is also found near the ends of the 
bar. Such wear demonstrates the amoimt of working and grind- 
ing which evidently takes place when a joint is depressed under 



O 



No. 5 



o^-^'o^^'B^f "o 



o. 3 trNo. 4 



^ 



Fig. 123. — Diagram op Ty^'es op Breaks of Angle-Bars. 

traffic. This is the only form of actual wear which occurs. 
24% of the 960 bars were removed for this cause. (2) When 
a joint bar is very long, the stresses in the bar may be reversed 
and there may be tension in the top and a break may start at 
the top and continue down, usually into a bolt hole. Less than 
5% of the failures were of this class. (3) and (4). Usually a 
crack starts at the bottom and may or may not extend to a 
bolt hole. Usually the crack starts from a spike slot or 
from the re-entrant angle at either end of a depending flange. 
If the cracked bar is permitted to remain in the track, the 
crack of (2), (3) or (4) develops into a complete break (5). 
44% of the 960 bars, or 59% of all but No. 1, were complete 
breaks. 

284. Standard angle-bars. An angle-bar must be so made 
as to closely fit the tails. The. great multiplicity in the designs 
t)f rails (refetred to in Chapter IX) results in a corresponding 
variety in the detailed dimensions of the angle-bars. The abso- 
lutely essential features required for a fit are (1) the angles 
(of the upper and lower surfaces of the bar where they fit against 
the rail, and (2) the height of the bar. The bolt-holes in the 
bar and rail must also correspond. The holes in the angle-bars 
are elongated or made oval, so that the track-bolts, which are 
made of corresponding shape immediately under the head, will 
not be turned by jarring or vibration. The holes in the rails 



318 



RAILROAD CONSTRUCTION. 



§284. 



are made of larger diameter (by about 3^") than the bolts, so as 
to allow the rail to expand with temperature. 

In Table XXIV and in Fig. 124 are shown the angles and< 
dimensions for angle-bars to fit the standard rail sections shown 
in § 267. Note that the dimension a for the angle-bar corre- 
sponds with dimension F for the rail and that Ri and the angle a are 
the same for both for each type of rail. These dimensions were 
copied from the 1916 Handbook of the Carnegie Steel Co. 
Although they correspond perfectly with the rail standards of the 




^. 



Fig. 124. — Standard Angle Bar. 



A. R. E. A., that association has not yet adopted any such 
definite standard dimensions for a rail-joint. 

The standard drilling for bolt-holes in angle-bar, as adopted 
by the A. R. E. A. in 1914, is as follows: 

For 6-bolt sphces, 5 spaces of 5| inches. 
For 4-bolt splices, 3 spaces of 5| inches. 

No definite recommendation was made by the Association as 
to the total length of angle-bars, but the committee recom- 
mended that, on the basis of the abo\ e spacing of holes, 24 inches 
is a satisfactory length for a 4-bolt splice and 32 inches for a 
6-bolt splice, in both cases using suspended joints. On this 
basis, the spacing from the center of the last hole to the end of 
the bar would be 3| inches for the 4-bolt splice and 2$ inches for 
the 6-bolt splice. 

In Plate VII are shown some of the many designs which have 
been competing for favor and which have been more or less 



§284. 



RAIL-FASTENINGS . 



319 



03 



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Am. 
Rwy. 

Eng. 


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all 



320 



RAILROAD CONSTRUCTION. 



§2^ 



extensively tried out for both steam and electric railroad wo 
While many thousands in the aggregate have been placed 
various roads, no one design has succeeded in displacing t\ 
angle-bar. There are necessarily as many variations in i 
details of the angle-bars as there are variations in the sizes 
rails, beside other slight variations, but all cross-sections a 
similar to that shown in Fig. 124. This general design pro 
ably represents the majority of all the angle-bars in the country. 
285. Specifications for steel angle-bars. Formerly these we 
made of either Bessemer or open-hearth steel. Now (1921), tl 
specifications of the A. E. R. A. require open-hearth steel excl 
sively. Three grades are used : " high carbon steel," " quench( 
carbon," and " quenched alloy steel." The special requirement 
in addition to the usual requirements about accuracy of wor' 
manship, branding, inspection, etc., are as follows: phosphon 
not to exceed 0.04%; quenched bars must have carbon betwee 
0.42 and 0.55%, but 1.00% of nickel or 0.35% of chromium w 
be considered the equivalent of 0.07% of carbon. The physic 
requirements are: 



High 

carbon 

steel. 



Quenched 
carbon. 



Quench ( 
alloy 
steel. 



Tensile strength, min., lbs. per sq. in. 

Elastic limit, lbs. per sq. in 

Elongation, per cent in 2 inches, min . 



85,000 
'16%" 



100,000 110,000 
70,000 I 85,000 
1,600,000-4- tens. str. 
min. 12% 



All grades: cold bending test, 90 
thickness of tested piece. 



on arc with diameter three tim( 



All punching, slotting and shaping is to be done at a tempei 
ature not less than 800° C. or 1470° F. Quenching shall be don 
in a bath of oil (or water, if specified) having a temperature o 
810° C. (1490° F.) and kept in the bath until cool enough t 
handle. 

TIE-PLATES. 

286. Advantages, (a) As already indicated in § 242, the lif 
of a soft-wood tie is very much reduced by " rail-cutting " an( 
" spike-killing," such ties frequently requiring renewal lon^ 
before any serious decay has set in. It has been practicallj 
demonstrated that the " rail-cutting " is not due to the merj 




BONZANO RAIL JOINT, 






L_iJUliL 



^ 



M^ ^ :^ - M 






-J-..|l!illll 




FISHER BRIDGE JOINT. 





WOLHAUPTER JOINT 



WEBER RAIL JOINT. 



Plate VII.— Some forms of Rail Joints. 
(Between pp. 320 and 321.) 




CONTINUOUS RAIL JOINT. 




ATLAS^SUSPENDED RAIL JOINT. 




«— a?i^H<- 6%- 



cp 






-24P 



-> 



TYPICAL 4-BOLT SPLICE BAR 




287. 



RAIL-FASTENINGS. 



321 



pressure of the rail on the tie, even with a maximum load on 
the rail, but is due to the impact resulting from vibration and 
to the longitudinal working of the rail. It has been proved 
that this rail-cutting is practically prevented by the use of tie- 
plates, (h) On curves there is a tendency to overturn the outer 
rail due to the lateral pressure on the side of the head. 
This produces a concentrated pressure of the outer edge of the 
base on the tie which produces rail-cutting and also draws the 
mner spikes. Formerly the only method of guarding against 
this was by the use of " rail-braces," one pattern of which is 
shown in Fig. 125. But shoulder tie-plates serve the purpose 
even better and rail-braces are 
chiefly used for guard rails and 
stock rails at switches, (c) Driv- 
ing spikes through holes in the 
plate enables the spiltes on each 
side of the rail to mutually sup- 
port each other, no matter in 
which (lateral) direction the rail 
may tend to move, and this prob- 
ably accounts in large measure 
for the added stability obtained 
by the use of tie-plates, (d) The 
wear in spikes, called "necking," 
caused by the vertical vibration 

of the rail against them, is very greatty reduced, (e) The cost is 
very small compared with the value of the added life of the tie, 
the large reduction in the work of track maintenance, and the 
smoother running on the better track which is obtained. It has 
been estimated that by the use of tie-plates the life of hard- 
wood ties is increased from one to three years and the life of 
soft-wood ties is increased from three to six years. From the 
very nature of the case, the value of tie-plates is greater when 
they are used to protect soft ties. 

287. Elements of the design. The Am. Rwy. Eng. Assoc, has 
stated these principles in its Manual, as follows : 

1. " Plates shall not be less than 6 inches in width, and as 
much v/ider as consistent with the class of ties to be used." The 
use of a wide tie presumes heavy traffic and heavy wheel loads 
and, therefore, a width as great as the face of the tie, up to at 
least eight inches, has been recommended. 




Fig. 125. — Atlas Brace K. 



322 RAILROAD CONSTRUCTION. § 287. 

2. " The length of the plates [parallel with the length of the 
tie] shall not be less than the safe-bearing area of the ties divided 
by the width of the plate, and, when made for screw spikes, shall 
be so shaped as to provide proper support for the screw spikes." 
335 lbs. per square inch is declared to be, by test, the minimum 
safe-bearing load. Tie-plates sometimes sink quickly and deeply 
into the tie, thus proving that the area is inadequate for the wheel 
loads and traffic on them. 

3. " The thickness of the plate shall be properly proportioned 
to the length." Tie-plates have been used as thin as -^ inch, 
but it is now being realized that the real function of the plate 
is to be a hearing plate which shall distribute the load, rather 
than a mere surface plate which shall protect the tie from abra- 
sion. The Track Committee of the A. R. E. A. recommended 
that the plates should be at least | inch thick under either edge 
of the rail. Although the Association refused to concur, the 
discussion developed the fact that the thin plates formerly used 
have been found to be too thin and that thicker plates are more 
satisfactory. 

4. Height of shoulder. The height of "at least l-inch " was 
recommended in the 1915 Manual. The Track Committee has 
since then recommended that the height should " not be less 
than j" nor more than f ". 

5. " Where treated ties are used or where plates are for screw 
spikes, a flat-bottom plate is preferable. Where ribs of any kind 
are used on base of plate, these shall be few in number and not to 
exceed i inch in depth." This specification is in direct contrast 
to the older designs which had been corrugations and even 
" claws " which were forced deeply into the tie, in order to anchor 
the plate immovably to the tie. But experience has proved that 
these corrugations hasten deterioration. In spite of this, the 
type using claws (see Fig. 126) is still the standard on some roads. 

6. " Punching must correspond to the slotting in the splice- 
bars and, where advisable, may be so arranged that the plates 
may be used for joints. Spike holes may be punched for varying 
widths of rail base where the slotting will permit such punching 
without the holes interfering with each other and when the plate 
is of such design that the additional holes will not impair the 
strength of the plate." 

Tie-plates are variously made of steel, wrought iron and malle- 
able iron. Tie-plates are peculiarly subject to rust, especially 



§287. 



RAIL FASTENINGS. 



323 



as an effect of brine drippings from refrigerator cars. The 
comparative immunity from rust of malleable iron explains its 
use for this purpose. The specifications for steel and wrought 
iron are similar to other physical tests for such a metal when 
toughness rather than high ultimate strength is desired. The 
malleable iron tie-plates have lugs cast on them for testing pur- 
poses. When this lug is broken off, it must not break easily, as 
cast iron, but must show toughness. The fracture must show a 
narrow band of white metal on the surface, the center portion 




WolLaupter 




Round grooved-tapered-flaft \ 
bottom-shoulder tie plato J 




P.R.R. flat bottom tie plate (1916) 



Claw and shoulder tie plate 

Fig. 126. — Various Forms of Tie-plates. 



being dark and fiberless. The plates must, when tested, bend 
sufficiently to prove thorough annealing. 

The holes in a tie-plate should be about xe" larger than the 
size of the intended spike. The length of the plate, perpendicular 
to the rail, should be 8| to 11 in., the extension on the outside 
of the rail base being f" to IJ" more than that on the inside. 
For very heavy traffic the thickness should be f " to f " ; for lighter 
traffic they may be as thin as \" . Flat-bottom plates should 
be at least J" thick; corrugated plates, being somewhat stiffer, 



324: RAILROAD CONSTRUCTION. | 2S8. 

may be thinner for the same Service. The tie-plates dveir the Joint, 
ties must be somewhat longer than thfe intermediates, iti atdi 
to allow for the extra length from out to out of the aiigle-plates.| 

288. Method of setting. A very important detail in the 
process of setting the tie-plates on the ties is that the plates 
should be rigidly attached to the ties in their intended position 
during the process of setting. If tie-plates with flat bottoms 
are used, the surface of the tie tiiust be adzed, so that it is not 
bnly plane but level, so that therei will be no danger that the 
plate will rock on the tie. When using tie-plates which are 
corrugated on the under surface, it is necessary to force them 
into the tie until the under side of the plate is flush with the 
surface of the tie. This requires a pressure of several thousand 
pounds. Sometimes trackmen have depended on the easy 
process of waiting for passing trains to force the corrugations 
into the tie ttiitil the plate is in its intended position. Until 
the plates are finally set the spikes cannot be driven home, 
and this apparently cheap and easy process generally results 
in loose spikes and rails. The best method for new work is 
to drive the plates into the tie before setting the tie in position. 
A tie-plate gauge holds both tie-plates in their proper relative 
position, and both plates may be driven by the hse of hfeavy 
beetles. When it is necessary to place the plate under the rail 
and drive it in, it is somewhat difficult to drive it by striking 
the plate with a swage on each side of the rail alternately. 
When it is struck on one side, the other side flies up unless held 
down by a wedge driven betweeii the plate and the rail on the 
other side of the rail. A straddler, which straddles the rail 
somewhat like an inverted U, is very useful for this purpose, 
since it makes it possible to stt-ike the head of the straddler and 
force down both sides of the plate at once. The Southern 
Pacific Railroad Company has rigged up a small pile-driver on 
a hand-car, which is used in connection with a straddler to drive 
the tie-plates into position. Some western railroads have even 
adopted the process of rigging up a flat car with a machine 
which will press the tie-plates into place in the ties before the 
ties are placed in the track. 

SPrKES. 

289. Requirements. The rails must be held to the ties by a 
fastening which will not only give sufficient resistance, but which 



§289. 



RAIL-FASTENINGS. 



325 



will retain its capacity for resistance. It must also be cheap 
and easily applied. The ordinary' track-spike fulfills the last 
requirements, but has comparatively small resisting power, com- 
pared with screws or bolts. Worse than all, the tendency to 




I 



re" 



J 

^ 



jy 





Fig. 127. 



Fig. 128. 



vertical vibration in the rail produces a series of upward pulls on 
the spike that soon loosens it. When motion has once begun 
the capacity for resistance is greatly reduced, and but little more 
vibration is required to pull the spike out so much that redriving- 
is necessary. Priving the spike to place again in the same hole 
is of small value except as a very temporary expedient, as its 
holding power is then very small. Redriving the spikes in new 
holes very soon "spike-kills" the tie. Many plans have been 
devised to increase the holding power of spikes, such as making 
them jagged, twisting the spike, swelling the spike at about the 
center of its length, etc. But it has been easily demonstrated 
that the fibers of the wood are generally so crushed and torn by 
driving such spikes thjat their holding power is less than that of 
the plain spike, and the durability is greatly diminished. 

The ordinary spike (see Fig. 127) is made with a square crossr 
section which is uniform through the middle of its length, the 
lower 1| in. tapering down to a chisel edge, the upper part swelling 
out to the head. The Goldie spike (see Fig. 128) aims to improve 
this form by reducing to a minimum the destruction of the 



326 RAILROAD CONSTRUCTION. § 290. 

fibers. To this end, the sides are made smooth, the edges are 
clean-cut, and the point, instead of being chisel-shaped, is ground 
down to a pyramidal form. Such fiber-cutting as occurs is thus 
accomplished without much crushing, and the fibers are thus 
pressed away from the spike and slightly downward. Any 
tendency to draw the spike will, therefore, cause the fibers to 
press still harder on the spike and thus increase the resistance. 
A series of tests made by a committee of the A. R. E. A. and 
reported to the 1914 Convention, established some very valuable 
conclusions with respect to the use of the ordinary cut spike. 
Spikes with sharp pyramidal points and with various degrees of 
bluntness, and also the ordinary chisel-pointed spike, were driven 
into ties and other timbers and were withdrawn by a testing 
machine. Then the timbers were cut so as to expose the holes 
to "their full length, so that the crushing of the fibers by the spike 
driving could be observed. A series of photographs illustrated 
this feature. In some cases the spikes were driven into f-in. 
bored holes, some of which were 2| ins. deep, but the most of them 
were 4 ins. deep. In other cases, the spikes were driven without 
previous boring. The following conclusions were unmistakable. 

1. The spike with a pyramidal point about 1 in. long (vir- 
tually the " Goldie " design Fig. 128), has greater holding power, 
not only when it first begins to yield, but also afterward while the 
spike is being drawn out. 

2. The long-pointed spilces crushed the fiber far less than any 
other type. 

3. The chisel-pointed spike, virtually as shown in Fig. 127, and 
which is the type now in most common use, has the least holding 
power and is more destructive in crushing the fibers. 

4. Spikes driven into f-in. bored holes have greater holding 
power than when driven without boring, and the crushing of the 
fiber is much less. This indicates the very real economy in bor- 
ing holes where the life of the tie is an economical consideration. 

290. Driving. The holding power of a spike depends largely 
on how it is driven. If the blows are eccentric and irregular 
in direction, the hole will be somewhat enlarged and the hold- 
ing power largely decreased. The spikes on each side of the 
rail in any one tie should not be directly opposite, but should 
be staggered. Placing them directly opposite will tend to split 
the tie, or at least decrease the holding power of the spikes. 
The direction of staggering should be reversed in the two pairs 



§ 291. 



RAIL-FASTENINGS. 



327 



of spikes in any one tie (see Fig. 129) 
vent any twisting of the tie in 
the ballast, which would other- 
wise loosen the rail from the tie. 
291. Screw spikes. The D., 
L. & W. R. R. began the general 
use of screw spikes for all new 
work and for extensive track 
renewals in 1910. In five years 
they used over 12,000,000 screw 
spikes. The design is shown in 
Fig. 130. From a report made 



This will tend to pre* 




Fia. 129. — Spike-deiving. 



Fig. 130. — Screw Spike, D. L. 
&. W. R. R. 



by Mr. G. .). Ray, Chief Engineer, to the A. R. E. A., the follow- 
ing facts and conclusions are deduced: 

1. The use of screw spikes, in conjunction with suitable tie- 
plates, is almost a necessity in order to fully utilize the durability 
of a treated tie. A treated tie is seldom removed on account of 
decay in the body of the tie. Its destruction is generally due to 
" spike-killing," rail cutting, or to the decay which comes im- 
mediately after mechanical injury to the wood under the rail. 
Screw spikes and tie-plates largely prevent this mechanical injury. 

2. " As a rule, with woods which it will pay to treat, the poorer 
the quality of the timber the more elaborate and expensive the 
fastening must be if the mechanical life of the tie is made to 
approach the life of the treated timber." 

3. " Tie-plates should be used on all ties where screw spikes 
are used." 



32S RAILROAD CONSTRUCTION. § 29L 

4. " Four holes should be provided for screw spikes, so that 
two extra holes will be available if needed." 

5. " The size of screw spikes and the design of the thread should 
be carefully considered before a screw spike is adopted. There- 
after no changes should be made; otherwise the new screw spikes 
cannot be used in old holes without damaging the wood fiber." 

6. " The screw-spike head should have tapering sides to pre- 
vent turning in the wrench socket after the size of the head has 
been diminished by rust." 

7. " When screw spikes are fully seated, no further strain 
should be pUt on them, as this will tend to destroy the threads 
in the wood or injure the spikes." 

8. "All ties should be bored at the treating plant before treat- 
ment. This can be done while the ties are being adzed, and not 
only insures that the holes are bored sufficiently deep, but provides 
for good treatment of allwood adjacent to the spike holes." 

9. " Where the ties are bored before treatment, the track must 
be to proper gauge before the ties can be placed." 

10. " The holes for screw spikes should be of proper dimensions 
for the class of wood used, with due regard to the size of Screw 
spike used." 

11. "A limited number of holes can be bored with one bit, after 
which its size will diminish so as to make it unfit for a hole of a given 
size." [The paper nowhere makes any statement as to the size of 
the bored hole in comparison with the diameter of the screw. The 
bored hole should have about the same diameter as the diameter of 
the screw at the base of the screw thread, but the hardness of thfe 
wood requires some variation, since, if the hole is too small, it will 
be impossible to turn the screw. The exact diameter must be de- 
termined for each kind of wood and must be strictly maintained;] 

12. " Holes should be bored somewhat deeper thaii the length 
of the screw spike. There is no serious objection to boring the 
holes clear through the ties." 

13. " Not only is the lateral and Vertical resistance of a screw 
spike greater than, that of a cut spike when both are first applied, 
but the lateral and vertical resistance of a loose screw spike is 
considerably greater than the lateral and vertical resistance of a 
loose cut spike." 

14. " When the threads in the tie are entirely destroyed, a 
screw lining (any one of several different varieties) may be used 
with good results." 



§292. 



HAIL-FASTENINGS . 



m 



15. "All ties should be bored and adzed before treatment. This 
■ insures good gauge, a perfect bearing for the tie-plates and good 

treatment under the rail seat and around the screw-spike holes." 

16. " In placing screw spikes, they should be driven by ham- 
mer only sufficient to make the threads take hold. If rigid in- 
structions are not carried out, laborers will continually overdrive 
spikes and thus destroy the wood fibers near the top of the holes." 

17. " The best results with the screw spikes can be expected in 
new construction, and where the number of screw spikes in tie 
renewals predominate over cut spikes." 

18. " The use of screw spikes for the past five years has not 
made it necessary to increase the number of sectionmen per mile 
of track." 

19. " Whether or not it will pay to use screw spikes will depend 
upon the cost of ties, their probable life and the amount of traffic." 

292. "Wooden spikes." Among the regulations for track- 
laying given in § 246, mention was made of wooden " spikes," 
or plugs, which are used to fill up the holes when spikes are 
withdrawn. The value of the policy of filling up these holes is 
unquestionable, since the expense is insignificant compared with 
the loss due to the quick and certain decay of the tio if these 
holes are allowed to fill with water and remain so. But the 
method of making these plugs is variable. On some roads they 
are "hand-made" by the trackmen out of otherwise use- 
less scraps of lumber, the work being done at odd mo- 
ments. This policy, while apparently cheap, is not 
necessarily so, for the hand-made plugs are irregular 
in size and therefore more or less inefficient. It is 
also quite probable that if the trackmen are required to 
make their own plugs, they would spend time on these 
very cheap articles which could be more profitably em- 
ployed otherwise. Since the holes made by the spikes 
are larger at the top than they are near the bottom, the 
plugs should not be of uniform cross-section but should 
be slightly wedge-shaped. The. "Goldie tie-plug" 
(see Fig. 131) has been designed to fill these require- 
ments. Being machine-made, they are uniform in 
size; they are of a shape which will best fit the hole; 
they can be furnished of any desired wood, and at a 
cost which makes • it a wasteful economy to attempt ^^q 131 
to cut them by hand. 




330 RAILROAD CONSTRtJCTlON. ' § 293. 



TRACK-BOLTS. 

293. Essential requirements. The track-bolts must have 
sufficient strength and must be screwed up tight enough to hold 
the angle-plates against the rail with sufficient force to develop 
the full transverse strength of the angle-bars. On the other 
hand the bolts should not be screwed so tight that slipping may 
not take place when the rail expands or contracts Avith tempera- 
ture. It would be impossible to screw the bolts tight enough to 
prevent slipping during the contraction due to a considerable fall 
of temperature on a straight track, but when the track is curved, 
or when expansion takes place, it is conceivable that the resist- 
ance of the ties i:i tho ballast to lateral motion may be less than 
the resistance at the joint. A test to determine this resistance 
was made by Mr. A. Torrey, chief engineer of the Mich. Cent. 
R. R., using SO-lb. rails and ordinary angle-bars, the bolts being 
screwed up as usual. If required a force of about 31000 to 
35000 lbs. to start the joint, which would be equivalent to the 
stress induced by a change of temperature of about 22°. But 
if the central angle of any given curve is small, a comparatively 
small lateral component will be sufficient to resist a compression 
of even 35000 lbs. in the rails. Therefore there will ordinarily 
be no trouble about having the joints screwed too tight. The 
vibration caused by the passage of a train reduces the resistance 
to slipping. This vibration also facilitates an objectionable 
feature, viz., loosening of the nuts of the track-bolts. The bolt 
is readily prevented from turning by giving it a form which is 
not circular immediately under . the head and making corre- 
sponding holes in the angle-plate. See Fig. 132. Note also 
the elongated and the round bolt holes in the standard angle 
bar shown on Plate VII. Half the nuts are thus on either side 
of the rail and the danger that all the bolts of a joint might be 
simultaneously sheared off by a derailment is somewhat mini- 
mized. 

" As a rule, as large track-bolts should be used as the rail and 
splice-bars will permit." [From 1915 Manual, A. R. E. A.] 
There is always some danger that a trackman may stretch a bolt 
beyond its elastic limit. A pull of 100 lbs. on a 33-inch track 
wrench will induce a stress of about 45000 lbs. per square inch 
in a |-inch track bolt. The same work on a 1-inch bolt would 
produce a stress of about 35000 lbs. per square inch. In order to 



§ 294. 



EAIL-F ASTENINGS . 



331 



obtain the necessary toughness, bolts must be made of low-carbon 
steel or of nickel-steel, untreated or heat-treated. When made 
of carbon steel, specifications require an elastic limit of at least 
35,000 lbs. per square inch but at the same time an elongation of 
25% in 2 inches and a reduction of area of at least 50%. A 
harder steel would have a higher elastic limit, but would not 
be sufficienlty ductile. Higher elastic limits, with sufficient 
ductility, may be obtained by using untreated nickel or other 
alloy steel (at least 45,000 lbs. per square inch), or heat- 
treated nickel or other alloy steel (at least 75,000 lbs. per square 
inch). The elastic limit shall not be less than 50% of the ulti- 
mate. Added strength can only be obtained by using larger 
bolts or a more expensive metal. 

294. Design of track-bolts. In Fig. 132 is shown a common 
design of track-bolt. In its general form this represents the 
bolt used on nearly all roads, 
being used not only with the 
common angle-plates, but also 
with many of the improved de- 
signs of • rail-joints. The varia- 
tions are chiefly a general in- 
crease in size to correspond with 
the increased weight of rails, 
besides variations in detail di- 
mensions which are frequently 
unimportant. The diameter is 
usually I" to |"; 1" bolts are 
used for 100-lb. rails. As to 
length, the bolt should not ex- 
tend more than |" outside of 
the nut when it is screwed up. 
If it extends farther than this it is Hable to be broken off by a 
possible derailment at that point. The lengths used vary from 
3 j", which may be used with 60-lb. rails, to 5", which is required 
with 100-lb. rails. The length required depends somewhat on 
the type of nut-lock used. 




Fig. 132. — Teack-bolt. 



NUT-LOCKS. 



295. Design of nut-locks. The designs for nut-locks may be 
divided into three classes: (a) those depending entirely on an 



332 



BAILEOAD CONSTRUCTION. 



§ 295. 





VEfiONA 



VULQANIZEQ FIBRE. 




IMPROVED VERONA 




NATIONAL 




Columbia.Nut Lock 







JONES 



Fig. 133. — ^Ttpes of NuT-iiOCKS, 



§ 295. RAIL-FASTENINGS. 333 

elastic washer which absorbs the vibration which might other- 
wise induce turning; (6) those which jam the threads of the 
bolt and nut so that, when screwed up, the frictional resistance 
is too great to be overcome by vibration; (c) the ''positive" 
nut-locks — those which mechanically hold the nut from turning. 
Some of the designs combine these principles to some extent. 
The "vulcanized fiber" nut-lock is an example of the first class. 
It consists essentially of a rubber washer which is protected by 
an iron ring. When first placed this lock is effective, but the 
rubber soon hardens and loses its elasticity and it is then ineffec-' 
tive and worthless. Another illustration of class (a) is the use 
of wooden blocks, generally 1" to 2'' oak, which ^extend the 
entire length of the angle-bar, a single piece forming the washer 
for the four or six bolts of a Joint. This form is cheap, but the 
wood soon shrinks, loses its elasticity, or decays so that it soon 
becomes worthless, and it requires constant adjustment to keep 
it in even tolerable condition. The "Verona" nut-lock is 
another illustration of class (a) which also combines some of the 
positive elements of class (c). It is made of tempered steel and, 
as shown in Fig. 133, is warped and has sharp edges or points. 
The warped form furnishes the element of elastic pressure when 
the nut is screwed up. The steel being harder than the iron of 
the angle-bar or of the nut, it bites into them, owing to the 
great pressure that must exist when the washer is squeezed 
nearly fiat, and thus prevents any backward movement, although 
forward movement (or tightening the bolt) is not interfered 
with. The " National" nut-lock is a type of the second class (b), 
in which, like the " Harvey" nut-lock, the nut and lock are com- 
bined in one piece. With six-bolt angle-bars and 30-foot rails, 
this means a saving of 2112 pieces on each mile of single track. 
The "National" nuts are open on one side. The hole is drilled 
and the thread is cut slightly smaller than the bolt, so that when 
the nut is screwed up it is forced slightly open and therefore 
presses on the threads of the bolt with such force that vibration 
cannot jar it loose. Unlike the " National" nut, the " Harvev " 
nut is solid, but the form of the thread is progressively varied so 
that the thread pinches the thread of the bok and the frictional 
resistance to turning is too great to be affected by vibration. 

The "Columbia" nut-lock is a two-piece nut, both parts of 
which must turn simultaneously. As shown in the figure, one 



334 RAILROAD CONSTRUCTION. § 295. 

section wedges into the other. The greater the tension in the 
bolt, the greater the wedging action and the greater the friction 
to prevent turning. 

The "Jones" nut-lock, belonging to class (c), is a type of a 
nut-lock that does not depend on elasticity or jamming of screw- 
threads. It is made of a thin flexible plate, the square part of 
which is so large that it will not turn after being placed on the 
bolt. After the nut is screwed up, the thin plate is bent over so 
that the re-entrant angle of the plate engages the corner of the 
nut and thus mechanically prevents any turning. The metal 
is supposed to be sufficiently tough to endure without fracture 
as many bendiiigs of the plate as will ever be desired. Nut- 
locks of class (c) are not in common use. 

The above types have been discussed in order to show the 
development of the various devices. With but few exceptions, 
the standard nut-lock is a steel spring ring of the same general 
class as the Verona. The A. R. E. A. have prepared specifica- 
tions for such nut-locks which include the following: 

" After the finished nut-lock has been subjected for one hour to 
pressure sufficient to compress it flat and has been released, its 
reaction shall be not less than two-thirds its height or thickness 
of section, provided thickness is less than width of section. If 
the section is square, the reaction must be not less than one-half 
its thickness. If height or thickness of section is more than 
width, the reaction shall be not less than the width of the section. 
The internal diameters naturally affect the percentage of reac- 
tion, and the above specifications apply to nut-locks of internal 
diameters from j% in. to 1 ^ ins. Owing to the difficulty of 
establishing a common rate of percentage that shall be uniformly 
applicable to any internal diameter of any nut-lock of any section 
it has been sought to cover the matter as above. Amount and 
durability of reactionary power under constant pressure is the 
true test of any spiral spring nut-lock. The percentage of reac- 
tion increases proportionately with the increased internal diam- 
eter of any given section." 

" With one end of the finished nut-lock secured in a vise, and 
the opposite end twisted to 45 degrees, there must be no sign of 
fracture. When further twisted until broken, the fracture must 
show a good quality of steel." 



CHAPTER XI. 

SWITCHES AND CROSSINGS. 
SWITCH CONSTRUCTION. 

296. Essential elements of a switch. Flanges of some sort are 
a necessity to prevent car-wheels from running off from the rails 
on which they may be moving. But the flanges, although a 
necessity, are also a source of complication in that they require 
some special mechanism which will, when desired, guide the 
wheels out from the controlling influence of the main-line rails. 
This must either be done by raising the wheels high enough 
so that the flanges may pass over the rails, or by breaking the 
continuity of the rails in such a waj'- that channels or "flange 
spaces" are formed through the rafls. An ordinary stub-switch 
breaks the continuity of the main-line rails in three places, two 
of them at the switch-block and one at the frog. The Wharton 
switch avoids two of these breaks by so placing inclined planes 
that the wheels, rolling on their flanges, will surmount these 
inclines until they are a little higher than the rails. Then the 
wheels on the side toward which the switch runs are guided 
over and across the main rail on that side This rise being ac- 
complished in a short distance, it becomes impracticable to 
operate these switches except at slow speeds, as any sudden 
change in the path of the center of gravity of a car causes very 
destructive jars both to the switch and to the roUing stock. The 
other general method makes a break in one main rail (or both) 
at the switch-block. In both methods the wheels are led to one 
side by means of the "lead rails," and finaUy one line of wheels 
passes through the main rail on that side by means of a "frog." 
There are some designs by which even this break in the main 
rail is avoided, the wheels being led over the main rail by means 
of a short movable rail which is on occasion placed across the 
main rail, but such designs have not come into general use. 

297. Frogs. Frogs are provided with two channel-ways or 
"flange spaces" through which the flanges of the wheels move, 

335 



336 



RAILROAD CONSTRUCTION. 



§297. 



Each channel cuts out a parallelogram from the tread area. 
Since the wheel-tread is always wider than the rail, the wing 
rails will support the wheel not only across the space cut out by 
the channel, but also until the tread has passed the point of the 
frog and can obtain a broad area of contact on the tongue of the 
frog. This is the theoretical idea, but it is very imperfectly 




Fig. 134. — Diagrammatic' Design of Frog. 



realized. The wing rails are sometimes subjected to excessive 
wear owing to "hollow treads" on the wheels — owing also to 
the frog being so flexible that the point " ducks" when the wheel 
approaches it. On the other hand the sharp point of the frog 
will sometimes cause destructive wear on the tread of the wheel. 
Therefore the tongue of the frog is not carried out to the sharp 
theoretical point, but is purposely somewhat blunted. But 
the break which these channels make in the continuity of the 
tread area becomes extremely objectionable at high speeds, 
being mutually destructive to the rolling stock and to the frog. 
The jarring has been materially reduced by the device of " spring 
frogs" — to be described later. Frogs were originally made of 
cast iron — then of cast iron with wearing parts of cast steel, 
which were fitted into suitable notches in the cast iron. This 
form proved extremely heavy and devoid of that elasticity of 
track which is necessary for the safety of rolling stock and 
track at high speeds. The present standard practice is to build 
the frog up of pieces of rails which are cut or bent as required. 
There are always four pieces for single-pointed frogs. Usually 
they are assembled by bolts running through the rail webs, 
which are properly separated by rolled steel filler blocks. Some- 
times they are enclosed by clamps held in place by wedges. 
Sometimes the rails are bolted or riveted to a base plate. For 
the hardest service, the wearing parts are made of manganese 



ii:. 






h++--t 



i|:!) 



o 
o 

« 

i 

o 

PQ 



4fl^ 



*>*■! 



ln^•^^ 



!f^r 



r- 



\ 




{To face page 336.) 

Plate VIII.— Some Types of Fbogs. 
(As made by Ramapo Iron Works.) 



§ 298. SWITCHES AND CROSSINGS. 337 

steel. For details, study Plate VIII. The operation of a 
spring-rail frog is evident from the figure. Since a siding is 
usually operated at slow speed, while the main track may be 
operated at fast speed, a spring-rail frog will be so set that 
the tread is continuous for the main track and broken for the 
siding. This also means that the spring-rail will only be 
moved by trains moving at a (presumably) slow speed on to 
the siding. For the fast trains on the main line such a frog is 
substantially a " fixed " frog and has a tread which is practically 
continuous. 

298. To find the frog number. The frog number (n) equals 
the ratio of the distance of any point on the tongue of the frog 
from the theoretical point of the frog divided by the width of 
!the tongue at that point, i.e. =hc-^ah (Fig. 134). This value 
may be directly measured by applying any convenient unit of 
measure (even a knife, a short pencil, etc.) to some point of the 
tongue where the width just equals the unit of measure, and then 
noting how many times the unit of measure is contained in the 
distance from that place to the theoretical point. But since c, 
the theoretical point, is not so readily determinable with exacti- 

jtude, it being the im.aginary intersection of the gauge lines, it 
may be more accurate to measure de, ah, and hs; then n, the frog 
i number, = hs -^ (ab + de) . If the frog angle be called F^ then 

n=hc-^ab=hs-i-(ah+de)^^ cot ^F; 
i.e., cot ^F=2n. 

299. Stub switches. The use of these, although once nearly 
universal, has been practically abandoned as turnouts from 
main track except for the poorest and cheapest roads. In some 
States their use on main track is prohibited by law. They have 
the sole merit of cheapness with adaptability to the circum- 
stances of very light traffic operated at slow speed when a con- 
siderable element of danger may be tolerated for the sake of 
economy. The rails from A to 5 (see Fig. 135*) are not fastened 

* The student should at once appreciate that in Fig. 135, as well as in 
nearly all the remaining figures in this chapter, it becomes necessary to 
use excessively large frog angles, short radii, and a very wide gauge in 
order to illustrate the desired principles with figures which are sufficiently 
small for the page. In fact, the proportions used in the figures are such 
that serious mechanical difficulties would be encountered if they were 
used. These difficulties are here ignored because they can be neglected 
in the proportions used ia practice, 



338 



RAILEOAD CONSTRUCTION. 



§299. 



to the ties; they are fastened to each other by tie-rods which 
keep them at the proper gauge; at and back of B they are 
securely spiked to the ties, and at A they are kept in place by 




Fia. 135. - Stub Switch. 



the connecting bar (C) fastened to the switch-stand. One great 
objection to the switch is that, in its usual form, when operated 
as a trailing switch, a derailment is inevitable if the switch is 
misplaced. The very least damage resulting from such a derail- 
ment must include the bending or breaking of the tie-rods of the 
switch-rail. Several devices have been invented to obviate this 
objection, some of which succeed very well mechanically, al- 
though their added cost precludes any economy in the total cost 
of the switch. Another objection to the switch is the looseness 
of construction which makes the switches objectionable at high 
speeds. The gap of the rails at the head-block is always con- 
siderable, and is sometimes as much as two inches. A driving- 




FiQ, 136. — Point Switch. 



wheel with a load of 20000 to 30000 pounds, jumping this gap 
with any considerable velocity, will do immense damage to the 



§300. 



SWITCHES AND CROSSINGS. 



339 



farther rail end, besides producing such a stress in the construc- 
tion that a breakage is rendered quite Hkely, and such a breakage 
might have very serious consequences. 

300. Point switches. The essential principle of a point switch 
is illustrated in Fig. 136. As is shown, one main rail and also 
one of the switch-rails is unbroken and immovable. The other 
main rail (from A to F) and the corresponding portion of the 
other lead rail are substantially the same as in a stub switch. 
A portion of the main rail {AB) and an equal length of the oppo- 
site lead rail (usually 16.5 to 22 feet long) are fastened together 
by tie-rods. The end at A is jointed as usual and the other end 
is pointed, both sides being trimmed down so that the feather 
edge at B includes the web of the rail. In order to retain in it 

as much strength as possible, the point- 
rail is raised so that it rests on the base 
of the stock-rail, one side of the base of 
the point-rail being nearly cut away. 
As may be seen in Fig. 137, although 
the influence of the point of the rail in 
moving the wheel-flange away from the 
stock-rail is really zero at that point, 
yet the rail has all the strength of the 
web, more than one-haK that of the 
base, and is also reinforced. The planing 
runs back in straight lines, until at about 
six or seven feet back from the point 
the full width of the head is obtained. The full width of 
the base will only be obtained at about 13 feet from the 
point. The A. R. E. A. standard switch rail is always cut on 
the basis that the distance between gauge lines at the heel of 
the switch (the distance MN in Fig. 143) is 6^ inches and that 
the " point " is j inch wide. Then, using four standard 
lengths, 11, 16i 22 and 30 feet, the angles vary from 2° 36' 19'' 
to 0° 57' 18" as shown in Table III. 




Fig. 137. 



301. Switch-stands. The simplest and cheapest form is the 
" ground lever," which has no target. The radius of the circle 
described by the connecting-rod pin is precisely one-half the 
throw. From the nature of the motion the device is practically 



340 



EAILROAD CONSTRUCTION. 



§301. 



self-locking ift either position, padlocks being only used to pre- 
vent malicious tampering. 




Fig. 138. — Ground Lever for Throwing a Switch, 



— I 
— I 



CTEEL shaft! ;3\^'*« 







Fig. 139. — Ramapo Patent Switch Stand. Non-automatic. 



"I 



§302. 



SWITCHES AND CROSSINGS. 



841 



In Fig. 139 is5 shown a design in which the arc of the throwing 
lever is parallel to the track, an important feature in quick 
switching Avork. 

302. Tie-rods. These are fastened to the webs of the rails by 
means of lugs which are bolted on, there being usually a hinge- 




A 



Fig. 140. — Forms of Tie-rods. 

joint between the rod and the lug. Two such tie-rods (three for 
a 30-foot switch) are generally necessary. The first rod is some- 
times made without hinges, which gives additional stiffness to 
the comparatively weak rail-points. The old-fashioned tie-rod, 
having jaws fitting the base of the rail, was almost universally 
used in the days of stub switches. One great inconvenience in 
their use lies in the fact that they must be slipped on, one by one, 
over the free ends of the switch-rails. 




j I ^GlUARD RAIL 

Fig. 141. — Standard Guard-rail. 



303. Guard-rails. As shown in Figs. 135 and 136, guard-rails 
are used on both the main and switch tracks opposite the frog- 
point.' Their function is not onlj'' to prevent the possibility of 
the wheel-flanges passing on the wrong side of the frog-point, 
but also to save the side of the frog-tongue from excessive wear. 
The flange-way space between the heads of the guard-rail and 
wheel-rail should equal If inches. Since this is less than the 
space between the heads of ordinary (say 80-pound) rails when 



342 



RAILROAD CONSTRUCTION. 



§304. 



placed base to base, to say nothing of the f " required for spikes, 
it becomes necessary to cut the flange of the guard-rail. The 
length of the rail should be 16 feet 6 inches, the middle portion 
being straight for a length of 3 feet 6 inches, and the ends, each 
being 6 feet 6 inches long, curved out so that the side of the 
rail head at each end is 4 inches from the main rail head, when 
the flange-way at the center is If inches. See Fig. 141. 



MATHEMATICAL DESIGN OF SWITCHES. 

In all of the following demonstrations regarding switches, 
turnouts, and crossovers, the lines are assumed to represent the 
gauge-lines — ^i.e., the lines of the inside of the head of the rails. 

304. Design with circular lead-rails. The simplest method 
is to consider that the lead-rails curve out from the main track- 




Fig. 142. 

rails by arcs of circles which are tangent to the main rails and 
which extend to the frog-point F. The simple curve from D to F 
is of such radius that {r-\-^g) vers F = g, in which F = ihe frog 
angle fir = gauge, L = the " lead " (BF), and r=the radius of the 
center of the switch-rails. 



.*. r+|fir = - 



g 



4-lso, 



vers F 

BF^BD = QO%W, BD = g; BF=L. 
.*. L=g cot F 



(69) 






(70) 



§ 304. SWITCHES AND CROSSINGS. 343 

Also, L = (r+i^)sinF; (71) 

QT = 2rsmhF (72) 

These formulae involve the angle F. As shown in Table III, 
the angles (F) are always odd quantities, and their trigonometric 
functions are somewhat troublesome to obtain closely with 
ordinary tables. The formulae may be simplified by substitut- 
ing the frog-number n, from the relation that n = | cot ^F. 
Since 

r—lg=L cot F and r-|-|gr=L cosec i^, 

then r = ^L (cot F+cosec F) 

= ^g cot |F(cot F+cosec F) 

= |gf cot^ ^F, since (cot a+cosec a) =cot fa 

=2gn^ . (73) 

Also, L=2gn, (74) 

from which r=nXL (75) 

These extremely simple relations may obviate altogether the 
necessity for tables, since they involve only the frog-number and 
the gauge. On account of the great simplicity of these rules, 
they are frequently used as they are, regardless of the fact that 
the curve is never a uniform simple curve from switch- block to 
frog. In the first place there is a considerable length of the 
gauge-line within the frog, which is straight unless it is pur- 
posely curved to the proper curve while being manufactured, 
which is seldom if ever done — except for the very large-angled 
frogs used for street-railway work, etc. It is also doubtful whether 
the switch-rails (BA, Fig. 135) are bent to the computed curve 
when the rails are set for the switch. The switch-rails of point 
switches aro straight, thus introducing a stretch of straight track 
which is about one-fifth of the total length of the lead-rails. The 
effect of these modifications on the length and radius of the lead- 
rails will be developed and discussed in the following sections. 

The throw (0 of a stub switch depends on the weight of the 
rail, or rather on the width of its base. The throw must be at 



344 RAILROAD CONSTRUCTION. § 305. 

least f" more than that width. The head-block should there- 
fore be placed at such a distance from the heel of the switch (B) 
that the versed sine of the arc equals the throw. These points 
must be opposite on the two rails, but the points on the two rails 
where these relations are exactly true will not be opposite. 
Therefore, instead of considering either of the two radii {r + ^g} 
and (r—^g), the mean radius r is used. Then (see Fig. 142) 

vers KOQ=t-^r, 

and the length of the switch-rails is 

QK=r sin KOQ (76) 

Stub-switches are generally used with large frog angles. For 
small frog angles (large frog-numbers) the values of QK are so 
great that the length of rail left unspiked is too great for a safe 
track. If this were obviated by spiking down a portion of the 
lead the theoretical accuracy of the switch would be lost. 

The use of stub switches may now be considered obsolete. 
But the above. demonstration has been retained in this edition 
for its educational value as an introduction to the more com- 
plicated method which is now the standard. J.'/ --u- 

305. Standard design, using straight frog-rails and straight 
point-rails. It becomes necessary in this case to find a curve 
which shall be tangent to both the point-rail and the frog-rail. 
The curve therefore begins at M, its tangent making an angle of 
a ( varying from 0° 52' to 2° 36') with the main rail, and runs to 
/. FJ = W= the length of the " wing-rail " from the theoret- 
ical point of the frog (F) to the toe, J or J'. FK^K — the 
length from the theoretical point to the heel of the frog.. MN 
= H = the " heel distance," or the distance of the gauge line of 
the switch-rail at the heel from the gauge line of the main track 
rail. 

The central angle of the curve equals (F — a). The. angle of 
the chord JM with the main rails is therefore 

iiF~a)-\-a = h{F+a)] 

g-W sin F-H 
"^ sin UF-\-a) ' 



§305. SWITCHES AND CROSSINGS. 345 

JM 



r-\-hg 



Xr,—. 



2sin|(i^-a) 

g-W dn F-H 
'2 sin K^+a) sin W-d) 
g-W sin F-H 



(77) 



cos a — COS F 

pN = s cos a, . (78) 

in which /Sf = length of switch-rail. 

BF=L = JM cos K/^+«)+TF cos F+S cos a 

= ig-W sin F-H) cot liF+a)+W cos F+ S cosa. . (79) 

It may be more simple, if (r+^g) has already been computed, 
to write 

L = 2(r-{-^g}sinUF-a)Gos^iF+a)-\-WcosF'\-Scosa 
«= (^ + ig) (sin F-sina) + W cosy+S cos a. . . . ■. (80) 

The above equations for L give the distance from the actual 
(blunt) point of the switch-rail to the theoretical point of the frog. 
The lead (L') given in Table III is the distance from the actual 
point of the switch-rail to the actual (blunt) point of the frog. 
The difference {U — L) is the " frog bluntness," which in each 
case equals the width of the frog point (| inch = .04166 foot) mul- 
tiplied by the frog number. The values of the frog bluntness 
for the various frogs is given in the second column of Part B, 
Table III. 

The value of MN = H has been standardized by the A. R. E. A. 
as 6i inches for all lengths of switch-rail and for all values of a. 
The point of the switch-rail (at D) is invariably |-inch thick. 
When it is necessary to calculate MN for other standards of 
construction, it may be computed (calling aS = length of switch- 
rail) to be 

MN=S sin a + (thickness of point of switch rail). 



346 



RAILROAD CONSTRUCTION. 



§305. 



The length to the blunt point of the frog {W = FJ) is given for 
each frog in the third column of Table III, Part B. The several 
values of F and a are also given in Table III. g is the gauge 
= 4 feet 8i inches = 4.7083 feet. 

The solution of Eq. 77-80 for various frog angles will give a 
series of " theoretical leads," as given in Table III. Part B. 
The "closure rails," between the switch points and the frog, 
will invariably have such odd total lengths that there must be 
at least one rail cutting (and some wastage of rail) for each 



,^1^ 



*^- 



F-a 

A— 



A- theoretical point of switch rail. 

j.:r...::::. 




Fia. 143. 



closure length. By shortening the radius of the connecting 
curve very slightly and inserting a very short length of tangent 
either between the curve and switch-rail at M, or between the 
curve and frog-rail at /, all of which will change very slightly 
the length of lead, the closure lengths can be made such that 
one rail cutting can be eliminated, and yet the combinations 
of curves and tangents are mathematically perfect. The detailed 
method of computing these combinations is tedious and will 
not be elaborated here, but a series of results developed by 
the A. R. E. A. is given under the heading of " practical leads " 
in Table III. Part C. 



§ 306. SWITCHES AND CROSSINGS. 347 

The above computations and tabular values assume that the 
two switch points (at B and D) are directly opposite. This 
would always mean that the straight rail {BF) is somewhat 
shorter than the curved rail from D to F. In the maximum case 
the difference is less than 4 inches. Therefore, assuming that 
rails are obtainable at even-foot lengths down to 27 feet, or 
24 feet for a No. 4 frog switch, the system of practical leads never 
requires more than one rail cutting. But even this is some- 
times avoided by using for the straight-rail closure the same 
number and lengths of uncut rails as are specified for the closure 
of the curved part. The chief effect of this is that the point of 
the switch-rail will be located a few inches below its normal posi- 
tion at B and that the gauge at the switch-point will be slightly 
widened when the switch is open. This effect is possibly an 
advantage rather than a disadvantage. 

306. Design for a turnout from the OUTER side of a curved 
track. Fig, 144 is a diagram of what the construction would be 




if the switch-rails were circular throughout. Before the inven- 
tion of point switches and when stub switches were in universal 
use, the lead-rails were considered to be circular, both for straight 
and for curved main track. If Eqs. 70 and 75 and the corre- 
sponding Eqs. 77 to 80 are solved for any given frog, it is found 
that the lead, when using straight switch-rails and straight frog- 
rails, is considerably less than when using circular lead-rails 
throughout; also the curvature is considerably sharper. But 
stub-rail switches are obsolete and the mathematical solutions 
used for them cannot be utilized, even approximately, for point 
switches. If such a diagram as Fig. 144 is Worked out in detail, 
as has been done in previous editions, it is found that 



348 EAILROAD CONSTRUCTION". § 307. 

(a) the lead (BF) is almost identical with that computed from 
Eq. 70 or 74, when the main line is straight. 

(6) the degree of curve {d) of the circular switch-rails would be 
very nearly equal to the degree of curve (d') of the circular 
switch-rails for a straight track minus the degree of curve (D) 
of the main track; or, d = d' — D. 

\ These statements are more exactly true when the degree of 
curvature of the main track is small. Even for a 10° curve on 
the main track the errors are not large. It has been found to be 
a needless refinement to compute the precise mathematical 
properties of the switch-rails from a curved main track, any- 
more than as given by the two principles stated above : There- 
fore 

(a) the length of the lead is assumed to be the same as that for 
a straight track, using the same frog, and 

(6) the degree of curve of the switch-rails is found as stated 
above — in principle (6). As the curvature of the main track 
sharpens, the curvature of the switch-rails becomes less until 
they become straight. For still sharper main track, the center 
of curvature is on the same side. This is illustrated in Fig. 145, 
if we consider the sharper curved track to be the main track 
and the easier curve the switch. The above rule is still appli- 
cable, the algebraic sign of the result showing the location of the 
center. 

307. Design for a turnout from the INNER side of a curved 
track. As in the previous section, Fig. 145 illustrates^ the dia- 




FiG. 145, 



gram for circular lead rails. It may be shown that the degree 
of the turnout {d) is nearly the sum of the degree of the main 



§308. 



SWITCHES AND CROSSINGS. 



349 



track (D) and the degree (d') of a turnout from a straight track 
when the frog angle is the same. The discrepancy in this case is 
somewhat greater than in the other, especially when the curva- 
ture of the main track is sharp. If the frog angle is also large, 
the curvature of the turnout is excessively sharp. If the frog 
angle is very small, the liability to derailment is great. Turn- 
outs to the inside of a curved track should therefore be avoided, 
unless the curvature of the main track is small. j 

308. Connecting curve from a straight track. The "con- 
necting curve " is the track lying between the frog and the side 



^j 



^-"¥" 



\ 



\ 



N 



N 



\ 



f 

1 




f7 




1 
1 

— Q 


— 




"s 


s 




/ 


I 


\ 




/ 


/ 


N 




f 


/ 


s 




/ 


/ 


K' 






/ 


l- 
J/ 


y 


/ 





Fig.. 146. 



track where it becomes parallel to the main track ( KS in Fig. 
146 or 147). Call d the distance between track centers. The 
angle KOiS=F (see Fig. 146). Call r' the radius of the con- 
necting curve. Then 

d—g —Ksinf ^^ _ 



(/-|y)=- 



vers F 



FQ = {r'-\g) sin F-]-K cos/ . 



(81) 



(82) 



In these equations (and in several that follow) K is the distance 
from the theoretical point of the frog to the heel. The length, 
for each standard frog, is found in Table III, Part B. 



350 



RAILROAD CONSTRUCTION. 



§309. 



309. Connecting curve from a curved track to the OUTSIDE. 

When the main track is curved, the required quantities are the 
radius of the connecting curve from K to <S, Fig. 147, and its 
length or central angle. 

The accuracy of all these computations on switches and frogs 
in curved main track is vitiated by the fact that the frog-rails 
are straight. The design might be mathematically more perfect 
if the main track curve were transformed into two curves on 
either side of the frog which had centers separated as far as the 




Fig. 147. 

length of the frog, but this would introduce a very great and 
needless complication and is never done. The more simple solu- 
tion is to consider that the frog-rail is a chord of the original 
curve, which (a) narrows the track gauge by an amount equal to 
the middle ordinate of that chord and which (6) is not tangent 
to the curve at either end. For all ordinary curvature neither 
of these theoretical defects is vitally objectionable or even appre- 
ciable. In Fig. 147 KC is practically perpendiculat to one frog- 
rail and KOi is exactly perpendicular to the other frog-rail. 
Therefore, the angle CKOi equals the frog angle i^.. While the 
following calculations are amply precise for practical purposes, 
the discrepancy from strict mathematical accuracy should be 
noted and properly valued, 
in the triangle CSK 

CS+CK:CS-CK ::tan i(CKS+CSK) :tan i{CKS-CSK); 



but ^{CKS-\-CSK) =^90-^xf^; and, since the triangle OiSK is 
isosceles, ^iCKS-CSK)^iF', 



310. 



SWITCHES AND CKOSSINGS. 



351 



.*. 2R+d-j-K sin F : d-g-K sin F::cot ^^|/ : tan ^F 

y.cot^F : tan^^; 
2n(d-g-K sin F) 



.*. tan 1^ = 



2R+d+K sin F 



(83) 



From the triangle COiK we may derive 

r-^g : i?+i^+ii: sin i^::sin ^ : sin {F-\-ip); 

sin \p 



Also 



r-hg = iR-\-h+K sin F)-. ^^ , . 

sin {F+\l/) 

KS = 2(r-ig)sinUF-\-rP). . . 



(84) 



(85) 



310. Connecting curve from a curved track to the INSIDE. 

As above, it may readily be deduced from the triangle CKS (see 
Fig. 148) that 




Fig. 148. 

CK+CS : CK-CS:: tan UCSK+CKS) : tan UCSK-CKS); 
(2R-d-KsinF) : (d-g-K sin F:: cot ^^ : tan fF; 

2n(d-g-K sin F) 



tan h\I/ = - 

2R-d-KsinF ' ' * 

From triangle COiK, 

OiK : Cii:::sin ^ : sin (F-^p); 
(r-^g) : {R-\g-K sin F) ::sin ^ : sin (F-^); 

sin \p 



(86) 



(^-k) = (^-i9'-XsinF) 



sin {F~\l/)' 



(87) 



352 
Also 



BAILKOAD CONSTRUCTION. 



KS = 2ir-ig)sm^{F-rl^). 



§310. 

. (88) 



c 



:5 




Fig. 149. 



Two other cases are possible, (a) r may increase until it 
becomes infinite (see Fig. 149), then F = \f/, In such a case 
we may write, by substituting in Eq. 86, 



2R-d-K sin F = 4:n^{d-g-K sin F). 



(89) 




Fig. 150. 

This equation shows the value of R which renders this case pos- 
sible. (6) \J/ may be greater than F. As before (see Fig. 150). 

{2R-d-K sin F) : {d-g-K sin F) ::cot ^t^ : tan ^F; 

2n(d-g-K sin F) 



tan ^xp = 
the same as Eq. 86, but 



2R-d-KBmF 



ir-\-hg = {R-^g-K sin f)- 



sin \p 



sin {yp-F) 



(90) 



§311. 



SWITCHES AND CROSSINGS. 



353 



Problem. To find the dimensions of a connecting curve run- 
ning to the INSIDE of a curved main track; number 9 frog, 4° 30' 
curve, d = 13', sr = 4'8r. 



Solution. 
[Eq. 86] d =13.000 
5.816 



K=-10'0"KamF =1.108 
^=4.708 



7.184 



5.816 



i2 =1273.6 

2fl =2547.2 
((f+Z sin F) =14.108 



2B-d-K sin F =2533.1 
log =3.40365 
co-log =6.59635 



Since F>\p, we must use Eq. 87, rather than Eq. 90» 

iff =2.354 i2-fff-ii: sin if =1270.1 
2rsinF = 1.108 (i^-,/') =1855"; log =3.26834 

4.68557 



sum 



=3.462 



log 2n =1.25527 



log 7.184 =0.85636 



co-log =6.59635 



log tan iiA =8.70799 

i^=2' 55*20" 
^=5*50' 40" 
F = 6°21'35" 



7.95391 
co-log =2. 04608 



F-^=0° 30' 55' 



log =3. 10384 
log sin ^=9.00787 



co-log =2.04608 
4.15779 



r =14383.5 
d=0°24' 



[Eq. 88]. 

HF-f) =927.5"; log=^2. 96731 

4.68557 



sin I (F-i^) =7.65289 



2 0. 30103 

r-lff 4.15779 
7.65389 



iTS =129.33 



2.11171 



311. Crossover between fwo parallel straight tracks. (See 
Fig. 151.) The turnouts are as usual. The cross-over track 
may be straight, or it may be a reversed curve. The reversed 
curve shortens the total length of track required, but is somewhat 
objectionable. The first method requires that both frogs 
must be equal. The second method permits unequal frogs, 
although equal frogs are preferable. The length of straight 
crossover track is FiT. 



354 




KAILROAD CONSTRUCTION. § 312. 

FiT sin Fi+g cos Fi = d-g; 

. (91) 



FiT=-r—^-g cot Fi. 



sin Fi 

The total distance along the track may 
be derived as follows: 

DZ = DiFi+D2F2+F2Y 

==DFi+D2F2+XY-XF2; 

XY=={d-g) cot Fi; 
XF2 = g ^sin F^; 



DiZ =2DiFi + {d-g)cotFv 



9 



Fig. 151. 



sin Fi 



(92) 




312. Crossover between two parallel curved tracks. Using 
a straight connecting curve. This solution has limitations. 

If one frog (Fi) is 
chosen, F2 must be 
determined, being a 
function of Fi. If 
Fi is less than some 
limit, depending on 
the width (d) between 
the parallel tracks, 
this solution becomes 
impossible. In Fig. 
152 assume Fi as 
known. Then KiN = g eec Fi. In the triangle NOK% we 
have 

sin NK^O : sin K2NO ::N0 : K2O; 

sin ZaiVO = cos Fi; NK^O^dO^+Fi; 

,'. sin NK20 = cos F2. 

NO-H+hd-hg-KisinFi-gsecFi] KiO^R-^d+^g 

-i-Kz sin Fi] 

R+id— ^g-Ki sin Fi-g sec Fi 



Fig. 152. 



/.COS Fi^Qos Fi- 



R-^d-{-^g-]-K2sinFi 



(93) 



§ 312. SWITCHES AND CROSSINGS. 355 

The solution of this equation involves the frog angle F2, which is 
the angle sought, but there is little error in considering in this 
solution that K2 sin F2 is numerically equal to Ki sin Fi and 
solving accordingly. If the computed value of F2 is very different 
from Fi, it would be more precise to recompute Eq. 93 by sub- 
stituting for Ki sin F2 the more exact quantities obtainable 
from the first trial solution. The relative position of the frogs 
Fi and F2 may be determined as follows : 

ATOaK = 180° - (90° -Fi) - (90° +i?'2) = Fi - F2. 
Then GFi = 2{R+^d-^g) sin UFi-F2)+KiCos Fi. . (94) 

There is a theoretical, but practically inappreciable, inac- 
curacy in Eq. 94, since the chord GFi is really the sum of two 
chords of which one is the chord from the point G to the point 
where ON produced intersects the gauge line. After locating 
G, the point radially opposite, on the outer gauge line of the inner 
track, may be located, from which the frog-point F2 is located 
at a distance of K2 cos F2. Note that these frog-points referred 
to are the theoretical points. Due allowance must be made 
during location for the " frog bluntness." 

In general, the value of F2 computed from Eq. 93 is not the 
angle of any standard number-frog, and a strict compliance with 
theory would require that the frog should be made to order. 
This is needlessly expensive and the nearest size frog may gen- 
erally be used without appreciable error. 

Example. A crossover between parallel tracks on a 6° curve, 
the track spacing d being 13 feet, i^i assumed a No. 9 frog. 



[Eq. 93] 

i? =955.37 
icf= 6.5 

961.87 


iff =2. 35 

iiTi sin Fi =1.11 

g sec Fi=4.74 


i5:i=10ft. 
sin A = . 11077 




- 8.20 
953 67 . 


8.20 


. log =2.97940 




3umed = to Ki sin 


• 




B =955.37 
iff= 2.35 
Kzsia Fi= 1.11 (as 

958 . 83 
-id = -6.5 

952.33 , 


. log =2. 97879 


F2=5°35'30" 




log cos Fi 
log cos 5° 


0.00061 
, . 9 . 99732 

35' 30' 9.99793 



356 EAILEOAD CONSTRUCTION. §313. 

This angle is within 8 minutes of the angle of a No. 10 frog, 
which could be used without appreciable error. The point K2 
would be shifted laterally .023 foot, or about \ inch, but there 
would be no visible irregularity in alinement. 

NOK2 =Fi -Fi =6° 21' 35" -5° 35' 30" =0° 46'. 

[Eq. 94] 22 +id =961.87 

-|g= -2.35 2 . . log =0.30103 

959.52 . log=2. 98205 

sin \NOK'i=siu 0° 23' = 7 . 82545 

12.84 log =1.10853 

i?icosFi= 9.94 



GFi =22.78 

It is instructive to note that if the same crossover problem is 
worked out for a straight track, as in § 311, using No. 9 frogs 
on both tracks, the distance between frog points, measured 
parallel with the track, is nearly the same as in the above prob- 
lem, especially when the distance 12.84, measured on the outer 
track, is reduced by bringing it in to the center line. This is 
analogous to the statement, previously made, that the lead of a 
switch on a curved track is nearly the same as that for a straight 
track. 

It is theoretically possible to find two standard frog angles 
which may be so located that the connecting curve consists of 
straight lines and circular curves, which connect tangentially, 
making perfect alinement, but such methods are very com- 
plicated and the above method is sufficiently exact for practical 
purposes. 

313. Practical rules for switch-laying. A consideration of 
the previous sections will show that the formulae are compara- 
tively simple when the lead-rails are assumed as circular; that 
they become complicated, even for turnouts from a straight 
main track, when the effect of straight frog and point rails is 
allowed for, and that they become hopelessly complicated when 
allowing for this effect on turnouts from a curved main track. 
It is also shown (§ 306) that the length of the lead is practically 
the same whether the main track is straight or is curved with 
such curves as are commonly used, and that the degree of curve 
of the lead-rails from a curved main track may be found with 
close approximation by mere addition or subtraction. From 
this it naay be assumed that if the length of lead (L) and the 



§313. 



SWITCHES AND CROSSINGS. 



357 



radius of the lead-rails (r) are computed from Eq. 77 and 80 for 
various frog angles, the same leads may be used for curved main 
track; also, that the degree of curve of the lead-rails may be 
found by addition or subtraction, as indicated in § 306, and that 
the approximations involved will not be of practical detriment. 
In accordance with this plan Table III has been computed from 
Eq. 77, 78 and 80. The leads there given may be used for all 
main tracks, straight or curved. The table gives the degree of 
curve of the lead-rails for straight main track; for a turnout to the 
inside, add the degree of curve of the main track; for a turnout 
to the outside, subtract it. 

But there are complications resulting from practical and eco- 
nomical switch construction. A committee of the A. R. E. A., 
in 1921, adopted certain standards 
in details, which, when applied to 
Eqs. 77 to 80 give the values for 
switch dimensions as quoted in the 
second section of Table III. They 
adopted four lengths of switch-rails. 
In each case the " point " is always 



K-R 



i" 



thick. The gauge line at the 




Fig. 153. 



other end is always to be placed 
Ql" from the gauge line of the main 
rail, and the planing is so done that 
when in this position the switch- 
rail lies against the main rail. 
Therefore the angle a. is always an 
apgle whose sine equals 6 inches (or 
0.5 foot) divided by the length of 
the switch-rail in feet. In Fig. 153, 
the point D is not on the gauge line of the main rail but at a 
point j" away from it, and the point M 6 j" away from it. The 
straight rail BF consists of a point-rail at one end, the "closure 
rails," and one of the toe rails of the frog at the other end. The 
closure rails will in general consist of one rail cut to a computed 
length and one or more rails from 24 to 33 feet long, the lengths 
being in even feet. The curved rail DF will also consist of a 
point-rail, a frog toe-rail, and one or more lengths of closure 
rail, but the closure rails in this case are slightly longer than 
those for the straight rail. Since it is always practically easier 
to measure to the " actual point " of a frog (see Fig. 134), rather 



358 



RAILROAD CONSTRUCTION. 



§313. 



than to the theoretical point, Table III gives the distance L', 
which is the distance L= BF, plus the "frog bluntness," 
which is found by multiplying |" ( = 0.0417 foot) by the frog 
number. 

The curvature for a curved switch-rail (for a straight track) 
is most readily determined by measuring off a series of ordinates 
whose origin is at the switch-point D, Fig. 153, the points being 
the center and the quarter points of the actual curve. More 
accurately, the origin is on the gauge line of the main rail, opposite 
D, which is J" from the gauge line. These ordinates, as com- 
puted on the basis of " practical leads," by the A. R. E. A. com- 
mittee, are quoted below. It should be remembered that the 
system of practical leads usually involves a very short tangent 
adjacent to either M or J, and that the line MJ for " practical 
leads " is not entirely an arc. 

TABLE XXV. -^ RECTANGULAR COORDINATES TO THE QUARTER 
AND CENTER POINTS ON THE GAUGE SIDE OF CURVED RAIL, 
REFERRED TO POINT OP SWITCH-RAIL AS ORIGIN. 



Frog 
No. 


Measured along main rail. 


Measured perpendicular to 
main rail. 
















X 


Xi 


X2 


Y 


1^1 


Y2 


5 


17.92 


24.83 


31.75 


0.97 


1.69 


2.69 


6 


19.19 


27.37 


35.56 


1.03 


1.79 


2.83 


7 


26.71 


36.92 


47.12 


0.98 


1.72 


2.76 


8 


28.10 


39.71 


51.31 


1.005 


1.77 


2.80 


9 


28.75 


40.98 


53.19 


1.02 


1.76 


2.75 


10 


30.28 


44.05 


57.81 


1.04 


1.79 


2.78 


11 


40.74 


56.47 


72.19 


1.08 


1.84 


2.87- 


12 


43.99 


60.65 


77.28 


1.15 


1.90 


2.91 


14 


41.10 


60.21 


79.31 


1.08 


1.87 


2.91 


15 


52.00 


74.00 


96.00 


1.03 


1.81 


2.86 


16 


53.23 


76.46 


99.69 


1.04 


1.83 


2.89 


18 


54.73 


79.46 


104 . 19 


1.06 


1.86 


2.91 


20 


57.75 


85.50 


113.25 


1.10 


1.91 


2.95 



If the position of the switch-block is definitely determined, 
then the rails must be cut accordingly; but when some freedom 
is allowable (which never need exceed 16.5 feet and may require 
but a few inches), one rail-cutting may be avoided. Mark on 
the rails at B, F, and Z); measure off the length DN and locate 
the point M at the distance 6j" from N. If the frog must be 
placed during the brief period between the running times of 



"ip 



§314. SWITCHES AND CROSSINGS. 85^' 

trains, it will be easier to joint up to the heel of the frog (the 
point K', Fig. 153), a piece of rail, the farther end of which will 
just reach the next joint and also joint up to the toe of the frog 
the straight closure rail and the point-rail. Then, when all is 
ready, the rails are loosened from the ties back to B, the joint 
beyond the frog is removed and the whole rail back to B is swung 
outward. The new combination is shoved into place and spiked, 
even the point-rail being temporarily spiked to hold it in place 
as a main track rail, until the other switch-rail and the tie rods 
can be placed. When the frog is thus in place, the point J 
becomes located. The curved closure rails; as called for in 
Table III, should prove to be just long enough, when properly 
curved, to fill in the gap between M and J. Using the proper 
pairs of values for X and Y as given above, the three values of X 
may be measured on the main track rail from the point D, and 
the corresponding offsets will give points on the curved switch- 
rail. The old main track rail which was bent outward from B 
may be utilized as the other switch-rail and set to gauge from the 
rail just located. 

Example. — Given a main track on a 4° curve — a turnout to 
the outside, using a No. 9 frog; gauge 4' 8|"; W = Q'.00; H = Qi"; 
/S = 16' 6'' and a = l° 44' 11" Then for a straight track r 
would equal 605.18 [d = 9° 28' 42"]. For this curved track d 
will be nearly 9° 29' -4° = 5° 29', or r will be 1045.3. L' for a 
straight track would be 72,28, and is here considered to be the 
same. The closure rails have a total arc length of 49.59, and 
will here be taken the same. Note that the curved and straight 
closure rails each have odd lengths which are made by one cut 
of a 33-foot rail. This avoids all rail waste and also one rail- 
cutting and the boring of holes. 

314. Slips. Track movements in crowded yards are facili- 
tated by using " slips " (see Fig. 154), which may be " single " 
or " double." The crossing of two rails is done either by oper- 
ating two movable rails or by using fixed " frogs," but a com- 
parison of the continuity of the running rails, using ordinary 
frogs (see Fig. 134) and these frogs, will show their radical 
difference. These slips can be used for frog angles from No. 6 
to No. 15. The levers are so connected that the several opera- 
tions necessary to set the rails for any desired train movement 
are accomplished by one motion, 



360 



RAILROAD CONSTRUCTION. 



§314. 




FiQ. 154. — Single and Double Slips. 



§316. 



SWITCHES AND CROSSINGS. 



361 



CROSSINGS. 

315. Two straight tracks. When two 
straight tracks cross each other, four frogs 
are necessary, the angles of two of them 
being supplementary to the angles of the 
other. Since such crossings are sometimes 
operated at high speeds, they should be 







11 J! MB 


1 /I 


1 


1 1 
^ 


1 




% 




t \ 



ca 
I 
< 

z 
o 
z 
o 

p 
o 

LJ 



very strongly constructed, and the angles should preferably be 
90° or as near that as possible. The frogs will not in general 
be " stock " frogs of an even number, especially if the angles are 
large, but must be made to order with the required angles as 
measured. In Fig. 155 are shown the details of, such a crossing. 
Note the fillers, bolts, and guard-rails. 

316. One straight and one curved track. Structurally the 
crossing is about the same as above,, but the frog angles are all 
unequal. In Fig. 156, K is known, and the angle M, made by 



362 



RAILROAD CONSTRUCTION. 



§317. 



the center lines of the tracks at their point of intersection, is 
also known. M = NCM. NC==R cos M. 

R cos M+l^ 



(R-hg) cos Fi = NC+h9; -'- cos Fi = 



CI- •, 1 ' r, ^ cos M+ig 
bimilarly cos F^ = — , cos r 3 



R-h 
R cos M — ^g 



R+hg 



cos F4 = 



R+ig ' 
R cos M — ^g 



R-hg 

F^F,= {R+\gysmF^-{R-\g) smF,; 1 
HF4 = {R- hg) (sin F^-sinFi). J 

I I 




(95) 



(96) 



Fig. 156. 

317. Two curved tracks. The four frogs are unequal, and 
the angle of each must be computed. The radii Ri and R2 
are known; also the angle M. ri, r2, r^ and r^ are therefore 
known by adding or subtracting ^g, but the lines are so in- 
dicated for brevity. Call the angle MCxd^Ci, the angle 
MC2Ci = C2, and the line CiC2 = c. Then 

KCi+C2)=90° -|M 

^^^ tanKCi-C2)=cot|M|p^. . . . (97) 



Ci and Ca then become known and 

c = CiCi =» R, 



Ri-\-Ri 



sin M 
sin Ci 



(98) 



§ 317. SWITCHES AND CROSSINGS. 363 

In the triangle F1C1C2, call |(c+ri+r4)=Si; S2=|(c+^2+r4); 




Fia. 157. 



S3 = Kc+ri+r3); and S4 = Kc+r2+r3). Then, by formula 29, 
Table XIV, 

2(si-rO(si-r,) 



Similarly 



vers F^ = 
vers F2 
vers F3 



r,r. 



1'4 



vers F^ ■■ 



2(s2-r2)(sa-r,) 

. 2(.'?3-ri)(g3-r3) 

2(54-^3) (84-^3) 



(99) 



sin(7A^4=sinF,-'; 

sinCA^2=sini?'2-*; 

.r. F/J,F,=^C,C,F,-C,C:,F.,, (100) 

BmFyC^C.,=BmFfi', 

sini^AC2=sini^2-, 

.'. FiCiF,==F,C,C,-F,C,C,; .... (101) 
from which the chords F1F2 and F^F^ are readily computed. 



364 



EAILROAD CONSTRUCTIONi 



§317. 



F^Fj^-nd i^2^4 ^^^ nearly equal. When the tracks are straight 
and the gauges equal, the quadrilateral is equilateral. 

Problem. Required the frog angles and dimensions for a cross- 
ing of two curves (Z)i=4°; Z)2=3°) when the angle of their tan- 
gents at the point of intersection =62° 28' (the angle M in 
Fig. 157). 
Solution 

7^1 = 1432.7; i22 = 1910.1; 
n =i22 + ^9' = 1910. 1+2. 35 = 1912. 45; 
ra =i2,-Jgr = 1910. 1-2.35 = 1907.75; 
n =i2i + i9' = 1432.7 + 2.35 = 1435.05; 
n =i2i-ig' = 1432. 7-2. 35 = 1430.35. 
Eq. 97. log cot iM= 0.21723 

R2-Ri=477 A; log =2.67888 

R2+Ri=S342.8; log=3. 52411; co-log = 6 . 47589 

KC1-C2) =13° 15' 07''; tan 13° 15' 07"=9. 37200 
hiC, + C,) =58° 46' U(C, + C,) =90°-p/l 



Eq. 98. 



C7i=72°01'07" 
Ca=45°30'53" 

logi22=3.28105 

log sin M= 9. 94779 

log sin Ci = 9 . 97825 ; co-log = . 02175 



c = C,C, = 1780.7; 


logCi(72=3.25059 - 


Eq. 99. 




c=1780.7 


c=1780:7 


c=1780.7 


c=1780.7 


ri = 1912.45 


r2=1907,75 


ri = 1912.45 


r2= 1907.75 


r4=1430.35 


r4= 1430.35 


r3=1435.05 


r3= 1435.05 


2|5123.50 


2|5118.80 


2|5128.20 


2|5123.50 


«i = 2561.75 


S2 = 2559.40 


S3 = 2564. 10 


84=2561.75 


«l-n= 649.30 


82 — r%= 651.65 


«3-ri= 651.65 


S4-r2= 654.00 


»i-r4= 1131.40 


-r4= 1129.06 


S3-r3 = 1129.05 


8,1 -r3= 1126.70 






log 2 = 0.30103 




(si -n) ; log 649 . 30 = 2 . 81244 




Ui-n); log 1131.40 = 3.05361 


ri = 1912.45; log = 3. 28159; 


co-log = 6. 7 1841 


n«* 1430.35; log = 3,15544; 


co-log = 6. 84456 


Fi=62° 25' 31"; 




log vers 62* 


25' 31" = 9. 73006 




log 2 = 0.30103 




(«2-r2); log 651.65 = 2.81401 




isi-r^)', log 1129.05 = 3.05271 


r2=-1907.75; log«=3. 28052; 


co-log = 6. 71948 


r4= 1430.35; log = 3. 15544 J 


co-log = 6. 84456 


F.,=Q2° 33' 55"; 




log vers 62" 


33' 55" = 9. 73180 



§ 317. 



SWITCHES AND CEOSSINGS. 



365 



ri = 1912.45; log = 3. 28159; 
r3= 1435.05; log = 3. 15686; 
Fa = 62° 21^ 57^'; 



r2 = 1907.75; log = 3. 28052; 
ra = 1435 . 05 ; log = 3 . 15686 ; 
F4 = 62° 30^ 14^^ 



log 2 = 0.30103 

(s3-ri); log 651.65 = 2.81401 

(sa-ra); log 1129.05 = 3.05271 

co-log = 6. 71841 

co-log = 6. 84313 

log vera 62" 21' 57" = 9 . 72930 



log 2 = 0.30103 

(s4-r2); log 654.00 = 2.81558 

(•54-^3); log 1126.70 = 3.05181 

co-log = 6. 71948 

co-log = 6.84313 

log vers 62° 30' 14" = 9. 73103 



As a check, the mean of the frog angles = 62° 27' 54 ', which is -within 6" of 
the value oi Al. 



Eq. 100. 

CiC2F4 = 45° 37' 51' 



log c = 3. 25059; 



Ci 02^2 = 45° 28' 17"; 

^2^2^4 = 45° 37' 51" -45* 28' 17" = 0° 09' 34' 



K0°09' 34") = 0°04'47' 



F.-F, = 5.309 ; 
Eq. 101. 

FiCiC2 = 72° 10' 22"3 



2r2CiC2=71° 57' 38"; 

F,C\F~2 = 72° 10' 22" -71° 57' 38" = 0° 12' 44' 



4(0° 12' 44") = 0°.06' 22"; log sin= f^- 

\2. 



log sin i?4 = 9.94794 

log r3 = 3. 15686 

co-log = 6.74940 

sin Ci(?2F4 = 9. 85^21 

log sin ^2 = 9. 948 18 

log r4 = 3. 15544 

co-log c = 6. 74940 

sin CiC2F2 = 9.85303 

log 2 = 0.30103 
log r2 = 3. 28052 

> log sin =^4 -68557 
, logsm-^ g^gygg 

log F2-P'4 = 0.72500 

sin Fi = 9. 94763 

log ri = 3. 281 50 

co-log = 6.74940 

sin F'iCiC2 = 9.97863 

sin ^2 = 9. 94818 

log r2 = 3. 28052 

co-log c = 6 ■ 74940 

sin F2C1C2 = 9.97811 

log 2 = 0.30103 
log r4 = 3. 15544 
4.68557 
58206 



^1^3 = 5.298; logFiF2 = 0.72411 

As a check, i^j^i and F^F^ are very nearly equal, as they should 
be. 



366 RAILROAD CONSTRUCTION. §317. 

The foregoing problems on switches, connecting curves and 
crossings cover only a few of the most common of the problems 
encountered by the engineer. For the solution of a far wider 
range of problems, the engineer is referred to " Track Formulae 
and Tables," by S. S. Roberts. [Wiley & Sons.l 



i 

i 



CHAPTER XII. 

MISCELLANEOUS STRUCTURES AND BUILDINGS. 
WATER-STATIONS AND WATER-SUPPLY. 

318. Location. The water-tank on the tender of a locomo- 
tive has a capacity of from 3000 to 10000 gallons — sometimes less, 
rarely very much more. The consumption of water is very vari- 
able, and will correspond very closely with the work done by 
the engine. On a long down grade it is very small; on a ruling 
grade, going up, using full stroke, an engine with 28-in. cylinders, 
30-in. stroke, 180 lbs. boiler pressure, will use 4.59 lbs. of steam, 
or water, per stroke or 18.36 pounds per revolution. With 
63-in. drivers, the circumference is 16.5 feet and there will be 
320 revolutions per mile. The engine will use 5875 lbs. or 700 
gallons of water per nule. This engine has a tank capacity of 
9000 gallons, which would permit running about 12 miles at full 
stroke. But it is very rare that a locomotive must work for 
such long distances at full stroke. After starting and attaining 
full normal speed, the valves may be set to cut off at one-fourth 
stroke, or even at one-fifth or one-sixth for high speed running. 
With ordinary grades, such an engine might average 200 gallons 
per mile, in both directions. A quoted numerical case is that of 
a 106-ton engine using 7,500,000 gallons during an annual mileage 
of 45000 miles. This means an average of 167 gallons per mile. 
Observations were taken in 1910, on the N. Y. Central R.R., 
where the grades are moderate, showing that the heavy pas- 
senger trains of eight to twelve cars consumed 80 to 100 gallons 
of water per mile and that freight trains of about fifty loaded 
cars consumed from 110 to 130 gallons per mile. These figures 
are far less than those given above, but the grades on the N. Y. 
Central are very light. 

Freight engines, running at lower speeds and longer cut-off, 
require more frequent water-tanks than passenger engines. 
Even before a road is built, the water-tank requirements and the 
minimum spacing may be computed on the basis of the steam 
consumption (see § 454), of the locomotives with which it is 
expected to handle the estimated traffic of the road. Usually 
tanks will be located at intervals of 10 to 20 miles. 

367 



868 RAILROAi) CONSTRUCTION. § 319. 

In the early history of some of the Pacific railroads it was nec- 
essary to attach one or more tank-cars to each train in order to 
maintain the supply for the engine over stretches of 100 miles and 
over where there was no water. Since then water-stations have 
been obtained at great expense by boring artesian wells. The 
individual locations depend largely on the facility with which a 
sufficient supply of suitable water may be obtained. Streams 
intersecting the railroad are sometimes utilized, but if such a 
stream passes through a limestone region the water is apt to be 
too hard for use in the boilers. More frequently wells are dug or 
bored. When the local supply at some determined point is 
unsuitable, and yet it is necessary to locate a water-station there, 
it may be found justifiable to pipe the water several miles. The 
construction of municipal water-works at suitable places along 
Ihe line has led to the frequent utilization of such supplies. In 
such cases the railroad is frequently the largest single consumer 
and obtains the most favorable rates. When possible, water- 
stations are located at regular stopping points and at division 
termini. 

319. Required qualities of water. Chemically pure water is 
unknown except as a laboratory product. The water supplied 
by wells, springs, etc., is always more or less charged with cal- 
cium and magnesium carbonates and sulphates, as well as other 
impurities. The evaporation of water in a boiler precipitates 
these impurities to the lower surface of the boiler, where they 
sometimes become incrusted and are difficult to remove. The 
protection of the iron or steel of a boiler from the fierce heat of 
the fire depends on the presence of water on the other side of the 
surface, which will absorb the heat and prevent the metal from 
assuming an excessively high temperature. If the water side 
of the metal becomes covered or incrusted with a deposit of 
chemicals, the conduction of heat to the water is much less 
free, the metal will become more heated and its deterioration or 
destruction will be much more rapid. An especially common 
effect is the production of leaks around the joints between tubes 
and tube-sheets and the joints in the boiler-plates. Such injury 
can only be prevented by the application of one (or more) 
of three general methods — (a) the mechanical cleaning of the 
boilers, (h) the chemical purification of the water before its intro- 
duction into the boiler, and (c) the use of some " boiler com- 
pound ^' which is introduced directly into the boiler and which 



§ 320. MISCELLANEOUS STRUCTURES AND BUILDINGS. 36Q> 

causes precipitation of the harmful ingredients as non-incrusting 
solids which can be readily blown out. 

320. Mechanical cleaning, as a sole dependence is impracticable 
except in the comparatively rare localities where the water is so 
" soft "that no in crusting deposits will be made and such pre- 
cipitation as does take pfece is of such a character that it is 
removable by blowing out the boiler. There are many rail- 
roads, especially the smaller ones, which do not give any chem- 
ical treatment to any of their engine water-supply, and yet 
which are not fortunate enough to obtain even approximately 
soft water. The only method by which such roads can prevent 
a great waste of heat and the rapid deterioration of boiler tubes 
and sheets is by frequent mechanical cleaning. 

321. Chemical purification before the water enters the boiler 
has the advantage of removing the troublesome ingredients, 
leaving nothing further to be done except the occasional removal, 
by blowing out, of the suspended matter or harmless matter 
precipitated by boiling. Sodium carbonate is the most common 
reagent. It is commercially sold as *' soda crystals, sal soda, 
washing soda, Scotch soda, concentrated crystal soda, sesqui-^ 
carbonate of soda, crystal carbonate of soda, black ash, soda ash 
and pure alkali." Although often chemically impure, it can 
now readily be obtained with a purity of 97 to 99%. The chem- 
icals which are most common as incrustants are calcium and 
magnesium carbonates and sulphates. The effect of sodium 
carbonate on calcium sulphate is to produce soluble sodium sul- 
phate — which is non-incrustant — and calcium carbonate, which 
precipitates into a sludge at the bottom of the water softener 
tank. The action on magnesium sulphate is similar. When 
this is done in a purifying tank, the purified water is drawn off 
from the top of the tank and supplied pure to the engines. The 
precipitants are drawn off from the settling-basin at the bottom 
of the tank. This purification, which makes no pretense of 
being chemically perfect, may be accomplished for a few cents 
per 1000 gallons. There are manufacturers which make a spe- 
cialty of machinery, working more or less automatically, which 
introduces into the raw water a measured amount of chemical 
which, by analysis, has been calculated to be necessary with 
that particular quality of water. In spite of the automatic 
features, such machinery needs constant attention, and the 
water, both raw and treated, needs frequent analysis to 



^6 



eailhoad construction. 



§321. 



insure efficiency, since the character of the raw water may 
change; 

Sodium hydrate, or " caustic soda," has the same general 
chemical effect as sodium carbonate, and acts more quickly and 
powerfully, but its caustic nature makes it somewhat objection- 
able to handle. Common lime, barium hydrate, and many other 
chemicals are also more or less used. 

In the following tabular form is given the quantities of reagents 
required per unit of scaling or corroding substance held in solu- 
tion, the table being copied from the 1915 Manual of the Amer. 
Rwy. Eng. Assoc. " Where the commercial product is not 
chemically pure, the proportion of reagents should be increased 
to correspond with an equivalent quantity of pure reagent. 
Given the analysis of a water, the pounds of incrusting or cor- 
rosive matter held in solution per 1000 gallons can be obtained 
by dividing the grains per gallon of each substance by seven, or 
the parts per 100,000 by twelve. In order to ascertain the full 
amount of lime necessary, the amount of free carbonic acid con- 
tained in the water should be determined, as well as the solids 
contained in solution, since this free acid must be eliminated in 



TABLE XXVI. QUANTITY OF PURE REAGENTS REQUIRED TO 
REMOVE ONE POUND OF INCRUSTING OR CORROSIVE MATTER 
FROM THE WATER. 



Incrusting or corrosive 

substance held in 

solution. 


Amount of reagent (pure). 


Foaming] 

matter 
increased 


Sulphuric acid 

Free carbonic acid 


0.57-lb. lime plus 1.08 lbs. 
1.27 lbs. lime. . •. . 


soda ash 


1.45 lbs. 
None 


Calcium carbonate 


0.56-lb. Ume 


None 


Calcium sulphate 

Calcium chloride 


0.78-lb. soda ash 


.1.04 lbs. 


0.96-lb. soda ash 


1.05 " 


Calcium nitrate 


0.65-lb. soda ash 


1.04 " 


Magnesium carbonate. . 


1.33 lbs. lime 


None 


Magnesium sulphate. . . 
Magnesium chloride . . . 
Magnesium nitrate. . . . 
Calcium carbonate 


0.47-lb. lime plus 0.88 lb. 
0.59-lb. lime plus 1.11 lbs 
0.38-lb. lime plus 0.72-lb. 
3.15 lbs. barium hydrate. 
3.76 lbs. barium hydrate 
2.62 lbs. barium hydrate. 
2.32 lbs. barium sulphate 


soda ash 

. soda ash 

soda ash 


1.18 lbs. 
1.22 " 
1.15 " 

None 


Magnesium carbonate. . 




None 


Magnesium sulphate . . . 
Calcium sulphate * . . . . 




None 
None 







* In precipitating the calcium sulphate, there would also be precipitated 
0.74 lb. of calcium carbonate or 0.31 lb. of magnesium carbonate, the 2.32 
lbs. of barium hydrate performing the work of 0.41 lb. of lime and 0.78 lb. 
of soda ash, or for reacting on either magnesium or calcium sulphate, 1 lb. 
of barium hydrate performs the work of 0.18 lb. of lime plus 0.34 lb. of 
soda ash, and the lime treatment can be correspondingly reduced. 



§ 322. MISCELLANEOUS STRUCTURES AND BUILDINGS. 371 

order to obtain efficient treatment of water and reduce scaling 
matter to the minimum," 

322. Foaming and priming. This phenomenon is the foaming 
or frothing of the water for a considerable height above its normal 
level in the boiler. The rapid flow of steam into the steam pipe 
in the dome mechanically carries some of this froth into the steam 
pipe and causes water to accumulate in the steam pipe and also 
in the cylinders, with considerable resulting loss in efficiency. 
Foaming in treated water is largely due to the presence of sodium 
salts as a result of treatment for incrusting sulphates, and this 
constitutes one of the objections to the use of soda in treating 
water. The presence of suspended matter in the water ag- 
gravates and even causes foaming. The constant withdrawal of 
the water from the boiler leaves these suspended solids in the 
boiler and they keep accumulating until the concentrations reach 
a critical point, which is about 100 grains per gallon. Beyond this 
point foaming will be experienced unless the water is changed, 
which is done by a systematic blowing-off and an occasional com- 
plete blowing-down and washing. But blowing-off involves the 
wastage of water which has been heated to boiler temperature 
and which has, perhaps, been chemically treated. Even the raw 
water costs something, perhaps several cents per 1000 gallons. 
The blowing-off required to keep the concentration below the 
proper limit may be so excessive that some anti-foaming agent 
may be necessary. The required effect is physical rather than 
chemical, the object being to reduce the surface tension, which 
is done chiefly by the use of oils, petroleum and castor oil being 
used. Tannic acids are also used for such a purpose. 

323. Boiler compounds. Chemical treatment at special plants 
along the road is unquestionably the most efficient method, but it 
is costly. The use of boiler compounds, often patented, obviates 
the erection of any plant, but, since the water at each water- 
supply station has its own characteristics and it is impracticable 
to vary the chemicals used at each supply-station according to 
the character of the water, the treatment is very imperfect. 
Minute instructions to enginemen to introduce definite amounts 
of chemical at each water-station have proved unsatisfactory 
and impractical. Sometimes the chemical is mixed with enough 
water to partially suspend it and then it is thrown into the ten- 
der tank, this method having the advantage that a considerable 
part of the precipitation takes place promptly and the sludge 



372 RAILROAD CONSTRUCTION. § 324. 

never enters the boiler. Sometimes a siphon attached to the 
feed-pipe outside of the injector, or, perhaps, a special injector, 
leads from a reservoir in which the chemical, suspended in water, 
has been placed. Sometimes a stick or " brick " of the chemical 
is placed directly in the boiler, through a hand-hole, during one 
of its periodical cleanings. In spite of the inefficiency of the 
method, 70% of replies to a circular inquiry reported the use of 
some kind of boiler compound. The chemicals used, some of 
which are patented compounds, are in general the same as those 
used in the outside chemical plants. Sodium carbonate is the 
most common constituent. 

324. Tanks. Height above rail. Whatever the source, the 
water must be led or pumped into tanks which are supported 
on columns so that the bottoms of the tanks are high enough 
above the track to force a flow of 2500 gallons of water per minute 
through a 12-inch spout. The frictional resistance in the pipes, 
elbows, valves, etc., are such that, allowing that the spout is 
12 feet above the rail, the bottom of the tank should be about 
16 feet above the rail. If the water flows from the tank into a 
■'stand-pipe," see § 327, there is additional frictional resistance, 
to allow for which the height of the support or " tower " is in- 
creased to perhaps 30 feet. The standard heights for towers 
are 16, 20 and 30 feet. Sub-structure. A standard plan, recom- 
mended by the Water Service committee of the A. R. E. A., 
is to support such tanks on twelve 12"X12" posts, arranged 
in a double cross, four posts in each line, each post resting on 
a concrete footing. The posts are suitably cross-braced as in 
trestle work, and are surmounted by cast iron caps. These 
support 12"X14" timber caps, which carry 4"X14" joists, 
spaced 14", which are immediately under the bottom of the tank. 
Size. Two sizes of tanks are standard. The "16X24" has a 
net height inside of 15' 4" and a net inside diameter of 24' 0". 
Although the capacity, brimming full, would be nearly 52,000 
gallons, it is called a " 50,000-gallon " tank since the outlet 
pipe must be several inches below the top. The "20X30" 
tank has a net inside diameter of 30' 0" and net height of 19' 4", 
It will contain 100,000 gallons when the water depth is slightly 
less than 19 feet. Since it is found that the 100,000-gallon 
tank costs but 10% more than a 75,000-gallon tank, the com- 
mittee recommended that the 50,000-gallon and the 100,000- 
gallon tanks should be considered the two standard sizes. 



§ 325. MISCELLANEOUS STRUCTURES AND BUILDINGS. 373 

Details. Cylindrical tanks are recommended, rather than 
tapered. The staves are machine-dressed so that the edges 
have the proper bevel toward the tank axis, and the outside is 
dressed to the proper convex cylindrical surface so that the hoops 
have a bearing for the full width of the stave. The " croze," 
2f" wide and f" deep, into which the bottom planks, 3" thick, 
slightly beveled at the ends, are inserted for a tight joint, is 
4" above the bottom of the staves. When the jointing edges 
are properly made, the tank will be water-tight without any 
plugging or caulking, which should not be permitted. The 
weight of the tank should be transmitted through the bottom 
planks and in no case by means of the staA^es. Round hoop- 
rods, rather than elliptical or flat, are recommended. They 
should be made of refined double-rolled wrought iron. Each 
hoop should have three sections for 16X24 tanks and four sec- 
tions for 20X30 tanks. On the basis of a maximum working 
stress of 12,500 pounds per square inch on the area at the base of 
the screw threads, the safe working load in pounds is as follows : 

I", 3750; I", 5250; l'', 6875; 1|'', 8625. 

The spacing of hoops may be computed from the formula: 

safe load for the given hoop in pounds 

Spacmg m mches = — -— — — ^ ..-. . — • 

2.6 diameter (it.) X depth m leet 

In the above formula, " depth " means the distance from top 
of stave to location of hoop. One hoop should be placed within 
two inches of the top and two hoops around the bottom opposite 
the croze. One of these is assumed to take up the bursting 
pressure due to the swelling of the bottom planks when water 
soaked, and that it does not withstand water pressure. The 
spacing should never exceed 21 inches. Hoop " lugs," made 
of cast or malleable iron, are used to connect the sections of the 
hoops. Each end of each rod should be threaded for 4^" and 
be provided with two hexagon nuts. 

325. Pumping, (a) Steam-pumps. When coal is very cheap 
or " when 100 lbs. of coal in the pumphouse is cheaper than one 
gallon of fuel oil in the storage tank," and especially when steam 
can be procured from the railroad repair-shop plant, direct-acting 
steam pumps may be preferable and more economical, but they 
always require skilled attendance. (6) Gasoline-engines. These 
have been so highly developed in recent years that they are very 
efficient and are nearly " fool-proof," so that they may be oper- 



374 



EAILROAb CONSTRirCTlON. 



§325. 





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OOOOOOOOO 

ooooooooo 
m 


'6 


14 lbs. 
Igal. 
12 cu. ft. 
8 cu. ft. 
Igal. 
.746 K.W. 
.746 " 
i gal. 




O 

• r-l 
U 


ocD»ciccocofo«:>co 

OrHtXNOOOrHO 


cqoooooooo 


d 

• r-t 


Bit. coal 
Gasoline 
111. gas 
Nat." 
Fuel oil 
Electric 

Gasoline 
Fuel oil 


a; 

a 


6 
a 
"m 
d 


oj o o 
13. - . .3 

>.5- - : g: .5: 

g d ^ -C d 


a 

a 

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bo • ; ; ; 

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ft '»-,"- 

o d 



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ated by unskilled labor, 
although skilled attention 
is periodically necessary. 
But the rising cost of gas- 
oline has directed atten- 
tion to other fuels, (c) Oil- 
engines. Crude petroleum, 
when refined, will give off 
approximately the follow- 
ing: Ether, 2%; gasoline, 
6%; naphtha and benzine, 
8%; kerosene, 44%; 39° 
power distillate, 10%; gas 
oil, 10%; lubricating oils 
and petrolatum, 15%, and 
"slops" 5%. The "fuel 
oil," as supplied for oil 
engines, is a mixture of the 
slops with enough of some 
other constituent, usually 
the " power distillate," 
which is at the time the 
cheapest, to make the 
gravity of the mixture 
about 29°. The fuel oil 
costs approximately 40% 
as much as gasoline. Gas- 
oline engines have been 
converted into fuel oil 
engines by attaching a 
mixing chamber in which 
the oil is heated by the 
exhaust of the engine. 
{d) Gas-engines, using 
natural gas. Where nat- 
ural gas is available at 
25 cents per 1000 cu.ft. or 
less, it is an economical 
fuel, (e) Electric power. 
Where this is obtainable 
at a low rate, it may be 



§ 326. MISCELLANEOUS STRUCTURES AND BUILDINGS. 375 

a cheaper source of power than steam, gasolene or fuel oil. The 
electric motor either operates a centrifugal pump, or a slow-speed 
motor is direct-connected to a triplex reciprocating pump. 

A Committee of the Amer. Rwy. Eng. Assoc, reported in 1915 
the comparative cost (see Table XXVIII) of pumping 240,000 
gallons per day of 10 hours. By comparing the data with 
that of any given locaHty a fair idea of relative costs and of 
the proper choice for that particular station may be made. 

326. Track tanks. These are chiefly required as one of the 
means of avoiding delays during fast-train service. A trough, 
made of steel plate, is placed between the rails on a stretch of 
'perfectly level track. A scoop on the end of a pipe is lowered 
from under the tender into the tank while the train is in motion. 
The rapid motion scoops up the water, which then flows into the 
tender tank. They should preferably be located on tangents, 
although the Penn. R. R. has track tanks at Atglen on a 2" 
curve where the track has 4 inches superelevation. Since the 
inside width of the tank (19") is almost exactly \ of the gauge, 
the water is about 1| inches deeper on the side toward the inner 
rail, but this much lack of symmetry does not seem to have 
interfered with successful operation. The length of the tanks 
varies from 1200 to 2500 feet; the net inside width is usually 
19 inches. The scoops are usually 12 to 13 inches wide, which 
gives allowance for swaying. The tanks are made of sheet steel 
■3^" to \" thick. The usual cross-section is that of a wide and 
shallow U, 19" wide, 6" to 7|" deep, reinforced on the sides with 
angles. The ties are usually dapped, especially for the deeper 
tanks, so that the upper edges will not be higher than the rail. 
At each end there is a double inclined plane on which the 
scoops may slide without catching if the scoop should be lowered 
too soon or if it is not raised before the far end of the tank is 
reached. Experiments have shown that, at a speed as low as 
2Q m.p.h., more water is wasted by slopping over the sides than 
the amount collected by the scoop. At a speed of 45 to 50 m.p.h. 
the amount wasted becomes minimum and the amount scooped 
up becomes maximum. At higher speeds the amount scooped 
up decreases and the wastage increases. The best results show 
a wastage of at least one-eighth of the total. These same tests 
showed that at 45 to 50 m.p.h. the 13" scoop in a 19" tank will 
scoop up about 625 gallons per inch of immersion per 1000 feet 
of tank, or say 2500 gallons per lOOQ feet for a 4-inch immersion. 



376 



RAILROAD CONSTRUCTION. 



§327. 



The amount scooped up is practically proportional to the depth 
of immersion when that depth is over 2f inches. Heating. 
The water must be heated in winter to prevent freezing. There 
are two general methods: (a) Live steam is forced into the 
tank through nozzles about 40 feet apart; (6) a " circulatory- 
system " by which steam is forced into a water main which feeds 
the tank in such a way that the water is in constant circulation 
through the main, into the tank and then back again into the 
main to be reheated. For the chmatic conditions of the N. Y. 
Central R. R, a steam capacity of 100 H. P. is considered essen- 
tail to heat 7000 sq. ft. of tank surface, which means about 4400 
Uneal feet of 19~inch tank, or two good-length tanks on a double 
track. On account of the great amount of water splashed over 
the track and its scouring action on any ordinary ballast, a 

large item in the cost of an in- 
stallation is the reconstruction 
of the track. The certainty of 
quick freezing in winter, at least 
in high latitudes, demands that 
a drainage system, to carry away 
the spilled water, shall be effec- 
tive and thorough. Scom-ing is 
prevented by a pavement of 
cobbles, 6-inch quarry spalls, or 
large flat stones, laid over the 
ballast. A layer of large stones 
under the ballast faciUtates 
drainage to numerous cross 
drains and to longitudinal drains 
laid between the tracks. For 
further details the student is re- 
ferred to a monograph by Geo. 
W. Vaughan, Eng. Main, of 
Way, N. Y. Central R. R.,inVol. 
XIV, Proc. Am. Rwy. Eng. Assoc. 
327. Stand-pipes. These are usually manufactured by those 
^ho make a specialty of such track accessories, and who can 
ordinarily be trusted to furnish a correctly designed article. In 
Fig. 159 is shown a form manufactured by the Sheffield Car Co. 
Attention is called to the position of the valve and to the device 
for holding the arm parallel to the track when not in use so that 




Ffo. 159. — Stand-pipe. 



§ 328. MISCELLANEOUS STRUCTURES AND BUILDINGS. 377 

it will not be struck by a passing train. When a stand-pipe is 
located between parallel trackSj the strict requirements of clear- 
ance demand that the tracks shall be bowed outward slightly. 
If the tracks were originally straight, they may be shoved over by 
the trackmen, the shifting gradually running out at about 100 
feet each side of the stand-pipe. If the tracks were originally 
curved, a slight change in radius will suffice to give the necessary 
extra distance between the tracks. 



BUILDIiSTGS. 

328. Station platforms. These are most commonly made of 
planks at minor stations. Concrete is used in better-class work, 
also paving brick. An estimate of the cost of a platform of paving 
brick laid at Topeka, Kan., was $4.89 per 100 square feet when 
laid flat and $7.24 per 100 square feet when laid on edge. The 
curbing cost 36 cents per linear foot. Cinders, curbed by timbers 
or stone, bound by iron rods, make a cheap and fairly durable 
platform, but in wet weather the cinders will be tracked into 
the stations and cars. Three inches of crushed stone on a 
cinder foundation is considered to be still better, after it is once 
thoroughly packed, than a cinder surface. 

Elevation. — ^The elevation of the platform with respect to 
the rail has long been a fruitful source of discussion. Some roads 
make the platforms on a level with the top of the rail, others 
? inches above, others still higher. As a matter of convenience to 
the passengers, the majority find it easier to enter the car from 
a high platform, but experience proves that accidents are more 
numerous with the higher platforms, unless steps are discarded 
altogether and the cars are entered from level platforms, ad is 
done on elevated roads. As a railroad must generally pay dam- 
ages to the stumbling passenger, they prefer to buUd the lower 
platform. Convenience requires that the rise from the platform 
to the lowest step should not be greater than the rise of the car 
steps. This rise is variable, but with the figures usually employed 
the application of the rule will make the platform 5 ins. to 15 ins. 
above the rail. 

Position with respect to tracks. — Low platforms are gen- 
erally built to the ends of the ties, or, if at the level of the top 
of the rail, are built to the rail head. Car steps usually extend 
4 ft. 6 ins. from the track center and are 14 ins. to 24 ins. above 



378 



BAILROAD CONSTRUCTION. 



§329. 



the rail. The platform must have plenty of clearance, and when 
the platform is high its edge is generally required to be 5 ft. 6 ins. 
from the track center. 

329. Minor stations. The Amer. Rwy. Eng. Assoc, recom- 
mend one general waiting room (without reference to separate 
waiting room for colored people), for a passenger station of 
medium size for the following reasons: (See 1915 Manual, p. 187). 

(1) It permits the general waiting room to be properly pro- 
portioned. 

(2) It permits proper development of a retiring room for 
women, with private entrance to the lavatory. 




Fig. 160. — Division of Floor Area Recommended for Passenger 
Stations with One General Waiting Room. 



(3) It readily admits of the other rooms being properly pro- 
portioned. 

(4) It permits ease of access from the agent's office to the 
trains, to the baggage room and to the waiting room. 

(5) It permits the ticket office to be of proper size and location 
for general office purposes. 

(6) It admits of the station being contracted in size without 
detriment to facihties. 

(7) It offers economy in heating. 

In the Southern States a separate waiting room for colored 
people is provided and is sometimes even required by law. The 
older design, combining a residence for the agent with the station, 
is now obsolete for new construction, although many such still 
exist. " Combination stations " (for both passenger and freight 
business) were formerly quite popular for very small stations and 



I 330. MISCELLANEOUS STRUCTURES AND BUILDINGS. 379 

are still considered desirable when all responsible freight and 
passenger business must be handled by one man. But it is 
desirable to separate them whenever the volume of business will 
justify the employment of two responsible men. 

In Gillette's Handbook of Cost Data (1910 ed.), is given in 
detail the cost of several station buildings. Such figures can be 
utilized when unit prices are given or can be derived. For exam- 
ple, in one case the building was 24X60 ft., exclusive of- plat- 
forms; there was no masonry foundation nor plastering. The 
summary was as follows: 



Materials. 


Total. 


Per cent. 


Per sq. ft. 
of floor. 


30,057 ft. B. M. at $13.23 (aver.) 

20 M shingles at $1.10. 


$296.97 
22.00 
55.75 
37.50 
16.10 
8.80 


33.2 
2.4 
6.1 
4.1 
1.8 
1.0 

48.6 

45.3 
0.6 

2.8 
1.8 
0.9 

51.4 
100.0 


21 ft. B. M. 


Mill work 


3 . 9 cents 


Hardware 


2.6 " 


23 gal. paint at 70 cents 


11" 


1100 brick, at $8.00 per M 








Total materials 


$437.12 

$406.38 

5.00 

24.50 

16.00 

8.50 


30.4 " 


Labor: 

176.2 days' labor, building at $2.32 

2 days' labor, put up ladders, at $2.50. . 

14 days' labor, painting at $1.75 

4 days' labor, building chimney, at $4.00, 
8 days' labor, filling cinders, at $1.20 ... 


28 . 2 cents 
1 . 7 cents 


Total labor 


$460.38 

$897.50 
55.00 
38.50 


31.9 *• 


Total, materials and labor. 




Freight, 55 tons, 200 miles ^c. ton-m. . . . 
Tools (excessive in this case) 








Grand total 


$990.00 


68.8 cents 







The cost of lumber was very low and even the unit cost of 
labor (carpentsrs, $2.50; masons, $4.00; average of all, $2.32), 
were lower than must frequently be paid. But the figures can 
be utilized by noting the percentages of the various items to the 
total and applying local unit costs for material and labor. The 
total cost per square foot ($0,688), is abnormally low, partly 
because of no masonry foundation nor cellar, which would add 
40 to 50 cents per square foot. Note also that no expenses were 
included for lighting, plumbing, or heating — except a chimney. 



FREIGHT HOUSES. 



330. Two types. The freight house, or freight room, at a sta- 
tion where the business is small, is merely a small ordinary build- 
ing or a room attached to the station building. As the business 



380 RAILROAD CONSTRUCTION. ^ § 331. 

becomes larger, efficient operation requires that two types of 
buildings must be designed — the inbound and the outbound 
freight house. These types agree in requiring certain details in 
common, but there are also differences. 

331. Fire-risk. A small freight house in the country usually 
has a minimum of actual fire-risk and of valuable freight stored 
at any one time. This may justify an inexpensive type of frame 
building which is in no sense fireproof. On the other hand, a 
building in the heart of a city, closely surrounded by other build- 
ings and stored with a large amount of valuable freight, justifies 
an expensive type of fireproof construction. The term "fire- 
proof " is only relative. Certain devices and added expendi- 
tures will reduce more and more the probability of destructive 
fires. Certain principles of construction which reduce fire-risk 
are as follows: (a) Use of noncombustible materials for floor, 
side walls and roof; (b) avoidance of space under wooden main 
floor, between foundations, where combustible rubbish ma^ 
accumulate; (c) fire-walls dividing large houses so that there is 
not more than 5000 square feet of floor between fire-walls; fire- 
walls to b^ never more than 200 feet apart; (d) minimum num- 
ber of doors through a fire-wall; no door larger than 80 square 
feet; all doors fireproof and automatically self-closing; (e) 
fireproofing protection of walls and roof for at least five feet 
each side of a fire-wall; (/) provision for fire stand-pipes and 
hose racks not more than 150 feet apart; the stand-pipe should 
run up about 8 feet above floor where there should be 50 feet of 
2-inch linen hose in a hose rack; the valve should be in a pit 
{always accessible), and so far below floor level that there is little 
or no danger of freezing, since freight houses are ordinarily not 
heated. 

332. Dimensions. A freight house usually has a track on one 
side and a vehicle driveway on the other, the floor being utilized 
for the more or less temporary storage of freight, which in this 
case is always in " less than carload " (L. C L.) lots, carload 
shipments being transferred directly between cars and vehicles. 
Since small shipments can usually be loaded into cars (outbound 
shipments) with less delay than the delivery of freight to vehicles 
(inbound shipments), the required space for outbound shipments 
can be less than that for inbound. Experience has shown that 
for outbound freight only, a width of 30 feet is desirable; for 
both outbound and inbound, the width may be 30 to 40 feet; 



§ 333. MISCELLANEOUS STRUCTURES AND BUILDINGS. 381 

for inbound only it should be 40 to 60 feet. Too great a width 
needlessly increases the amount of hand-trucking. The length 
is indefinite and should correspond to the amount of business to 
be handled. Freight houses are usually single-storied, except 
where galleries or partial second stories are built to accommodate 
offices, file and stationery rooms, toilet and locker rooms, the 
room for " over, short and damaged " freight and the cooperage 
room for repairing broken packages. 

333. Platforms. The platform on the track side should 
preferably be 8 to 10 feet wide, which will avoid the necessity 
of spotting cars with their doors directly in front of freight-house 
doors. The platform should be not more than 4 feet above the 
top of the rail. Even this would be too high to permit opening 
the doors of refrigerator cars, which swing outward. An occa- 
sional refrigerator car could be handled, even with a high plat- 
form, by opening the doors before placing the car. The M. C. B. 
standard, for regular use of refrigerator cars, is " not more than 
3 ft. 8 ins." The P. R. R. standard is 3 ft. 5 ins. The minimum 
distance from track center to edge of platform is 5 ft. 9 ins. 
The P. R. R. standard is 6 ft. If ins. If there is a platform on 
the driveway side, it should be 3 to 4 feet above the driveway 
level. At an outbound house, where the froight is delivered 
from the vehicle into the freight house, the height should be not 
more than 3 feet. Platforms should slope away from the house 
with a grade of about 1 in. to 8 ft. for drainage. 

334. Floors. The designed floor loading should be 250 lbs. 
per square foot. In § 347 are described several types of floors 
suitable for engine houses, many of which are also suitable for 
freight houses. In selecting a type, it should be remembered 
that hand-trucking is apt to be concentrated along certain 
rather narrow paths and that this wears out the floor surface, 
requiring premature renewals along these paths, unless these 
paths are overlaid with iron or steel plates. When a solid type 
of floor is used (supported on sub-soil), the flooring should be 
independent of the side walls, which avoids trouble due to floor 
settlement. For inbound freight houses the floor should slope 
about 1 inch in 8. feet from the track side toward the driveway 
side, the slope continuing to the outer edge of the driveway plat- 
form, since this is in the direction of traffic and aids it, but the 
track platform must slope the other way for drainage. For out- 
bound freight houses, the slope is exactly reversed. 



382 RAILROAD CONSTRUCTION. § 335. 

^ 335' Doors. Ordinary swinging doors are unsuitable. Lift- 
ing doors, counterbalanced, which sometimes fold as they lift, 
are used. Rolling metal shutters are, perhaps, most satisfaqtory, 
but are expensive. Sliding doors require that a guarded space 
be made so that stored freight does not interfere with the 
sliding. They also limit the possible total door width to less 
than half the side of the house. All lifting types permit opening 
up the whole side of the house (if desired), except the space oc- 
cupied by the posts. Continuous doors are particularly neces- 
sary when there is no platform between the house and the 
track. Doors should be at least 8 feet high. On the track side 
this is sufficient, since the car door cannot be higher. On the 
driveway side a greater height might be desirable. 

336. Roofs projecting over platforms. These are desirable as 
a protection when loading or unloading during storms. That 
over the driveway platform should be at least 10 feet above the 
platform or 14 feet above the driveway. When not forbidden 
by State laws, the roof may be extended beyond the edge of 
the track platform, but it should be, at least, 17 feet above the 
rail and 18 inches from the track center, thus leaving a walking 
space on top of the car. 

337. Lighting. Dayhght lighting should be obtained by win- 
dows through the side-walls above the doors, or by vertical sashes 
in a monitor roof, which will also provide for ventilation. Sky- 
lights, especially when nearly flat, are expensive both for con- 
struction and for maintenance. Artificial lighting should be 
obtained from electricity, with wires run according to the strictest 
specifications of the National Board of Underwriters. Platforms 
should be illuminated. A series of push plugs should be placed 
along the platform wall face, from which extension cords with 
bulbs may be run to light car interiors. 

338. Scales. Outbound houses need scales, with capacity of 
8000 lbs., to weigh outgoing freight. " From 50 to 80 feet apart 
is good practice." 

339. Ramps. These are slopes from the driveway level to 
the car level which facilitate the loading or unloading of agricul- 
tural implements and all heavy vehicles running on their own 
wheels. They are usually built at the end of an extension of 
the platform, with as low a grade as the circumstances will 
permit. 

" Buildings and Structures of American Railroads," by Walter 



§ 340. MISCELLANEOUS STRUCTURES AND BUILDINGS. 383 

G. Berg, although now (1916) somewhat old, contains many 
plans, showing considerable detail, of station and other buildings. 
" Railroad Structures and Estimates " by J. W. Orrock, also 
shows some plans. 

340. Section houses. These are houses built along the right- 
of-way by the railroad company as residences for the trackmen. 
The liability of a wreck or washout at any time and at any part 
of the road, as well as the convenience of these houses for ordinary 
track labor, makes it all but essential that the trackmen should 
live on the right-of-way of the road, so that they may be easily 
called on for emergency service at any time of day or night. 
This is especially true when the road passes through a thinly 
settled section, where it would be difficult if not impossible to 
obtain suitable boarding places. It is in no sense an extrava- 
gance for a railroad to build such houses. Even from the direct 
financial standpoint the expense is compensated by the corre- 
sponding reduction in wages, which are thus paid partly in free 
house rent. And the value of having men on hand for emergen- 
cies will often repay the cost in a single night. Where the coun- 
try is thickly settled the need for such houses is not so great, and 
railroads will utilize or perhaps build any sort of suitable house, 
but on Southern or Western roads, where the need for such 
houses is greater, standard plans have been studied with great 
care, so as to obtain a maximum of durability, usefulness, com- 
fort, and economy of construction. (See Berg's Buildings, etc., 
noted above.) On Northwestern roads, protection against cold 
and rain or snow is the chief characteristic; on Southern roads 
good ventilation and durability must be chiefly considered. 
Such houses may be divided into two general classes — (a) those 
which are intended for trackmen only and which may be built 
with great simplicity, the only essential requirements being a 
living room and a dormitory, and (b) those which are intended 
for families, the houses being then distinguished as " dwelling- 
houses for employees." 

ENGINE HOUSES.* 

- 341. Form. When not more than three or four engines are 
to be housed at once and when no turntable is to be provided, 

* Condensed and abbreviated from Cojnmittee Report, Am, Ry. Eng. 
Assoc, 1915, 



384 RAILROAD CONSTRUCTION. § 342. 

the rectangular form is preferable. All large engine houses are 
" circular," with a turntable at the center of the circle, except 
some very large houses, which are really repair shops, where it 
seems advisable to install a transfer table. 

342. Doors. The clear opening should be not less than 13 
feet wide by 16 feet high. The doors should fold outward and 
should have such a design that a pilot door may be inserted. 

343. Length. The length of stall along the center line of the 
track should be 15 feet greater than the overall length of the 
longest locomotive, which will provide a walkway behind the 
tender, a trucking space in front of the pilot and a sufficient dis- 
tance in which to stop the engine so that the side rods will be in 
any desired position. 

344. Materials of construction. Wood was formerly very 
commonly used, but it is too inflammable. The walls should be 
made of brick, stone, or plain concrete — not reinforced, at least 
*' for that portion of the wall directly in line of track where 
engine is liable to run into it." The roof is the difficult problem, 
since wood is inflammable and iron or steel, even for framing, is 
very rapidly corroded by coal gas from the engines. Rein- 
forced concrete is the only thoroughly satisfactory material but 
" when the roof is of reinforced concrete, the columns and roof 
beams should be of the same material," i. e., it is useless to sup- 
port a reinforced concrete slab on steel beams. 

345. Engine pits. These " should be liot less than 60 feet 
in length, with convex floor, with drainage toward the turntable. 
The walls and floors may be of concrete. Proper provision 
should be made for the support of the jacking timbers." The 
engine should stand with its tender toward the turntable. 

346. Smokejacks. Locomotives leave an engine house under 
their own steam, which requires starting their fires considerably 
beforehand, and the smoke must be removed. The precise 
position of the locomotive on the track is variable, since it must 
be adjusted to the place where the side rods are in a proper 
position for repairs.. A smokejack is essentially a funnel whose 
base is at the minimum height above the track which will give 
the smokestack a proper clearance. The base should be 42 
inches wide and long enough for the adjustment as stated above, 
which means at least 10 feet. The sides should slope upward 
gradually to a flue whose area should be not less than 7 square 
feet. There should be a drip trough around the base of the jack. 



§ 347. MISCELLANEOUS STRUCTURES AND BUILDINGS. 385 

The material should be " non-combustible," but the choice is 
troublesome. Sheet iron, even when heavily painted, corrodes 
rapidly. Wood, covered with " fireproof paint," has been tried. 
Cast iron has been tried but is exceedingly heavy as well as ex- 
pensive. Asbestos is being used on several important roads. 
Patented designs, of which there are several, are used on the 
majority of roads. 

347. Floors, (a) Stone screenings. Subsoil should be good; 
all soft spots cleaned out and filled with good material; subsoil 
rolled. Foundation of cinders or gravel, 6 ins, thick. Top coat, 
2 inches of stone screenings, perhaps mixed with a httle clay or 
crude oil, the surface being thoroughly rolled. Special founda- 
tions for machinery necessary. Surface is not good for heavy 
wheeling, (h) Planks. Subsoil same as above; 6 ins. cinders 
or gravel, with 4"X6" creosoted sleepers, spaced about 3 feet, 
embedded in upper surface of cinders; then 3-inch plank. 
Again, special foundations for machinery and at jacking-up 
places are necessary, (c) Creosoted wood-block. The wood 
blocks, 4 ins. deep, fiber vertical, should be laid on a 1-inch 
cushion coat of sand which is supported by a 6-inch layer of 
concrete. A 6-inch layer of cinders, as specified above, is also 
recommended as a bed for the concrete, but this may depend on 
the character of the subsoil. The joints should be filled with 
asphaltic mastic, and an expansion joint 1 inch wide should be 
provided every 50 feet, (d) Wood floor on concrete. Sleepers, 
spaced about 3 feet, trapezoidal, 4-inch top, 6-inch bottom, 4 
inches deep, embedded in a 6-inch layer of concrete, so that the 
sleepers project ^ inch above concrete. Then layer of 2-inch 
plank, covered with l|-inch maple flooring, (e) Brick. Same 
as (c) except that bricks are used in place of wood block. (/) 
Concrete. Same foundation as above; 6-inch course of concrete 
overlaid with 1-inch surface coat (1:2) laid on before base has 
taken initial set. (g) Asphalt. Same as (/) except that surface 
coat is 1| inches of rock mastic. Expert workmen are needed 
for satisfactorily mixing and laying the asphalt, but the floor is 
ideal. 

348. Drop pits are necessary, where pairs of truck, driving and 
trailer wheels may be dropped from their journals and removed 
from the engine for repairs or renewals. 

349. Heating. The primary object of heating is to thaw out 
the engines so that they may be returned to service as quickly 



386 RAILROAD CONSTRUCTION. § 350. 

as possible, rather than to heat the building, whose general tem- 
perature should be kept at 50° to 60°. Therefore heat should 
be concentrated at the pits. Hot air should be forced through 
permanent ducts, preferably laid under the floor. The outlets 
should have dampers, which may be closed when men are working 
in the pits. Fresh air should be drawn from outdoors and no 
recirculation permitted. The air should be heated by passing 
over coils containing exhaust steam, supplemented by live steam, 
if necessary. The air passes out of the building through annular 
openings around the smokejacks, and also through openings 
between the wall plates and the roof rafters. These openings 
should extend entirely around the building. 

350. Window lighting. Skylights are undesirable because of 
preponderant disadvantages. The windows in the outer walls 
should be as large, wide and as high as safe construction will 
permit, the sill not more than 4 feet from the floor. Windows 
should be placed over the locomotive doors. Windows set into 
locomotive doors cause heavy maintenance charges on the doors. 

351. Electric lighting. Numerous lights should be provided 
to avoid shadows. Plugged outlets for incandescent lights in 
alternate spaces between pits should be provided. 

352. Piping. Pipes for air, steam and water supply should be 
provided, and where desired, piping for a washout and refilling 
system should be installed. Where this system is installed, the 
blow-off lines should be led to a central reservoir; where it is not 
used, the blow-off lines should be led outside the house. The 
steam outlet should be located near the front end of the boiler. 
The blow-off pipe, the air, the washout and refilling water and the 
cold water connections should be near the front end of the fire- 
box. Connections need only be provided in alternate spaces 
between stalls. 

353' Tools. There should ordinarily be facilities provided for 
hand tools and for the location of a few machine tools, prefer- 
ably electrically driven. 

354. Hoists. Hoists with differential blocks are generally 
used for handling heavy repair parts, and suitable provision 
should be made for supporting them. 

355. Turntables. The turntable should be long enough to 
balance the engine when the tender is empty. The deck form is 
preferable to the through form. Power should be provided at 
turntables having great service. Electric power is best and least 



§ 356. MISCELLANEOUS STEUCTURES AND BUILDINGS. 387 

expensive when it is available. Compressed air, supplied either 
by a pumping plant or by the locomotive itself, is sometimes 
used. The turntable pit should be thoroughly drained and prefer- 
ably paved. The circle wall should be of concrete or brick, with 
proper supports and fastenings for rails on the coping. The cir- 
cle rail should preferably bear directly on concrete base. The 
use of wood ties and tie-plates supported by masonry is desirable 
for the circle rail under some conditions. Easy access to the 
parts of a turntable for the oiling of bearings, painting and inspec- 
tion should be provided in the design of the turntable pit, unless 
ample provision is made in the turntable itself. 



LOCOMOTIVE COALING STATIONS. 

356. Hand shoveling. For roads of the smallest traffic, par- 
ticularly at terminals where locomotives lie overnight, hand 
shoveling direct from coal cars or from platforms provided with 
a jib crane and one-ton buckets, is the most economical. 

357. Locomotive crane. A locomotive crane, equipped with 
buckets, provides an efficient method of transferring coal from 
the coal car to a tender, particularly when the crane can be 
profitably employed at other times. 

358. Coaling trestle. This method requires a trestle with an 
approach not exceeding 5%, so that coal may fall from bottom- 
dumping cars into a pocket and then be discharged through 
chutes into the tender on a track on either side of the trestle. 
This method is satisfactory when two coalmg tracks are sufficient 
and when there is available space for the approach track. 

359. Coal conveyors. When more than two coaling tracks 
are essential, a conveyor system may be preferable. The coal is 
brought to the plant in bottom-dumping gondola cars, which 
dump the coal on to a conveyor which conveys it up and drops 
it into the bin, from which it may fall either into the tender or 
into an elevated conveyor car which runs it across a system of 
parallel tracks and dumps it into a tender, spotted there for the 
purpose. Incidentally, such a plant usually has also an ash 
conveyor onto which ashes are dumped from the engine. This 
conveyor carries the ashes to a place where the conveyor buckets 
dump them into a waiting gondola car, which when full is hauled 
away. 



388 



RAILROAD CONSTRUCTION. 



§360. 



360. Oil houses * should be fireproof and should be separated 
from other buildings. Above ground there should be a masonry 
building, 20'X40', or perhaps ]ess, with one fireproof door and 
one.or more windows, having wire glass. This room contains a 
row of pumps, one for each kind of t)il; also a series of inlet pipes 
in the floor leading to tanks in the basement. The floor should 
be 4 feet above the track rail outside and there should be a 



COMPOSITION ROOJ' 



PITCH 1:12 




16 



VI ABOUT I 
^FORCED CONCRETE BEAMS. 

20'o:: 



WIRE GLASS WINDOW 



i 



REINFORCED CONCRETE 






X 



OPENING FOR VENTIUAJION 
WITH WIRE NETTING. 



Us* 



■13 BRICK WALL. 



.SLIDING DOOR 
TIN CLAD 



\ FOR PjPING OIlJ 



SLOPE V2y 



FROM TANK CARS. 
ONE PIPE FOR EACH 
KIND OF OIL, 






::f" 



BASE OF RAIL 



CONCRETE 6s 






■ . - i -v-' . ■:-■■■- , ■ . •■y|- I ■■■- | : 
riNfiPRft fi" ».-— J 



*2'4* 



'-k 



Fig. 161. — Cross-section of Typical Oil-house. 

platform between the house and the track. The storage space 
for oil is entirely in the basement and includes the area under the 
floor and also the area under the platform. The height depends 
on the required storage space for tanks. A series of pipes, one 
for each kind of oil, pass through the outer vertical face of the 
platform, for the convenient emptying of tank cars into the 
storage tanks. The inlet pipes through the floor are only for 
small quantities of oil drawn from barrels. 

The delivery system from the storage tanks to the faucets 
should be such that the oil can be delivered quickly and measured 
automatically. The delivery should also be such that there will 



* Condensed from the Manual of the Am. Rwy. Eng. Assoc, 1915 Ed. 



§ 361. MISCELLANEOUS STRUCTURES AND BUILDINGS. 389 

be a minimum of dripping at the faucet and that the dripping 
may drain back to the storage tanks. Openings for ventilation 
should be provided above the level of the top of the tanks. 
Lighting, when required, should be by electricity and heating by 
steam . For fire protection purposes a live-steam line should be run 
to the oil storage space, controlled by a valve outside the house. 

361. Section tool houses. For small-traffic roads these 
should be 10'X14', the short dimension parallel with the track, 
with double swinging doors, swinging out on the end nearest the 
track. For roads of larger traffic the dimension parallel with the 
track should be 18 to 20 feet and the other dimension 12 to 14 
feet. There should be a sliding door, 8 feet in clear, at extreme 
end, on track side, to permit the storing of hand car. A sliding 
wooden shutter (instead of glass) may serve as a window for 
fair weather. It should not be made so convenient and com- 
fortable that it will become a lounging place for trackmen in 
stormy or wintry weather. The building should be of wooden 
frame construction, resting on wooden posts, or on masonry piers 
if the location can be considered permanent. Drop siding on 
the sides and some kind of prepared roofing will usually be most 
economical. 

362. Sand houses. Sand is a necessity in the operation of 
locomotives. Ordinarily it is obtained in a more or less moist 
and caked condition. It must be made thoroughly dry, so that 
it will flow readily through a pipe having sufficient slope. The 
plant consists essentially of a " wet storage bin," about 12'X16', 
which adjoins a " drying room " of about the same size. This 
room contains a screen, which is usually necessary to screen out 
the coarser particles; also a furnace to dry the sand, and a coal 
bin. For small traffic roads it may be sufficient to store the dry 
sand in a bin or even in buckets which are lifted by hand to the 
engine. For heavier traffic it may be justifiable to raise the 
sand to a bin or hopper whose lowest point is at least 22 feet 
above the rail, from which the sand may flow through a jointed 
pipe, somewhat similar to a water-supply pipe, directly into the 
sand box on the engine. Of course the bottom of the hopper 
must have sufficient slope so that the sand will always flow over 
it. The sand is hoisted to the hopper, either by some mechan- 
ical conveyor system, or is forced through a pipe by compressed 
air. The building should be located about 8 feet from the nearest 
track center. 



390 EAILEOAD CONSTRUCTION. § 363. 

363. Ash pits. A locomotive must dump the ashes from its 
ash pan at frequent intervals. The operation is usually timed 
to be done at terminal or divisional points, just before taking 
on water, coal, etc. These several plants are, therefore, grouped 
together in the yard. When there are no facilities for removing 
ashes by a conveyor at the same time that coal is being loaded on 
to the tender (see §§ 356-359), the ashes are dumped into a pit. 
The poorest roads dump them on the track under the engine, but 
this burns the ties, is dangerous, and is uneconomical, since they 
must be immediately removed. The simplest form of ash pit 
is made by dropping the ties about a foot, and then laying the 
rails on a pair of stringers about 12"X12". The stringers and 
ties must be covered with sheet iron to protect them from hot 
ashes. The capacity of such a pit is so small that the ashes must 
be removed quite frequently, which must usually be done by 
hand shoveling over the side of a gondola car on an adjacent 
track. The next development is a deeper pit, with concrete 
walls. Even then, the rails must be fastened to longitudinal 
wooden stringers, protected with sheet iron, or to cast-iron chairs 
which are embedded in the concrete. The ashes may be shov- 
eled out by hand after the locomotive has passed, or they may be 
dropped from the ash pan into buckets or small cars, which run 
on a narrow track at the bottom of the pit, and which may be 
lifted out by a jib crane. Another development is to widen the 
pit, running one rail on one wall and the other rail on a series 
of cast-iron columns. The pit has much greater capacity 
and the ashes may be hoisted out at any time, even if the loco- 
motive is still on the ash track. Great economy in the disposal 
of ashes is obtained when it is practicable to construct a de- 
pressed track, with its track center about 14 feet away from the 
ash track and 9 feet or more lower. The ashes may then be 
dropped onto a platform about 3 feet below the ash track, the 
platform extending to the top of a vertical retaining wall whose 
face is 5 ft. 6 ins. from the center of the depressed track, and from 
there the ashes are easily shoveled over the side of a gondola car 
placed on the lower track. No lifting of the ashes by hand is 
necessary. As in the previous plan, one rail of the ash track is 
supported by a wall, while the rail toward the depressed track is 
supported on cast-iron columns. The platform space is thus 
10 to 11 feet wide. 

Ashes should be quenched promptly after being deposited. 



§ 364. MISCELLANEOUS STRUCTURES AND BUILDINGS. §91 

SO as to reduce their heating effect even on metal and masonry. 
This requires a hose and a water supply. The pits should be 
graded so as to drain to a sump, which should have an overflow 
sufficiently above the bottom so that periodical cleaning out will 
suffice to keep the drain pipe from getting clogged with detritus 
from the ashes. 

SNOW STRUCTURES. * 

364. Snow-fences. Snow structures are of two distinct 
kinds — fences and sheds. A snow-fence implies drifting snow — ■ 
snow carried by wind — and aims to cause all drifting snow to be 
deposited away from the track. Some designs actually succeed 
in making the wind an agent for clearing snow from the track 
where it has naturally fallen. A snow-fence is placed at right 
angles to the prevailing direction of the wind and 50 to 100 feet 
away from the tracks. When the road line is at right angles to 
the prevailing wind, the right-of-way fence may be built as a 
snow-fence — high and with tight boarding. Hedges have some- 
times been planted to serve this purpose. When the prevailing 
wind is oblique, the snow fences must be built in sections where 
they will serve the best purpose. The fences act as wind break- 
ers, suddenly lowering the velocity of the wind and causing the 
snow carried by the wind to be deposited along the fence. 
Portable fences are frequently used, which are placed (by per- 
mission of the adjoining property owners) outside of the right- 
of-way. If a drift forms to the height of the portable fence the 
fence may be replaced on the top of the drift, where it may act 
as before, forming a still higher drift. When the prevailing 
wind runs along the track line, snow-fences built in short sec- 
tions on the sides will cause snow to deposit around them 
while it scours its way along the track line, actually clearing 
it. Such a method is in successful operation at some places on 
the White Mountain and Concord divisions of the Boston & 
Maine Railroad. Snow-fences, in connection with a moderate 
amount of shoveling and plowing, suffice to keep the tracks 
clear on railroads not troubled with avalanches. In such cases 
snow-sheds are the only alternative. 

365. Snow-sheds. These are structures which will actually 
keep the tracks clear from snow regardless of its depth outside. 
Fortunately they are only necessary in the comparatively rare 
situations where the snowfall is excessive and where the snow 



392 



KAILROAD CONSTRUCTION. 



§365. 



is liable to slide down steep mountain slopes in avalanches. 
These avalanches frequently bring down with them rocks, trees, 
and earth, which would otherwise choke up the road-bed and 
render it in a moment utterly impassable for weeks to come. 
The sheds are usually built of 12^' X 12'' timber framed in about 
the same manner as .trestle timbering; the " bents " are some- 
times placed as close as 5 feet, and even this has proved insuffi- 
cient to withstand the force of avalanches. The sheds are there- 




TYPICAL SHED 




P^^^ 



.fall shed 



Fig. 1G2. — Snow-sheds — Canadian Pacific Railroad, 

fore so designed that the avalanche will be defieded over them 
instead of spending its force against them. Although these 
sheds are only used in especially exposed places, yet their length 
is frequently very great and they are liable to destruction by fire. 
To confine such a fire to a limited section, ''fire-breaks" are 
made — i.e., the shed is discontinued for a length of perhaps 100 
feet. Then, to protect that section of track, a V-shaped de- 
flector will be placed on the uphill side which will deflect all 
descending material so that it passes ovei; the sheds. Solid crib 



§ 366. MISCELLANEOUS STRUCTURES AND BUILDINGS. 393 

work is largely used for these structures. Fortunately suitable 
timber for such construction is usually plentiful and cheap 
where these structures are necessary. Sufficient ventilation 
is obtained by longitudinal openings along one side immediately 
under the roof. "Summer" tracks are usually built outside 
the sheds to avoid the discomfort of passing through these semi- 
tunnels in pleasant weather. The fundamental elements in 
the design of such structures is shown in Fig, 162, which illus- 
trates some of the sheds used on the Canadian Pacific Railroad. 

FENCES. 

366. Wire fences. The following is condensed from the con- 
clusions adopted by the Amer. Rwy. Eng. Assoc, and incor- 
porated in their 1915 Manual. The recommended standard 
right-of-way fence is a wire fence, supported on wood or concrete 
posts. The wiring is to consist of five to nine longitudinal 
strands, with vertical stay wires spaced 12 to 24 inches apart. 
The longitudinal and vertical wires are to be locked or fastened 
with a mechanical lock which will prevent slipping either longi- 
tudinally or vertically, or the wires shall be electrically welded. 
The wire shall be galvanized so as to stand the following test: 
" The galvanizing shall consist of an even coating of zinc, which 
shall withstand one-minute immersion tests in a solution of 
commercial sulphate of copper crystals and water, the specific 
gravity of which shall be 1.185 and whose temperature shall be 
from 60° to 70° F. Immediately after each immersion the 
sample shall be washed in water and wiped dry. If the zinc is 
removed, or a copper-colored deposit formed at the end of the 
fourth immersion, the lot of material from which the sample is 
taken shaU be rejected. The fence shall be so fabricated as not 
to remove the galvanizing or impair the tensile strength of the 
wire." Electrically welded fencing should be galvanized after 
it has been fabricated. 

367. Types. Class A fence has 9 horizontal smooth wires 
whose spacing, starting at the ground, is 5, 4, 4|, 5, 5|, 6, 7, 8 
and 9 inches. To make it " hog-tight " the bottom space (5") 
is reduced to 3 inches and a barbed wire is inserted midway in 
the 3-inch space. The top and bottom smooth wires are No. 7 
gauge wire and the 7 intermediate wires are No. 9. The ver- 
tical stay wires, spaced 12 inches, shall be No. 9 gauge. 



394 EAILROAD CONSTRUCTION. § 368. 

Class B fence has 7 horizontal wires, with vertical wires spaced 
18 inches — all wires No. 9 gauge. The spacing, starting at the 
ground, is 7, 6^, 7, 7|, 8, 8^ and 9 inches. 

Class C fence has 5 horizontal wires, with vertical wires spaced 
24 inches — all wires No. 9 gauge. The spacing, starting at the 
ground, is 9, 7|, 8, 8| and 9 inches. 

Class D fence has 5 horizontal wires and no vertical stay 
wires, the wires being No. 9 gauge. The spacing, starting at 
the ground, is 10, 10, 10, 12 and 12 inches. 

368. Posts. End, corner, anchor and gate posts shall be at least 
8 feet long and set 3 feet 4 inches in the ground, even if blasting 
must be resorted to. Intermediate posts shall be at least 7 feet 
long and set 2 feet 4 inches in the ground. Where rock is en- 
countered at intermediate post holes, the intermediate posts, if 
of wood and not more than two in succession, may be set on sills, 
6"X6"X4'0", braced on both sides by braces 2"X6"X3'0". 
End, corner, anchor and gate posts, when of wood, shall be 8 
inches in diameter at the small end; when of concrete, shall be 
6 inches square at the top, 8 mches square at the base and shall 
be remforced with four |-inch square twisted rods. Intermediate 
wood posts shall be at least 4 inches in diameter at the small end; 
intermediate concrete posts shall be 4 inches thick at the top, 
5 1 inches at the bottom and reinforced with three (or four, 
dependmg on design) |-inch square twisted rods. 

369. Braces. End, corner, anchor and gate posts shall be 
braced by 4" X4" sawed lumber, or round posts at least 4 inches 
in diameter, or by concrete struts, 4"X4", reinforced with four 
5-inch twisted rods. The strut braces shall extend from a point 
about 12" below the top of the braced post to a point about 12" 
from the ground line at the adjacent intermediate post. In 
addition, a tie, made of a double strand of No. 9 galvanized soft 
wire, looped around the end, corner, anchor or gate post near the 
ground line, and around the next intermediate or line post about 
12 inches from the top, shall be put on and twisted until the top 
of the next intermediate or line post is drawn back about 2 
inches. 

370. Concrete posts. These are recommended. They may 
be made of one part of cement to four parts of pit gravel; or 
one part cement, two parts sand and four parts of stone of low 
absorption or screened gravel, the aggregate in any case being 
not less than j" nor more than |". The molds should be oiled 



§ 371. MISCELLANEOUS STRUCTURES AND BUILDINGS. 395 

or soaped and should be vibrated while concrete is poured to 
make the concrete more compact. The concrete should have a 
" quaking " consistency. The pouring should not be done out 
of doors in freezing weather. The concrete should not be ex- 
posed to sun, should be sprinkled every day for 8 or 10 days 
and should have 90 days for curing. They should be packed 
in sawdust or straw for shipment. Posts are usually made taper- 
ing and the cross-section is variously a square, a rectangle, or 
an isosceles triangle, the corners being chamfered. The rein- 
forcement should be placed not more than |" from the surface 
and should be wired by bands spaced about 12''. The fencing 
is sometimes fastened to the posts merely by wires tied tightly 
about the post or may be fastened to metal lugs which are 
embedded in the soft concrete during molding. 

371. Construction details. Wood posts shall be anchored 
by gaining and spildng two cleats, 2"X6"X2' 0", on the side of 
the post below the ground line. Staples shall be 1 inch long for 
hard wood, and 1^ inch for soft wood, made of No. 9 galvanized 
steel wire. They shall be driven diagonally with the grain of 
the wood, the top wires double-stapled. Staples, No. 9 wire, 
1 inch long, weigh 108 to th*e pound; 1^ inch long, 72 to the 
pound. 

Wire. No. 7 wire is 0,177 inch in diameter, weighs 439 pounds 
to the mile, or 12.05 feet to the pound. No. 9 wire is 0.148 inch 
in diameter, weighs 306 pounds to the mUe or 17.24 feet to the 
pound. Smooth wire is preferable to barbed. A heavy smooth 
wire or a plank should be used at the top of a barbed-wire fence. 
Wires shall be placed on the side of the post away from the 
track. Splicing shall be done as follows: " The ends of the 
wires shall be carried 3 inches past the splicing tools and 
wrapped around both wires backward from the tool for at least 
five turns, and after the tool is removed, the space occupied by 
it shall be closed by pulling the ends together." After erection, 
wood posts should be sawed off, on a one-fourth pitch, the high 
side being next to the wire and 2 inches above it. 

Gates should be hinged to swing away from the track; should 
be at least 12 feet wide and 4 feet 6 inches above the ground; 
should swing shut by gravity, and the free end should overlap 
the post so that it cannot be swung open toward the track. 
AU-metal construction is preferable, 



396 



RAILROAD CONSTRUCTION. 



§372. 



SIGNS. 

372. Highway signs. The crossing sign recommended by 
the Amer. Rwy. Eng. Assoc, is essentially as follows: Two 
wooden blades, 12 inches wide, 8 feet long, with mitered ends, 
are placed diagonally, with an angle of 50° between the blades, 
on an 8"X8"X16' 0" wooden post sunk 4 feet in the ground. 
The lower 9 feet is painted black, the upper 7 feet white. The 
blades are painted white with black letters and a |-inch black 
border around the blades. The border and lettering is on both 
sides. The lettering is Egyptian style 9 inches high with the 
exception of the connecting terms, as " for the " in the recom- 
mended sign, which should be 4 inches high. The recommended 
wording is " RAILROAD CROSSING " on one blade and 
" LOOK OUT FOR THE LOCOMOTIVE " on the other blade. 
The width of band of the letters is 1^ inches. If two railroads 
parallel each other within 400 feet, another blade marked 
" TWO CROSSINGS " should be added. The laws in some 
states prescribe what the lettering shall be. 

f 373. Trespass signs. . The specifications for these signs are 
applicable to many other public warnings which must be dis- 
played. A cast-iron plate, I inch thick, stiffened on the back by 
f-inch diagonal cast ribs and having the letters and border cast 
on the front by raising the surface about | inch, is set on an iron 
post 10 feet long, which is embedded 2 feet in a block of con- 
crete, which serves as foundation. The letters should be about 
2 inches high. A socket is cast on the rear side of the plate of 
such dimensions that it will set on the pipe and be fastened with 
a ^-inch set screw. The posts may be made of 2|-inch wrought 
iron pipe or of good second-hand boiler tubes, which should 
be filled with cement grout. The face of the letters and the 
borders should be painted black while the background is painted 
white. The tablet will usually be about 30 inches wide by 18 
inches high with rounded corners, although the dimensions will 
vary in accordance with the lettering to be placed on it. The 
following trespass signs frequently need to be displayed: 



RAILROAD PROPERTY 

TRESPASSING 

FORBIDDEN UNDER 

PENALTY OF LAW 



DANGER 
DO NOT 
TRESPASS ON THE 
RAILROAD 



§ 374. MISCELLANEOUS STRUCTURES AND BUILDINGS. 397 



DANGER 


DO NOT 


TRESPASS ON THIS 


BRIDGE 



374. Marker posts. Mile posts are most economically made, 
considering their durability, of skeletonized cast iron. The post 
is made up of two slabs of cast iron | inch thick, 8 feet long, the 
width tapering from 10 inches to 12 inches, the two slabs being 
formed in one piece and connected at intervals by |-inch webs 
and a top and bottom plate. They should be set 3 feet 6 inches 
in the ground and have a 4-inch slab of concrete or a heavy, flat 
stone as a base. The mile post numbers should be cast in raised 
letters on the face, the letters being 4| inches high. The two 
faces should be at right angles with each other and should each 
stand at an angle of 45° with the track. They should be set at 
least 8 feet from the gauge line of the nearest rail and 11 feet 
away, where it is practicable. The numbers should be so set 
that, on approach, the distance to the terminus or division point 
beyond will be indicated. 

The separating line between divisions is indicated to track 
men by an iron sign, called a division post, which is structurally 
the same as that of the mile posts. The two divisions are 
indicated by raised lettering on the faces of the posts. Of 
course there must be a variation in the lettering or numbering 
and a special post must be cast for each location of division post 
or mile post. 

Whistle signs are made similarly except that there is but one 
slab, suitably reinforced with ribs, and which faces in the desired 
direction. The letter W 7| ins. high is cast in raised letters 
near the top. The ring sign is made similarly by using the letter 
R. The separating line between sections is indicated to the 
trackmen by a cast-iron sign, called a section post, which is made 
similarly to the Trespass Signs, except that the tablet is much 
smaller. Such a sign will have two consecutive numbers, for 
example, 24-25, to indicate that the sign is at the separating 
line between section 24 and section 25. 

375. Bridge warning. When possible the headroom beneath 
overhead bridges is made at least 22 ft., which will make it 
safe for a trainman to stand on the top of a freight car which is 



398 



RAILROAD CONSTRUCTION. 



§375. 




§ 375. MISCELLANEOUS STRUCTURES AND BUILDINGS. 399 

passing under the bridge, but it is not always possible to have 
that amount of headroom. Under such circumstances, a 
warning for trainmen is necessary. These are made by sus- 
pending " ticklers," which are a series of ropes spaced 6 ins. 
apart which are suspended over the track at a sufficient distance 
from the bridge or tunnel so that the trainman shall have suffi- 
cient warning if he is struck by the dangling ropes. For a 
single track road the tickler may be suspended from a horizontal 
arm fastened to a pole planted at least 10 ft. from the track 
center, the arm being braced by a tie from the top of the pole 
and also by a short strut imderneath. When several tracks are 
to be spanned, two poles will be used and a catenary cable, 
between the tops of the poles, supports a horizontal cable by 
means of a pair of suspenders over each track. The standard 
on the Pennsylvania Railroad has 19 ticklers 6 ins. apart over 
each track. The bottoms of the several ropes are 6 ins. below 
the bottom fine of the bridge, the ropes having a length varying 
from 3 ft. to 5 ft. 3 ins. The ropes are fastened to | in. or f in. 
iron rods which swing on ring-bolts which are run through a 
wooden arm or hanger. The distance from the warning to the 
bridge or tunnel should be about 100 to 200 ft., depending 
somewhat on the grade, since that affects the time of the average 
freight train in passing the interval. 



CHAPTER XIII. 

YARDS AND TERMINALS. 

376. Value of proper design. A large part of the total cost of 
handling traffic, particularly freight, is that incurred at terminals 
and stations. It amounts to about 15% of the total operating 
expenses of a railroad. Freight arrives at any one of the hun- 
dreds of thousands of freight stations of the country, to be 
shipped to any other one of those stations. It may consist of 
a single package or several carloads of bulk freight. It may 
have to be transferred from car to car, or the car itself trans- 
ferred from road to road. In any case, the classification and 
handling of the freight, whether in individual packages or in 
carloads, is complicated and expensive and any device for reduc- 
ing the labor of handling such freight, or which saves time in 
doing it, has a definite money value. Assume that an improve- 
ment in the design of the yard will permit a saving of the use 
of one switching engine, or for example, that the work may 
be accomplished with three switching engines instead of four. 
Assuming a daily cost of $25, we have in 313 working days an 
annual saving of $7825, which, capitalized at 5%, gives $156,500, 
enough to reconstruct any ordinary yard. 

377. Definitions. (Compiled from Proc, Amer. Rwy. Eng. 
Assoc.) 

Yard. A system of tracks within defined limits provided for 
making up trains, storing cars, and other purposes, over which 
movements not authorized by timetable or by train order may 
be made, subject to prescribed signals, rules and regulations. 

Receiving yard. A yard for receiving trains. 

Classification yard. A yard in which cars are classified or 
grouped in accordance with requirements. 

Departure or forwarding yard. A yard in which cars are 
assembled in trains for forwarding. 

Storage yard. A yard in which cars are held awaiting dis- 
position. 

Summit or hump yard. A yard in which the movement of 
cars is accomplished by pushing them over a summit, beyond 
which they run by gravity. 

400 



§ 378. YARDS AND TERMINALS. 401 

Body track. Each of the parallel tracks of a yard, upon 
which cars are switched or stored. 

Ladder track. A track connecting successively the body 
tracks of a yard. 

Lead track. An extended track connecting either end of a 
yard with the main track. 

Running track. A track reserved for movement through a yard. 

Crossover track. A track connecting two adjacent tracks. 

Stub track. A track connected with another at one end only. 

Spur track. A stub track of indefinite length diverging from 
a main line or track. 

House track. A track alongside of (or entering) a freight 
house; used for cars receiving or delivering freight at the house. 

Team track. A track where freight is transferred directly 
between cars and wagons. 

378. General principles. It should be recognized at the start 
that at many places an ideally perfect yard is impossible, or at 
least impracticable, generally because ground of the required 
shape or area is practically unobtainable. But there are some 
general principles which may and should be followed in every 
yard and other ideals which should be approached as nearly as 
possible. Nevertheless every yard is an independent problem. 

Body tracks should be spaced 13 feet to 14 feet center to center, 
under ordinary conditions, and where they are parallel to main 
track or other important running track, the first body track 
should be spaced not less than 15 feet center to center from 
such main or important track. 

Ladder tracks should be spaced not less than 15 feet center 
to center from any parallel track. Frogs of greater angle than 
No. 8 should not be generally used, and the angle between the 
ladder track and body tracks will be governed by the distance 
on ladder tracks required for a turnout. 

To faciUtate train movements the connections of lead tracks 
with the main track should be interlocked. 

Running tracks should be provided for movements in either 
direction to enable yard engines to pass freely from one position 
of the yard to the other; also to enable road and yard engines 
to pass to and from the engine house and other points where 
facilities are provided. 

Crossover tracks should be located at most convenient points 
where they will least interfere with regular movements. 



402 RAILROAD CONSTRUCTION. § 378. 

Caboose tracks should be so located, where conditions permit, 
that cabooses can be placed on and removed from trains in the 
order of their arrival, and should be so constructed that cabooses 
can be dropped by gravity onto the rear of trains made up for 
departure. 

Scale tracks should be so located that weighing can be done 
with least delay and without drilling over scale. Where many 
cars are to be weighed they should pass separately over the scale 
by gravity, being weighed while in motion. 

Coaling, ashpit, sand and engine tracks should be located 
on the route leading to and from the engine house and should 
provide sufficient storage for the reception of engines by the 
hostler. They should be so arranged that water, coal and sand 
can be taken and ashes disposed of in convenient rotation, and 
that switching engines may clean fires, take coal, water and 
sand and pass around waiting engines. 

Bad-order tracks. Where cars are classified, one or more 
classification tracks, easy of access, should be provided for setting 
off cars in bad order, from which they may be readily removed 
to the repair tracks. 

Repair tracks should preferably be connected at both ends 
and have a maximum capacity of about 15 cars each, spaced 
alternately 16 feet and 24 feet center to center and be connected 
conveniently to bad-order tracks. 

Icing tracks should be so located that the work of shifting 
out, icing and classifying cars for movement can be performed 
in the least time. 

The Main tracks of both single and double track roads 
should be located, if it is possible to so arrange, on the outside 
of yard, and the engine house, coaling station, etc., should be 
centrally located. 

The Coach cleaning yard should be located near the terminal 
station. The tracks should be of sufficient, length to hold full 
trains, with a car cleaners' repair and supply building adjacent 
thereto. 

Roadways. Where the freight house is on one side and a 
wall on the other, the minimum width of roadway should be 
30 feet; but where a freight house is on one side and a team 
track or another freight house is on the other, the minimum 
clear width of roadway should be 40 feet. 

A Transfer Station should be located at a point where traffic 



§378. 



YARDS AND TERMINALS. 



403 



is concentrated 
and where a ne- 
cessity exists for 
consolidating 
freight into a 
less number of 
cars for move- 
ment to a cer- 
tain destination, 
or for separating 
and reloading 
freight into a 
greater number 
of cars or into 
system cars for 
further move- 
ment to final 
delivery. 

The car capac- 
ity of freight 
tracks should be 
computed on the 
basis of 42 feet 
for each car. 

Frogs. Al- 
though not ab- 
solutely neces- 
sary, there is an 
advantage in 
having all frog 
numbers and 
switch dimen- 
sions uniform. 
No. 8 frogs are 
recommended. 
Sharper - angled 
frogs make easier 
riding, lessresist- 
ance and less 
chance of derail- 
ment, but on the 




404 



EAILROAD CONSTRUCTION. 



378. 




quire 
leads 



other hand re- 
longer 
and more 
space. No. 7 
and even No. 6 
frogs are some- 
times used on 
account of econ- 
omy of space, 
but they have 
the disadvan- 
tages of greater 
tractive resist- 
ance, greater 
wear and tear on 
track and rolling 
stock, and great- 
er danger of de- 
railment. 

The design of 
an existing yard 
is best studied 
by first picking 
out the ladder 
tracks and the 
through tracks 
which lead from 
one division of' 
the yard to an- 
other. These 
are tracks which 
must always be 
kept open for 
the passage of 
trains, in contra- 
distinction to the 
tracks on which 
cars may be left 
standing, even 
though it is only 
for a few mo- 



§379. 



Yards ai^d Terminals. 



405 



ments, while drilling 
is being done. Such 
a set of tracks, which 
may be called the 
skeleton of the yard, 
is shown by heavy 
lines in Fig. 164. 
Each line indicates a 
pair of rails. The 
tracks of the storage 
yards are shown by 
the lighter lines. 

379- Minor freight 
yards. Fig. 165 il- 
lustrates a freight 
yard on the New 
York harbor front to 
which cars are brought 
on floats. Ten team 
tracks, for the trans- 
fer of freight between 
cars and teams have 
been provided in a 
very limited space. 
Great ingenuity is 
often required to ob- 
tain the desired facil- 
ities without the use 
of excessively sharp 
curvature. The lim- 
iting radius which 
will permit cars to 
pass a curve with- 
out adjacent corners 
touching is about 
175 feet. Extension 
coupler bars, al- 
though inconvenient, 
will make possible 
the use of still sharper 
curves. 




406 EAILROAD CONSTRUCTION. § 380. 

380. Hump yards. The operation of hump yards makes it 
possible to develop the necessary potential energy for car move- 
ment by a switching engine with the maximum of economy, 
while the classification is accomplished in the minimum of time. 
The cars are pushed up the grade and over the summit, from 
which they begin immediately to descend on a grade which is 
preferably 4%. As each " cut" of one or more cars reaches 
the 4% grade, gravity accelerates its motion and it separates 
automatically from the cars behind it. Each cut then passes 
down the ladder track until it reaches the particular body track 
on which it is desired to be run. Grades. In Chapter XVI, 
it is elaborated that track resistance is greater in winter than 
in summer, and also that it is much greater on switch tracks 
than on straight imbroken track. The difference between cold- 
weather and warm-weather resistance is so great that the length 
or rate of the acceleration grade required to furnish the neces- 
sary energy varies with the temperature or climate. The Amer. 
Rwy. Eng. Assoc, in 1917 adopted three typical profiles for 
humps, designed for " cold, moderate and warm climates." 
The designs also include the location of track.scales (see § 382) 
which modify the grading. Some of the grades are only nominal 
since the transition from one grade to another requires such 
long vertical curves (see §§ 84-87) that they occupy the entire 
length of the nominal grades, and the profile over the hump, 
and for some distance beyond, consists of a series of compounded 
vertical curves. For example, the profile which is recommended 
for warm climates is shown in Fig. 166. Nominally the summit 
is reached by a short length of 1.5% grade, with a level grade 
at the summit followed by 25 feet of 4% down-grade and then 
77 feet of 0.6% down-grade, on which is located the track scales. 
But Fig. 166 shows that a vertical curve of 674 feet radius starts 
from the 1.5% grade, is tangent to the level grade at the summit, 
and reaches the 4% grade, where it reverses into an up-curving 
674-ft. curve which joins the 0.6% grade. The recommended 
profiles for " moderate and cold " climates can be constructed, 
similar to that in Fig. 166 from the data in the tabular form. 
Note that the length or steepness of the acceleration grades 
and of the ladder track is increased as the climate is colder. If 
the grades are too low the cars will not reach their desired desti- 
nations; if too steep, there must be an unnecessary use of brakes 
or a destructive bumping of cars on the body tracks. Never- 



§380. 



YARDS AND TERMINALS. 



407 



Locality. 


Hump- 
level 
length. 


Accel, 
grade. 


Scale 
grade. 


Accel, 
grade. 


Ladder 
track. 


Radius 

vert. 

curve. 


Warm climate 
Moderate ' ' 
Cold 


18.58' 
28.75' 
39 . 30' 


25' 4% 
37.5' 4 
50' 4 


77' 0.6% 
89' 0.8 
100' 1.0 


100' 2.5% 
100' 3.0 
100' 4.0 


1.0% 

1.25 

1.5 


674' 
1040' 
1428.6' 



theless, as will be shown in § 438, the actual resistance of cars 
through switches is so variable that an excess of power must be 
provided to prevent the stalling of some cars before they reach 
their destination. The grade from the receiving track to the 
hump should be such that one engine can push the maximum 
train over the hump. Since empty cars have a greater tractive 
resistance per ton than loaded cars, they require a steeper grade 
to maintain the same velocity, and, therefore, when tracks are 
set aside for the use of empty cars, the grade leading to such 
empty tracks should be increased if possible. Operation. Ta 
operate such a hump efficiently, the yard clerk makes up a triple 
(or quadruple) list for each freight train arriving at the yard 
for distribution. One of these lists is given to the man cutting 
off the cars at the top of the hump, and one to the towerman, 
if the switches are operated from the tower, or one to each switch 
tender if the switches are hand-operated. Each list contains 
in the first column the consecutive number of the cut, in the 
second column the number of the track on which that cut of 
cars is to be placed, and in the third column the number of cars 
cut. Cut No. 1 is the first car (or cars) to go over the hump. 
A brakeman, or " rider," accompanies each car, or group of cars. 
To avoid the great waste of time required for these riders to 
walk back to the hump, it has been found economical in some 
large yards to have a track for the exclusive use of a car, especi- 
ally fitted for easy jumping on or off, operated, perhaps, by a 
switching engine, or possibly by gasoline, which picks up the 
riders and carries them back to the hump. The aggregate 
time saved justifies the expenditure. The scale grade has been 
designed in each case so that each car will pass over the scale 
with a maximum velocity of four miles per hour, which means 
that the car shall be entirely on the scale platform for a minimum 
time of three seconds. Although the grade over the scales may 
be as high as 1% for motion weighing, the weighing mechanism 
must be installed on a level plane and the weighing rails are 
blocked up to the desired grade. 



408 



RAILROAD CONSTRUCTION. 



§381. 



381. Ladder tracks. Twenty-seven types of ladder tracks 
are shown in the 1917 Committee report to the A. R. E. A., but 
nearly one-half of the ladders reported in actual use belong to 
type a, Fig. 166a, and about one-half of the remainder belong to 
types h and c. The other twenty-four types are chiefly expansions 
and developments of the three types shown. Note that in types 
a and c, the switches are, in each case, in a straight line along 




Fig. 166a. — Types of Ladder Tracks. 

one of the tracks, which simpHfies the working of the switches, 
whether they are worked from a tower or on the ground by hand. 
382. Track scales. I'he standard design for a hump yard, 
§ 380, shows a track-scale grade, as an integral part of the design, 
located just beyond the hump. It has been found that it is 
practicable to weigh cars with suflicient accuracy while the cai:s 
are in motion, provided the speed does not exceed 4 miles per 
hour, or 5;87 feet per second, and provided that the lergth of 
the scale is such that the car is entirely and alone on the scale 
for a minimum of three seconds. This condition will be ful- 
filled when the scale is 17.6 feet longer than the distance from 
front to rear axle of the car. Scales with lengths of 50, 56 and 60 
feet are considered standard. The sensibility reciprocal is the 
weight required to be added or removed from the live rails to 
turn the beam frorr a horizontal position of equilibrium in the 



§ 383. YARDS AND TERMINALS. 409 

center of the trig loop to a position of equilibrium at either 
limit of its travel; such weight shall not exceed 50 lbs. in any 
case. The tolerance to be allowed on the first field test, after 
installation corrections, of all new railroad track scales, shall 
not exceed -gV of !%> or 50 lbs. per 100,000 lbs. for any position 
of the test-car load on the scale. The minimum test-car load 
shall be 30,000 lbs. Location. The scale should be elevated 
above the other tracks of the yard so that surface drainage 
shall not drain into the pit. The location of the scale near a 
hump summit fits in with this requirement. The foundations 
should be made of concrete. The finished floor of the pit should 
be at least 7 feet below the base of the rails; the floor should 
be at least 6 inches thick and as much thicker as a soft sub- 
soil might demand. The concrete of the walls and floor should 
be effectively waterproofed to exclude sub-soil water. A sump, 
with provision for drainage outfall, should be provided to dis- 
pose of any rainfall or other drainage which might accumulate 
in the pit. Approach. There should be at least 50 feet of 
tangent track on each approach. The approach tracks should 
be carried on approach walls or piers extending 15 to 25 feet from 
the end walls of the pit, so that accurate line and surface of the 
approach tracks is maintained and so that the approach rails 
may be absolutely anchored against creeping. Dead rails, off- 
setted 16" from the live rails, will carry cars over the scale pit, 
when so desired, without any stress or influence on the scale 
mechanism. One dead rail may be supported on the side wall 
of the pit and the other on pedestals or on transverse floor beams 
which are spaced (usually) 2' 6" and which are independent of 
the weighing platform. Details must conform to the some- 
what varying plans of various manufacturers. 

383. Transfer cranes. These are almost an essential feature 
for yards doing a large business. The transportation of built- 
up girders, castings for excessively heavy machinery, etc., which 
weigh 5 to 30 tons and even more, creates a necessity for 
machinery which will easily transfer the loads from the car to 
the truck and vice versa. An ordinary " gin-pole " will serve 
the purpose for loads which do not much exceed 5 tons. A 
fixed framework, covering a span long enough for a car track and 
a team space, with a trolley traveling along the upper chord, is 
the next design in the order of cost and convenience. Increasing 
the span so that it covers two car tracks and two team spaces 



410 



RAILROAD CONSTRUCTION. 



§384. 




will very materially in- 
crease the capacity. Mak- 
ing the frame movable so 
that it travels on tracks 
which are parallel to the 
car tracks, giving the 
frame a longitudinal mo- 
tion equal to two or three 
car lengths, and finally 
operating the raising and 
traveling mechanism by 
power, the facility for 
rapidly disposing of heavy 
articles of freight is greatly 
increased. Of course only 
a very small proportion 
of freight requires such 
handling, and the business 
of a yard must be large or 
perhaps of a special char- 
acter to justify and pay 
for the installation of such 
a mechanism. A transfer 
crane, evidently of the 
fixed type, is indicated in 
Fig. 165. 

384. Engine Yards or 
Terminals. These should 
be located so that there 
is easy access to both the 
main line and the various 
yards, with the fewest 
possible reverse or con- 
flicting movements. The 
yards must contain all 
the tracks, buildings, 
structures, and facilities 
which are necessary for 
the maintenance, care, 
and storage of locomotives 
and for providing them 



Fie. 167. — Engine Yakd and Shops, 
Ukbana, III. 



/ELEVATOR 



ri^sr 



-H 



ifL><! 



::i 



'g^ . 



.J^SL 



PLATFORM 



^ELEVATOR 



100 



B 



200 



300 feet 








_« i 1 • •_ 



' eAQQjTGE ahS expRtss plaVform' 



PLATFORM SHELTE:R-N^ 



TRAIN FLpOR-^ I 1 

■VEL fIoR BAgIgAGE, (ilAIL AftlD 



■pR BAaGAGE, MAILAND 



SECTIONAL ELEVATION 



' 




Fig. 167a. — Dead-end Passenger Terminal. 



§ 384a. YARDS AXD TERMINALS. 410a 

with all needed supplies. The supplies are fuel, water, sand, 
oil, waste, tallow, etc. Ash-pits are generally necessary for the 
prompt and economical disposition of ashes; engine-houses 
are necessary for the storage of engines and as a place where 
minor repairs can be quickly made. A turntable is another all 
but essential requirement. The arrangement of all these 
facilities in an engine yard should properly depend on the form 
of the yard. In general they should be grouped together and 
should be as near as possible to the place where through 
engines drop the trains just brought in and where they couple 
on to assembled outgoing trains, so that all unnecessary run- 
ning light may be avoided. Switching engines should be able 
to dump ashes, take their supplies and pass around waiting 
road engines. In Figs. 164, 167 and 167a are shown designs 
which should be studied with reference to the relative arrange- 
ment of the yard facilities. 

384a. Passenger terminals. The word terminal is applied 
not only to a railway station at an actual terminus, beyond 
which no trains axe run, but also to an important intermediate 
station, where trains are assembled, assorted, classified and 
reluyed. The two types are called dead-end and through term- 
inals. The Am. Rwy. Eng. Assoc, has adopted standard plans 
for each of these two types. Even when there is good repson 
for modifying some of the details, certain principles should be 
observed, as far as possible. Some of these principles, which 
sometimes apply to both types, are as follows : 

(a) Dead-end terminals. See Fig. 167a. The track level and 
train floor is raised above the street level, so as to permit any 
intersecting cross streets to Tun under the tracks. A ramp on 
an easy grade is indicated in the section of the terminal building. 
Each platform serves a pair of tracks whose centers are 28' 
apart. Allowing 5' 6" from the track center to the edge of the 
platform, the platforms themselves are 17' wide. The length 
of the platforms vary from about 600 to over 1100 feet, but the 
length and their number should depend on the extent of business 
to be handled. The intermediate platforms are protected for 
about 500 feet of their length by " butterfly " roofs supported 
on a line of columns, the roofs draining inward to longitudinal 
gutters in the center, which discharge into leaders alongside the 
columns. Two sets of ladder tracks, with single or double slips 
(§ 314) connect with each one of the platform tracks, so that 



§ 384ar YARDS AND TERMINALS. 411 

either main track may be directly connected with any platform 
track. The space under the tracks, and at the street level, 
is utilized for rooms for baggage, mail and express, which are 
carried to the track level by elevators, one to each platform. 
The coach cleaning yard has a series of parallel tracks 13' to 18' 
apart c.c. between a pair of parallel ladder tracks. The engine 
yard has a sand and coaling station, ash-pit with ash-car track, 
oil-house, water tank, engine supply house, turntable, shop 
and shop-yard tracks. 

(b) Through terminals. See Fig. 1676, As above, the train 
floor level is above the street level. A passage way runs trans- 
versely under the tracks, from which two pairs of stairways run 
in each direction to the station platforms. As before, the bag- 
gage, mail and express rooms are on the street level, under the 
tracks, and connect with the platforms by elevators. The two 
middle tracks are main tracks, which may be used by any trains, 
through freight or passenger, which do not stop at the station. 
The two platforms each have two tracks, one on each side. 
The three tracks of each group run into one main track for each 
direction of movement, at either end of each platform. The 
two main tracks are connected by two crossovers, arranged for 
direct and reversed movement. The figure also shows an 
arrangement of switches for the junction of a branch line with 
the main line, with three car-yard tracks in the Y of the junc- 
tion. 



CHAPTER XIV. - 

BLOCK SIGNALING. 
GENERAL PRINCIPLES. 

385. Two fundamental systems. The growth of systems of 
block signaling has been enormous within the last few years — 
both in the amount of it and in the development of greater per- 
fection of detail. The development has been along two general 
lines: (a) the manual, in which every change of signal is the 
result of some definite action on the part of some signalman, but 
in which every action is so controlled or limited or subject to 
the inspection of others that a mistake is nearly, if not quite, 
impossible; (6) the automatic, in which the signals are oper- 
ated by mechanism, which cannot set a wrong signal as long as the 
mechanism is maintained in proper order. The fundamental 
principles of the two systems will be briefly outlined, after which 
the chief details of the most common systems will be pointed out. 

386. Manual systems. Small traffic roads are usually operated 
on the basis of the "train-order system." A "train dispatcher" 
controls the movement of every train on his division and telegraphs 
orders to men (who are frequently station agents) at various points 
along the line, who transmit these orders to the trainmen as the 
trains reach these points. A train-order signal station, whether at 
a regular traffic station or in a special cabin, has " train-order 
signals " which, when in the stop position, inform the engineman 
and conductor that they are to receive orders at the telegraph 
office; the clear position informs them that there are no orders 
for them. When more than one train is allowed on a single 
track between two consecutive train-order stations, the engine- 
man and conductor of each train has strict orders with reference 
to the other train, for example, that the trains are to pass at 
some siding where there is no telegraphic station. A very strict 
code of rules has been developed which, when literally followed, 
ensures safety of operation, but these rules cannot eliminate 
the human element, or the liability of personal negligence 
or error. When such a system is applied to a double-track 
road, or even to a single-track road, with train-order signal 

412 



§ 387. BLOCK SIGNALING. 413 

stations located so frequently that only one train will be al- 
lowed between two consecutive offices at . once, it virtually 
becomes a block system even though it is not called such. 
When such a system is adhered to rigidly, it is called an absolute 
block system. But when operating on this system, a delay of 
one train will necessarily delay every other train that follows 
closely after. A portion, if not all, of the delay to subsequent 
trains may be avoided, although at some loss of safety, by a 
system of permissive blocking. By this system an operator 
may give to a succeeding train a "clearance card" which per- 
mits it to pass into the next block, but at a reduced speed and 
with the train under such control that it may be stopped on 
very short notice, especially near curves. One element of the 
danger of this system is the discretionary power with which it 
invests the signalmen, a discretion which may be wrongfully 
exercised. A modification (which is a fruitful source of colli- 
sions on single-track roads) is to order two trains to enter a 
block approaching each other, and with instructions to pass 
each other at a passing siding at which there is no telegraph- 
station. When the instructions are properly made out and 
literally obeyed, there is no trouble, but every thousandth or 
ten thousandth time there is a mistake in the orders, or a mis- 
understanding or disobedience, and a collision is the result. The 
telegraph line, a code of rules, a corps of operators, and sig- 
nals under the immediate control of the operators, are all that 
is absolutely needed for the simple manual system. 

387. Development of the manual system. One great diffi- 
culty with the simple system just described is that each operator 
is practically independent of others except as he may receive 
general or specific orders from a train-dispatcher at the division 
headquarters. Such difficulties are somewhat overcome by a 
very rigid system of rules requiring the signalmen at each station 
to keep the adjacent signalmen or the train-dispatcher in- 
formed of the movements of all trains past their own stations. 
When these rules (which are too extensive for quotation here) 
are strictly observed, there is but little danger of accident, and 
a neglect by any one to observe any rule will generally be appar- 
ent to at least one other man. Nevertheless the safety of trains 
depends on each signalman doing his duty, and a little careless-v 
ness or forgetfulness on the part of any one man may cause an 
accident. The signaling between stations may be done by 



414 EAILROAD CONSTRUCTION. § 387. 

ordinary telegraphic messages or by telephone, but is frequently 
done by electric bells, according to a code of signals, since these 
may be readilj'- learned by men who would have more difficulty 
in learning the Morse code. 

In order to have the signalmen mutually control each other, 
the "controlled manual" system has been devised. The first 
successful system of this kind which was brought into exten- 
sive use is the "Sykes" system, of which a brief description 
is as follows: Each signal is worked by a lever; the lever is 
locked by a latch, operated by an electro-magnet, which, with 
other necessary apparatus, is inclosed in a box. When a signal 
is set at danger, the latch falls and locks the lever, which cannot 
be again set free until the electro-magnet raises the latch. The 
magnet is energized only by a current, the circuit of which is 
closed by a ''plunger" at the next station ahead; just above 
the plunger is an ''indicator," also operated by the current, 
which displays the words clear or blocked. (There are varia- 
tions on this detail.) When a train arrives at a block station 
(A), the signalman should have previously signaled to the station 
ahead (B) for permission to free the signal. The man ahead (B) 
pushes in the "plunger" on his instrument (assuming that the 
previous train has already passed him), which electrically opens 
the lock on the lever at the previous station (A). The .signal 
at A can then be set at "safety." As soon as the train has 
passed A the signal at A must be set at " danger," A further 
development is a device by which the mere passage of the train 
over the track for a few feet beyond the signal will automati- 
cally throw the signal to "danger." After the signal once goes 
to danger, it is automatically locked and cannot be released 
except by the man in advance (B), 'who will not do so until the 
train has passed him. The "indicator" on i?'s instrument 
shows "blocked" when A's signal goes to danger after the train 
has passed A, and B's plunger is then locked, so that he can- 
not release A's signal while a train is in the block. As soon as 
the train has passed A , B should prepare to get his signals ready 
by signaling ahead to C, so that if the block between B and C 
is not obstructed, B may have his signals at "safety" so that 
the train may pass B without pausing. The student should 
note the great advance in safety made by the Sykes system; 
a signal cannot be set free except by the combined action of 
two men, one the man who actually operates the signal and 



§ 388. block: signaling. 415 

the other the man at the station ahead, who frees the signal 
electrically and who by his action certifies that the block im- 
mediately ahead of the train is clear. 

A still further development makes the system still more " auto- 
matic" (as described later), and causes the signal to fall to dan- 
ger or to be kept locked at danger, if even a single pair of wheels 
comes on the rails of a block, or if a switch leading from a main 
track is opened. 

388. Permissive blocking. "Absolute" blocking renders ac- 
cidents due to collisions almost impossible unless an engineer 
runs by an adverse signal. The signal mechanism is usually 
so designed that, if it gets out of order, it will inevitably fall to 
"danger," i.e., as described later, the signal-board is counter- 
balanced by a weight which is much heavier. If the wire breaks, 
the counterweight will fall and the board will assume the hori- 
zontal position, which always indicates " danger." * But it some- 
times happens that when a train arrives at a signal-station, the 
signalman is unable to set the signal at safety. This may be 
because the previous train has broken down somewhere in the 
next block, or because a switch has been left open, or a rail has 
become broken, or there is a defect of some kind in the electrical 
connections. In such cases, in order to avoid an indefinite 
blocking of the whole traffic of the road, the signalman may 
give the engineer a "caution-card" or a "clearance card," 
which authorizes him to proceed slowly and with his train under 
complete control into the block and through it if possible. ^ If 
he arrives at the next station without meeting any obstruction 
it merely indicates a defective condition of the mechanism, 
which will, of course, be promptly remedied. Usually the next 
section will be found clear, and the train may proceed as usual. 
On roads where the "controlled manual" system has received 
its highest development, the rules for permissive blocking are 
soi rigid that there is but little danger in the practice, unless 
there is an absolute disobedience of orders. 

389. Automatic systems. By the very nature of the case, 
such sj^stems can only be used to indicate to the engineers of 
trains something with reference to the passage of previous 

* This was written on the basis of the older system, in which the sema- 
phore swings through the lower right-hand quadrant. The most recent 
practice swings the semaphore through the upper right-hand quadrant. 
A break in the wire holding the semaphore vertical will cause it to f^U 

to horizoutal position without the aid of a counterweight, 



416 RAILROAD CONSTRUCTION. § 389. 

trains. The complicated shifting of switches and signals which 
is required in the operation of yards and terminals can only be 
accomplished by " manual " methods, and the only automatic 
features of these methods consist in the mechanical checks 
(electric and otherwise), which will prevent wrong combina- 
tions of signals. But for long stretches of the road, where it 
is only required to separate trains by at least one block length, 
an automatic system is generally considered to be more relia- 
ble. As expressed forcibly by a railroad manager, "an auto- 
matic system does not go to sleep, get drunk, become insane, 
or tell lies when there is any trouble," The same cannot always 
be said of the employes of the manual system. 

The basic idea of all such systems is that when a train passes 
a signal-station (A), the signal automatically assumes the "dan- 
ger" position. This may be accomplished electrically, pneu- 
matically, or eA^en b}'- a direct mechanism. When the train 
reaches the end of the block at B and passes into the next one, 
the signal at B will be set at danger and the signal at A will be 
set at safet}^ The lengths of the blocks are usually so great 
that the only practicable method of controlling from B a 
mechanism at A is by electricity, although the actual motive 
power at A may be pneumatic or mechanical. At one time 
the current from A to B was run only through wires. This 
method has the very positive advantage of reliability, definite 
resistance to the current, and small probability of short-circuit- 
ing or other derangement. But now all such systems use the 
rails for a track circuit and this makes it possible to detect the 
presence of a single pair of wheels on the track anywhere in the 
block, or an open switch, or a broken rail. Any such circum- 
stances, as well as a defect in the mechanism, will break or 
short-circuit the current and will cause the signal to be set at 
danger. To prevent an indefinite blocking of traffic owing to 
a,,signal persistently indicating danger, most roads employing 
such a system have a rule substantially as follows : When a train 
finds a signal at danger, after waiting one minute (or more, 
depending on the rules), it may proceed slowly, expecting to 
find an obstruction of some sort; if it reaches the next block 
without finding any obstruction and finds the next signal clear, 
it may proceed as usual, but must promptly report the case to 
the superintendent. Further details regarding these methods 
will be given later. See § 394. 



§ 390. BLOCK SIGNALING. 417 

390. " Distant " signals. The close running of trains that 
is required on heavy-traffic roads, especially where several 
branches combine to enter a common terminal, necessitates the 
use of very short blocks. A heavy train running at high speed 
can hardly make a "service" stop in less than 2000 feet, while 
the curves of a road (or other obstructions) frequently make 
it difficult to locate a signal so that it can be seen more than a 
few hundred feet away. It would therefore be impracticable 
to maintain the speed now used with heavy trains if the engi- 
neer had no foreknowledge of the condition in which he will 
find a signal until he arrives within a short distance of it. To 
overcome this difficulty the '^ distant" signal was devised. This 
is placed about 1800 or 2000 feet from the ''home" signal, and 
is interlocked with it so that it gives the same signal. The dis- 
tant signal is frequently placed on the same pole as the home 
signal of the previous block. When the engineer finds the 
distant signal ''clear," it indicates that the succeeding home 
signal is also clear, and that he may proceed at full speed and 
not expect to be stopped at the next signal; for the distant 
signal cannot be cleared until the succeeding home signal is 
cleared, which cannot be done until the block succeeding that 
is clear. A clear distant signal therefore indicates a clear track 
for two succeeding blocks. When the engineer finds the distant 
signal blocked, he need not stop (providing the home signal is 
clear). It simply indicates that he must be prepared to stop 
at the next home signal and must reduce speed if necessary. 
It may happen that by the time he reaches the succeeding home 
signal it has already been cleared, and he may proceed without 
stopping. This device facilitates the rapid running of trains, 
with no loss of safety, and yet with but a moderate addition to 
the signaling plant. 

391. "Advance " signals. It sometimes becomes necessary 
to locate a signal a few hundred feet short of a regular passen- 
ger-station. A train might be halted at such a signal because 
it was not cleared from the signal-station ahead — perhaps a 
mile or two ahead. For convenience, an "advance" signal 
may be erected immediately beyond the passenger-station. 
The train will then be permitted to enter the block as far as 
the advance signal and may deliver its passengers at the station. 
The advance signal is interlocked with the home signal back 
of it, and cannot be cleared until the home signal is cleared and 



418 RAILROAD CONSTRUCTION. § 392. 

the entire block ahead is clear. In one sense it adds another 
block, but the signal is entirely controlled from the signal station 
back of it.^ 

MECHANICAL DETAILS. 

308. Signals. The primitive signal is a mere cloth flag. A 
better signal is obtained when the flag is suspended in a suit- 
able place from a fixed horizontal support, the flag weighted 
at the bottom, and so arranged that it may be dra^vn up and 
out of sight by a cord which is run back to the operator's office. 
The next step is the substitution of painted wood or sheet metal 
for the cloth flag, and from this it is but a step to the standard 
semaphore on a pole, as is illustrated in Fig. 168. The simple 
flag, operated for convenience with a cord, is the signal em- 
ployed on thousands of miles of road, where they perhaps make 
no claim to a block-signal system, and where the trains are runi 
on the " train-order system." 

Semaphore boards. These are about 5 feet long, 8 inches 
wide at one end, and tap6red to about 6 inches wide at the hinge 
end. The boards are fastened to a casting which has a ring to 
hold a red glass which may be swung over the face of a lantern, 
so as to indicate a red signal. "Distant" signal-boards usually 
have their ends notched or pointed; the "home" signal-boards 
are square ended. The boards are always to the right of the 
hinge when a train is approaching them. The "home" signals 
are generally painted red and the "distant" signals green, 
although these colors are not invariable. The backs of the 
boards are painted white. Therefore any signal-board which 
appears on the lejt side of its hinge will also appear white, and 
is a signal for traffic in the opposite direction, and is therefor 3 
of no concern to an engineman. 

Poles and bridges. When the signals are set on poles, they 
are always placed on the right-hand side of the track. When 
there are several tracks, four or more, a bridge is frequently 
built and then each signal is over its own track. The signals 
for two tracks, operated in the same direction, may be placed 
on one pole by having a cross-piece which supports two " masts," 
see Fig. 168. In that figure the signals on the left-hand mast 
control the second track at the left of the signal; those on the 
right-hand mast control the track just to the left of the signal. 



(To face page 418.) 




Courtesij of the Union Switch and Signal Co. 

Fig. 168. — Semaphoee«. 



{To face page 418.) 




Courtesij of the Union Switch and Signal Co. 

Fig. 170. — " Banjo " Signals, 



§ 393. BLOCK SIGNALING. 419 

A train movement, from the switch track at the right of the sig- 
nal on to the main track, is controlled by the " dwarf " signal 
at the right of the switch track. The signals controlling the 
two tracks at the extreme left are not shown. The building at 
the left of the track in the extreme background is apparently 
v4he signal tower controlling this signal. 

In Fig. 169 is shown a " bridge " and the two signals (home 
and distant), for each track. The two pairs of signals on the 
two right-hand poles are extended to the right and show that 
the movement of trains on those tracks is away from the observer. 
The darkness of the blades in the picture shows that they are 
painted dark, probably orange or red. The other blades show 
light (because painted white), and extend to the left but would 
appear to the right to an engineman on either left-hand track 
coming toward the observer. Incidentally the picture shows, 
over the two right-hand tracks, the ropes of a " tickler " (see 
§ 375), to protect brakemen on the tops of cars which will enter 
the tunnel shown in .the background. «; 

"Banjo" signals. This name is given to a form of signal, 
illustrated in Fig. 170, in which the indication is taken from the 
color of a round disk inclosed with glass. The great argument 
in their favor is that they may be worked by an electric current 
of low voltage, which is therefore easily controlled; that the 
mechanism is entirely inside of a case, is therefore very light, 
and is not exposed to the weather. The argument urged against 
them is that it is a signal of color rather than form or position, 
and that in foggy weather the signal cannot be seen so easily; 
aLo that unsuspected color-blindness on the part of the engine- 
man may lead to an accident. Notwithstanding these objections, 
this form of signal is used on thousands of miles of line in this 
country. 

393. Wires and pipes. Signals are usually operated by levers 
in a signal-cabin, the levers being very similar to the reversing- 
lever of a locomotive. The distance from the levers to the siff- 
nals is, of course, very variable, but it is sometimes 2000 feet. 
The connecting-link for the most distant signals is usually 
No. 9 wire; for nearer signals and for all switches operated 
from the cabin it may be 1-inch pipe. When not too long, one 
pipe will serve for both motions, forward and back. When 
wires are used, it is sometimes so designed (in the cheaper s}'s- 
tems) that one wire serves for one motion, gra\'ity being de- 



420 RAILROAD CONSTRUCTION.^ § 393. 

pended on for the other, but now all good systems require two 
wires for each signal. 

Compensators. Variations of temperature of a material with 
as high a coefficient as iron will cause very appreciable differ- 
ence of length in a distance of several hundred feet, and a 
dangerous lack of adjustment is the result. To illustrate: A 
fall of 60° F. will change the length of 1000 feet of wire by 

1000 X 60 X .0000065 = 0.39 foot = 4.68 inches. 

A much less change than this will necessitate a readjustment 
of length, unless automatic compensators are used. A com- 
pensator for pipes is very readily made on the principle illus- 
trated in Fig. 171. The problem is to preserve the distance 
between a and d constant regardless of the temperature. Place 
the compensator half-way between a and d, or so that ah=cd. 
A fall of temperature contracts ah to ah'. Moving h to h' will 
cause c to move to c', in which 66' = cc'. But cd has also short- 
ened to c'd) therefore d remains fixed in position. 

The regulations of the Am. Rwy. Eng. Assoc, require that 
"A compensator shall be provided for each pipe line over fifty 
(50) feet in length and under eight hundred (800) feet, with 
crank-arms eleven by thirteen (11X13) inch centers. From 
eight hundred (800) to twelve hundred (1200) feet in length, 
crank-arms shall be eleven by sixteen (11X16) inch centers. 
Pipe lines over twelve hundred (1200) feet in length shall be 
provided with an additional compensator. 

"Compensators shall have one sixty (60) degree and one one 
hundred and twenty (120) degree angle-cranks and connecting 
link, mounted in ca'st iron base, having top of center pins sup- 
ported. The distance between center of pin-holes shall be 
twenty-two (22) inches." 

The compensator should be placed in the middle of the length 
when only one is used. When two are used they should be 
placed at the quarter points. Note that in operating through 
a compensator the direction of motion changes; i.e., if a moves 
to the right, d moves to the left, or if there is compression in ah 
there is tension in cd, and vice versa. Therefore this form of 
compensator can only be- used with pipes which will withstand 
compression. It has seemed impracticable to design an equally 
satisfactory compensator for wires, although there are several 
degiglis on the market, 



I 



§393. 



BLOCK SIGNALING. 



421 



The change of length of these bars is so great that allowance 
must be made for the temperature at the time of installation. 
On the basis of 50°, as the mean temperature, the pipes are so 
adjusted that the distance between the points b and c of Fig. 171 
is made greater or less than 22 inches, according to the tem- 
perature of installation. For example, if the temperature wert: 
80° and the length of the piping were 900 feet, the length of the 
pipes should be adjusted so that be is less than 22 inches by an 
amount equal to 900 X (80°- 50°) X .0000065 = 0.1755 feet = 



a Ij b 




Fi«. 171. — ^Standard Pipe Compensator. 



2.106 inches. The length should therefore be 19.9 inches in- 
stead of 22 inches. If the mean temperature was very different 
(say in Florida) some higher temperature should be taken as 
normal, so that the extreme range above and below the normal 
shall be approximately the same. 

Guides around curves and angles. When wires are required 
to pass around curves of large angle, pulleys are used, and a 
length of chain is substituted for the wire. For pipes, when 
the curve is easy the pipes are slightly bent and are guided 
through pulleys. When the angle is sharper, "angles" are 
used. The operation of these details is self-evident from an 
inspection of Fig. 172. 



422 



RAlLBOAD CONSTHtfCTtON. 



SM. 



394. Track circuit for automatic signaling. The fundamental 
principle of the track circuit method of indicating a track obstruc- 
tion or breakage, using direct current, is as follows: A current 
of low potential is run from a battery at one end of a section 
through one line of rails to the other end of the section, then 
through a relay, and then back to the battery through the other 
line of rails. To avoid the excessive resistance which would 
occur at rail joints which may become badly rusted; a wire 




Fig. 172. — Deflecting-kods and Angle. 



suitably attached to the rails is run around each joint. In 
order to insulate the rails of one section from the rails at either 
end and yet maintain the rails structurally (Continuous, the ends 
of the rails at these dividing points are separated by an insulator 
and the joint pieces are either made of wood or have some 
insulating material placed between the rails and the ordinary 
metal joint. The bolts must also be insulated. When the 
relay is energized by a current, it closes a local circuit at the 
signal-station, which will set the signal there at " safety." The 
resistance of the relay is such that it requires nearly the whole 
current to work it and to keep the local circuit closed. There- 
fore, when there is any considerable loss of current from one 
rail to the other, the relay will not be sufficiently energized, the 
local circuit will be broken, and the signal will automatically 
fall to danger. This diversion of current from one rail to the 
other before the current reaches the relay may be caused in 
several ways: the presence of a pair of wheels on the rails any- 
where in the section will do it; also the breakage of a rail; also 
the opening of a switch anywhere in the section; also the pres- 
ence of a pair of wheels on a siding between the " fouling point " 
and the switch. (The " fouling point " of a siding is that point 
where the rails first commence to approach the main track.) 
In Fig. 173 is shown all of the above details as well as some others. 



§394. 



BLOCK SIGNALING. 



423 



lJ-5 



Ml 



At A, B, and the " fouling point " are shown the insulated joints. 
The batteries and signals are arranged 
for train motion to the right. When Ss 

a train has passed the points near A, 
where the wires leave the rails for the 
relay, the current from the " track 
battery " at 5 will pass through the 
wheels and axles, and although no 
electrical connection is broken, so 
much current will be shunted through 
the wheels and axles that the weak 
current still passing through the relay 
is not strong enough to energize it 
against its spring and the " signal- 
magnet " circuit is broken, and the 
signal A goes to " danger." At the 
turnout the rails between the fouling 
point and the switch are so connected 
(and insulated) that a pair of wheels 
on these rails will produce the same 
effect as a pair of the main track. This 
is to guard against the effect of a car 
standing too near the switch, even 
though it is not on the main track. 
When the train passes B, if there is 
no other interruption of the current, 
the track battery at B again energizes 
the relay at A, the signal-magnet 
circuit at A is closed, and the signal 
is drawn to " safety." 

About 1903 the application of alter- 
nating current to signaling circuits was 
invented. This not only permits the 
substitution of a. c. circuit for track 
batteries, but also makes it possible to 
utilize the track circuit method to in- 
dicate obstructions or rail breakages 
even when the track is the return cir- 
cuit for an electrified road. But an 
explanation of this development would 
be too long for this text-book. It is Fig. 173. 



<ll 




<0 



424 RAILROAD CONSTRUCTION. § 394. 

given in a 548-page book called "Alternating Current Signaling," 
published by the Union Switch & Signal Co., Swissvale, Pa. 

This chapter also omits all references to " interlocking plants," 
which are essential features of the operation of large terminal 
yards. Even an elementary treatment of the present develop- 
ment of signaling and interlocking would require a large text- 
book, and, therefore, nothing more than the above brief outline 
will be here given. 



CHAPTER XV. 



ROLLING-STOCK. 



(It is perhaps needless to say that the following chapter is 
in no sense a course in the design of locomotives and cars. Its 
chief idea is to give the student the elements of the construc- 
tion of those vehicles which are to use the track which he may 
design — to point out the mutual actions and reactions of vehicle 
against track and to show the effect on track wear of varia- 
tions in the design of rolling-stock. The most of the matter 
given has a direct practical bearing on track-work, and it is con- 
sidered that all of it is so closely related to his work that the 
civil engineer may study it with profit. For " Stresses in Track," 
see Chap. XXV.) 

WHEELS AND RAILS. 

395. Effect of rigidly attaching wheels to their axles. The 
wheels of railroad rolling-stock are invariably secured rigidly 
to the axles, which therefore revolve with the wheels. The 
chief reason for this is to avoid excessive wear 
between the axles and the wheels. 

Any axle must always be somewhat loose in 
its journals. A sidewise force P (see Fig. 174) 
acting against the circumference of the wheel 
will produce a much greater pressure on the 
axle at S and S', and if the wheel moves on 
the axle, the wear at S and S' will be exces- 
sive. But when the axle is fitted to the wheel 
with a "forced fit" and does not revolve, 
the mere pressure produced at S is harmless. 
When two wheels are fitted tight to an axle, " p 

as in Fig. 175, and the axle revolves in the jour- Fig. 174. 

nals aa, a sidewise pressure of the rail against the wheel flange 
will only produce a slight and harmless increase of the journal 
pressure Q, although at Q there is sliding contact. Twist- 

425 




426 



KAILROAD CONSTRUCTION. 



§396. 



ing action in the journals is thus practically avoided, since a 
small pressure at the journal-boxes at each end of the axle 
suffices to keep the axle truly in line. 




EZZ3 



a 



flC,^ 



Fig. 175. 




On the other hand, when the wheels are rigidly" attached to 
their axles, both wheels must turn together, and when rounding 
curves, the inner rail being shorter than the outer -rail, one 
wheel must slip by an amount equal to that difference of length. 
I'he amount of this slip is readily computable : 



Longitudinal slip = ^{r^ ~'^'^^3m 



^a° = (7a°, . (102) 



in which C is a constant for any one gauge, and g= the track 
<5auge = (r2— ri). For standard gauge (4.708) the shp is .08218 
foot per degree of central angle. This shows that the longitu- 
dinal slipping around any curve of any given central angle will 
be independent of the degree of the curve. The constant (.08218) 
here given is really somewhat too small, since the true gauge 
that should be considered is the distance between the lines of 
tread on the rails. This distance is a somewhat indeterminate 
and variable quantity, and probably averages 4.90 feet, which 
would increase the constant to .086. The slipping may occur 
by the inner wheel slipping ahead or the outer wheel slipping 
back, or by both wheels slipping. The total slipping will be 
constant in any case. The slipping not only consumes power, 
but wears both the wheels and the rail. But even these dis- 
advantages are not sufficient to offset the advantages resulting 
from rigid wheels and axles. 

I 396. Effect of parallel axles. Trucks are made with two or 
three parallel axles (except as noted later), in order that the 
axles shall mutually guide each other and be kept approximately 



306. 



ROLLING-STOCK. 



427 



perpendicular to the rails. If the curvature is very sharp and 
the wheel-base comparatively long (as is notably the case for 
four-wheeled street cars passing around street corners), the front 



CASE1 



CASE 2 








Fig. 177. 



a 



Fig. 179. 



and rear wheels will stand at the same angle (a) with the track, 
as shown in Fig. 177, which also applies to easy curvature when- 
ever the rear outer wheel-flange is forced against the rail, which 
is claimed by some to be the normal position. Others claim 
that for ordinary curvature the rear axle will take a position 
normal to the curve, as shown in Fig. 178. But it is certain that 
track irregularities cause the rear wheels to sway within the limits 
of the play of the gauge and that the angle a varies. For Case 
1, sin a = t-ir2r; for Case 2, sin a — t-^r. 

When the two parallel axles are on a curve (as shown), the 
wheels tend to run in a straight line. In order that they shall 
run on a curve they must slip laterally. The principle is illus- 
trated in an exaggerated form in Fig. 179. The wheel tends to 
roll from a toward b. Therefore in passing along the track from 
a to c it must actually slip laterally an amount be which equals 
ac sin a. The lateral slipping per unit of distance traveled there- 
fore equals sin a. For Case 1, both front wheels slip laterally 
toward the curve center, apd both rear wheels slip laterally 
away from the center. For Case 2, both front wheels slip lat- 
erally toward the center, but tj3,e slip per unit of forward dis- 
tance is only one-half that of Case 1, while the rear axle, being 
radial does not slip laterally at all. Neither Case 1 nor Case 2 
(nor any other combination) is constantly applicable. 

From the above it might be inferred that the flanges of the 
forward wheels will have much greater wear than those of the 
rear wheels. Since cars are drawn in both directions about 
equally, no difference in flange wear due to this cause will occur, 
but locom-otives (except switching-engines) run forward almost 



428 



RAILROAD CONStRUOTION. 



397. 



exclusively, and the excess wear of the front wheels of the pilot - 
and tender-trucks is plainly observable. 

For a given curve the angle a (and the accompanying resist- 
ance) is evidently greater the greater the distance between 
the axles. On the other hand, if the two axles are very close 
together, there will be a tendency for the truck to twist and 
the wheels to become jammed, especially if there is consider- 
able play in the gauge. The flange friction would be greater 
and would perhaps exceed the saving in lateral slipping. A 
general rule is that the axles should never be closer together 
than the gauge ; usually it is considerably more. 

Although the slipping per unit of length along the curve varies 
directly as the degree of curvature, the length of curve necessary 
to pass between two tangents is inversely as the degree of curve, 
and the total slipping between the two tangents is independent 
of the degree of curve. Therefore when a train passes between 

two tangents, the total slipping 

^^ . ^^ of the wheels on the rails, lon- 

I gitudinal and lateral, is a quantity 

which depends only on the central 
angle and is independent of the 
radius or degree of curve. 

397. Effect of coning wheels. 
The wheels are always set on the 
axle so that there is some "play" 
or chance for lateral motion be- 
tween the wheel-flanges and the 
rail. The treads of the wheel are 
also " coned." This coning and play 
of gauge are shown in an exagger- 
ated form in Fig. 180. When the 
wheels are on a tangent, although there will be occasional oscil- 
lations from side to side, the normal position will be the sym- 
metrical position in which the circles of tread hh are equal. 
When centrifugal force throv^'s the wheel-flange against the rail, 
the circle of tread a is larger than 6, and much larger than c; 
therefore the wheels will tend to roll in a circle whose radius 
equals the slant height of a cone whose elements would pass 
through the unequal circles a and c. If this radius equaled the 
radius of the track, and if the axle were free to assume a radial 
position, the wheels would roll freely on the rails without any 




Fig. 180. 



§ 398. ROLLING-STOCK. 429 

slipping or flange pressure. Under such ideal conditions, 
coning would bo a valuable device, but it is impracticable to 
have all axles radial, and the radius of curvature of the track 
is an extremely variable quantity. It has been demonstrated 
that with parallel axles the influence of coning diminishes as 
the distance between the axle increases, and that the effect is 
practica,lly inappreciable when the axles are spaced as they are 
on locomotives and car-trucks. The coning actuall}'- used is 
very slight (see Chapter XV, § 420) and has a different object. 
It is so slight that even if the axles were radial it would only 
prevent the slipping on a very light curve — say a 1° curve. 

398. Effect of flanging locomotive driving-wheels. If all the 
wheels of all locomotives were flanged it would be practically 
impossible to run some of the longer types around sharp curves. 
The track-gauge is always widened on curves, and especially 
on sharp curves, but the widening would need to be excessive 
to permit a consolidation locomotive to pass around an 8° or 
10° curve if all the drivers were flanged. The action of the 
wheels on a curve is illustrated in Figs. ,181, 182, and 184. All 
small truck-wheels are flanged. The rear drivers are always 
flanged and four-driver engines usually have all the drivers 
flanged. Consolidation engines have only the front and rear 
drivers flanged. Mogul and ten-wheel engines have one pair 
of drivers blank. On Mogul engines it is always the middle 
pair. On ten-wheel engines, when used on a road having sharp 
curves, it is preferable to flange the front and rear driving- 
wheels and use a "swing bolster" (see § 399); when the curva- 
ture is easy, the middle and rear drivers may be flanged and 
the truck made with a rigid center. The blank drivers have 
the same total width as the other drivers and of course a much 
wider tread, which enables these drivers to remain on the rail, 
even though the curvature is so sharp that the tread overhangs 
the rail considerably. 

399. Action of a locomotive pilot-truck. The purpose of 
the pilot-truck is to guide the front end of a locomotive around 
a curve and to relieve the otherwise excessive flange pressure 
that would be exerted against the driver-flanges. There are 
two classes of pilot -trucks — (a) those having fixed centers and 
(6) those having shifting centers. This second class is .again 
subdivided into two classes, which are radically different in 
their action — Q){) four-wheeled trucks having two parallel axles 



430 



RAILROAD CONSTRUCTION. 



§399. 



and (62) two-wheeled trucks which are guided by a "radius- 
bar." The action of the four-wheeled fixed-centered truck (a) 
is shown in Fig. 181. -Since the center of the truck is forced 




Fig. 181. — Fixed Center Pilot-truck. 

to be in the center of the track, the front drivers are drawn 
away from the outer rail. The rear outer driver tends to roll 
<iway from the outer rail rather than toward it, and so the effect 




Fig, 182, — Four-wheeled Truck — Shifting Center. 

of the truck is to relieve the driver-flanges of any excessive 
pressure due to curvature. The only exception to this is the 
case where the curvature is sharp. Then the front inner driver 
may be pressed against the inner rail, as indicated in Fig. 181. 

This limits the use of this type of 
wheel-base on the sharper curves. 
The next type — (bi) four-wheeled 
trucks with shifting centers — is 
much more flexible on sharp 
curvature; it likewise draws the 
front drivers away from the outer 
rail. The relative position of the 
wheels is shown in Fig. 182, in 

/. v^ which c' represents the position 
/^ of center-pin and c the displaced 
: / truck center. The structure and 

action of the truck is shown in 
Fig. 183, The "center-pin" (1) is 
supported on the "truck-bolster" (2), which is hung by the 
"links" (4) from the "cross-ties" (3). The links are therefore 




Fig. 



183. — Action of Shifting 
Center. 



§399. 



ROLLING-STOCK. 



431 



in tension and when the wheels are forced to one side by the 
rails the links are inclined and the front of the engine is 
drawn inward by a force equal to the Aveight on the bolster 
times the tangent of the angle of inclination of the links. This 
assumes that all links are vertical when the truck is in the 
center. Frequently the opposite links are normally inclined to 
each other, which somewhat complicates the above simple relation 
of the forces, although the general principle remains identical. 

The two-wheeled pilot-truck with shifting center is illus- 
trated in Fig. 184. The figure shows the facility with which 




HSZ|=31itf=i 



Fig. 184. — Two-wheeled Truck — Shifting Center. 




Fig. 185. — Action of Two- 
wheeled Truck. 



an engine with long wheel-base may be made to pass around 
a comparatively sharp curve by omitting the flanges from the 
riiiddle drivers and using this form of pilot-truck. As in the 
previous case, the eccentricity of 
the center of the truck relative 
to the center-pin induces a cen- 
tripetal force which draws the 
front of the engine inward. But 
the swing-truck is not the only 
source of such a force. If the 
"radius-bar pin" were placed at 0' (see Fig. 185), the truck- 
axle would be radial. But the radius-bar is always made some- 
what shorter than this, and the pin is placed at 0, a considerable 
distance ahead of 0', thus creating a tendency for the truck 
to run toward the inner rail and draw the front of the loco- 
motive in that direction. This tendency will be objectionably 
great if the radius-bar is made too short, as has been practically 
demonstrated in cases when the radius-bar has been subse- 
quently lengthened with a resulting improvement in the running 
of the engine. This type of pilot truck is used on both Mogul 
and Consolidation locomotives and explains why these long 
engines can so easily operate on sharp curves. 



432 



RAILROAD CONSTRUCTION. 



§400. 



400. Types of locomotive wheel-bases. The variations in 
locomotive service have developed all conceivable types as to 
total weight, ratio of total weight to weight on drivers, types of 
running gear, relation of steaming capacity to tractive power, 
etc. The method of classification on the basis of the running 
gear is very simple. The number of wheels on both rails of the 
pilot truck, if any, is placed as the first of three numbers. If 
there is no pilot truck, the character is used. This is followed 
by the number of drivers and then by the number of trailing 
wheels, if any. For example,, a Pacific type engine has four 
wheels on the pilot truck, six driving wheels, and two trailing 
wheels under the rear of the boiler. The wheel-base is symbolized 
as 4-6-2. The most common types of locomotives, with their 
popular names and wheel base symbols, are 



American 4-4-0 

Columbia 2-4-2 

Atlantic 4-4-2 

Mogul 2-6-0 

Prairie 2-6-2 

Ten-wheel 4-6-0 

Pacific . , 4-6-2 

Six-wheel switcher 0-6-0 



Consolidation. 

Mikado 

Mastodon . . . . 
Santa Fe 



2-8-0 

2-8-2 

4-8-0 

2-10-2 



Mallet A-B-B-A 

A = truck wheels, usually 2 or 
B = drivers, varying from 4 to 10 



The " Mallet " type of locomotive is one which combines 
sufficient flexibility to operate on ordinary railroad curves, wheel . 
loads on the drivers which are not excessive, a very great increase 
in the total tractive power and yet operated by one engineman. 
In one respect it is like coupling two or three locomotives together, 
but the saving consists in reducing the number of enginemen 
and firemen which would be needed to run the two or three 
locomotives. Excluding freak variations, they are usually 
*' four-cylinder compounds," one pair of cylinders discharging 
into the other pair and then exhausting. This type has from 
five to ten driving axles and has a length of engine wheel-base 
up to about 60 ft., but this wheel-base is flexible, so that it will 
bend on a curved track. Sometimes the boiler is made flexible 
by having a set of accordion-shaped steel rings forming a joint 
in the boiler shell. The boiler itself is on one side of this flexible 
joint and the feed-water heater, the reheater, and perhaps the 
superheater are on the other side of the joint. In this case each 
half of the flexible boiler is carried on a frame supported by one 
of the sets of driving wheels, the two frames bemg connected by a 
suitable joint. The boiler shell is made rigid; one end is rigidly 
attached to the frame carrying the high-pressure cylinders and 



§ 401. KOLLiNG-STOCK. 433 

the other end is supported on a bearing on the truck frame which 
carries the low-pressure cyHnders and the drivers operated by 
them. The low-pressure truck frame swings around a pivot in 
the fixed frame. This flexibility has been made so great that 
these locomotives are operated successfully on 20° curves. The 
Baldwin Locomotive Works have developed this type still 
further by building a locomotive for the Erie R. R. which has 
three wheel frames, mutually flexible with each other, the third 
frame being under the tender. Each wheel frame has eight 
driving wheels. The total load carried by the twenty-four 
drivers is 761,600 lbs. or an average of 31,733 lbs. per driver. 
There are six cylinders of equal size. The two cylinders on the 
center frame use high-pressure steam and exhaust into the other 
four cylinders. The total weight of locomotive and tender is 
853,050 lbs. On a test trip it pulled a train with a total length 
of 8547 ft. or 1.6 miles, the total weight of the train being 18,338 
tons. The maximum draw-bar pull, registered by the dyna- 
mometer car, was 130,000 lbs. The adhesion between the 
drivers and the rails must have been considerably more. Such 
engines are chiefly used for hauling long trains of slow-speed 
freight. Their boilers cannot produce steam fast enough to 
develop their enormous tractive power at high speeds and the 
power falls off rapidly with increase in speed. They are fre- 
quently equipped with automatic stokers for burning coal, or 
with oil-burning outfits, since the great amount of power devel- 
oped can only be produced by the consumption of a corresponding 
amount of fuel, and a fireman would be physically incapable of 
shoveling coal as rapidly as the production of such an amount of 
power would demand. - 

LOCOMOTIVES. 
GENERAL STRUCTURE, 

401. Frame. The frame or skeleton of a locomotive con- 
sists chiefly of a collection of forged wrought-iron bars, aa 
shown in Figs. 186 and 187. These bars are connected at the 




Fig. 186. — Engine-frame. 
front end by the "bumper" (c), which is usually made of wood. 



434 



RAILROAD CONSTRUCTION. 



§402. 



A little further back they are rigidly connected at 66 by the 
cjdinders and boiler-saddle. The boilers rest on the frames 
at aaaa by means of "pads/' which are bolted to the fire-box, 
but which permit a free expansion of the boiler along the frame. 
This expansion is sometimes as much as -f^". On a "con- 
solidation" engine (frame shown in Fig. 187) it is frequently 




Fig. 187. — Engine-frame — Consolidation Type. 

necessary to use vertical swing-levers about 12'' long instead 
of "pads." The swinging of the levers permit all necessary 
expansion. At the back the frames are rigidly connected by 
the iron "foot-plate." The driving-axles pass through the 
"jaws" dddd, which hold the axle-boxes. The frame-bars 
have a width (in plan) of 3" to 4". The depth (at a) is about 
the same. Fig. 1S6 shows a frame for an "American" type 
of locomotive; Fig. 187 shows a frame for a " Consolidation" 
type (see §400). 

402. Boiler. A boiler is a mechanism for transferring the 
latent heat of fuel to water, so that the water is transformed 
from cold water into high-pressure steam, which by its expan- 
sion will perform work. The efficiency of the boiler depends 
largely on its ability to do its work rapidly and to reduce to 
a minimum the waste of heat through radiation. The boiler 
contains a fire-box (see Fig. 188), in which the fuel is burned. 
The gases of consumption pass from the fire-box through the 
numerous boiler-tubes into the "smoke-box" S and out through 
the smoke-stack. The fire-box consists of an inner and outer 
shell separated by a layer of water W to 5'' thick. The ex- 
posure of water-surface to the influence of the fire is thus very 
complete. The efficiency of this transferal of heat is somewhat 
indicated by the fact that, although the temperature of the 
gases in the fire-box is probably from 3000° to 4000° F., the 
temperature in the smoke-box is generally reduced to 500° to 
600° F. If the steam pressure is 180 lbs., the temperature of 
the water is about 380° F., and, considering that heat will not 
pass from the gas to the water unless the gas is hotter than the 
water, the water evidently absorbs a large part of the theo- 
retical maximum. Nevertheless gases at a temperature of 



§ 403. 



ROLLING-STOCK. 



435 



600° F. pass out of the smoke-stack and such heat is utterly- 
wasted. 

The tubes vary from Ij' to 2", inside diameter, with a thick- 
ness of about O'MO to 0'M2. The aggregate cross-sectional 



^^ 




Fig. 188.— Locomotive-boilee. 



area of the tubes should be about one-eighth of the grate area. 
The number will vary from 140 to 375. The length varies from 
11' to 21', but the length is virtually determined by the type and 
length of engine. 

403. Fire-box. The fire-box is surrounded by water on the 
four sides and the top, but since the water is subjected to the 




Fig. 189. 



Fig. 190. 



boiler pressure, the plates, which are yg" to f '' thick, must be 
stayed to prevent the fire-box from collapsing. This is easily 
accomplished over the larger part of the fire-box surface by 



436 



RAILROAD CONSTRUCTION. 



§403. 



having the outside boiler-plates parallel to the fire-box plates 
and separated from them by a space of 3" to 5". The plates 





ooooo 
joooioooo 

lOOOOOOOOOO 

ooooiooooooo 

lOOOOOOOOOO 0(i 

, _ ooodoooooooc',, ., 

^_ J:>0 00 00 000 00 0(iiE:J 

^'c3; iF oooooioooooooc'i 
^^~ -tg>oooooooooooo 
!iooooo;ooooooo 



It 



e • 



• e 



e • 



o 




g gfF^ Wi 4J 



are then mutually held by " stay-bolts." See Fig. 189. These 
are about |" in diameter and spaced 4" to 4|". The ^" hole, 
drilled U" deep, indicated in the figure, will allow the escape 



§403. 



ROLLING-STOCK. 



437 



of steam if the bolt breaks just behind the plate, and thus calls 
attention to the break. The stay-bolts are turned down to a 
diameter equal to that at the root of the screw-threads. This 
method of supporting the fire-box sheets is used for the two 
sides, the entire rear, and for the front of the fire-box up to the 
boiler-barrel. The "furnace tube-sheet" — the upper part of 
the front of the fire-box — is stayed by the tubes. But the top 
of the fire-box is troublesome. It must always be covered 
with water so that it will not be "burned" by the intense heat. 
It must therefore be nearly, if not quite, flat. There are three 
general methods of accomplishing this. 




Thru AB \ Thru €D 
Half-sections. 

Fig. 192. — "Belpaire" Fire-box. 



(a) Radial stays. This construction is indicated in Fig. 190. 
Incidentally there is also shown the diagonal braces for resist- 
ing the pressure on the back end of the boiler above the fire- 
box. It may be seen that the stays are not perpendicular to 
either the crown-sheet or the boiler-plate. This is objection- 
able and is obviated by the other methods. 

(b) Crown-bars. These bars are in pairs, rest on the side 
furnace-plates, and are further supported by stays. See Fig, 
191. 

(c) Belpaire fire-box. The boiler above the fire-box is recti 
angular, with rounded corners. The stays therefore arc per- 
pendicular to the plates. See Fig. 192. 

Fire-brick arches. These are used, as shown in Fig. 193,' to 
force all the gases to circulate through the upper part of the fire- 
box. Perfect combustion requires that all the carbon shall be turned 
into carbon dioxide, and this is faciUtated by the forced circulation. 



43g 



RAILROAD CONSTRUCTION. 



§404. 



Water-tables. 'The same object is attained by using a water- 
table instead of a brick arch — as shown in Fig. 191. But it has 
the further advantages of giving additional heating-surface and 
' avoiding the continual expense of maintaining the bricks. One 
feature of the design is the use of a number of steam- jets which 
force air into the Qre-box and assist the combustion. 





Fig. 193.— F'ire-brick AncH. 



Fig. 194. — Wootten Fire-box. 



404. Area of grate. The older types of engines, as represented 
by the "American," " Mogul " or " Consolidation " type, 
always had the fire-box set between the drivers, which practically 
meant that the maximum effective inside width of the fire-box 
was' limited to about 3 ft. 5 ins. for standard-gauge locomotives. 
The maximum distance over which a fireman can properly 
control a fire is perhaps 10 to 11 ft., but such extreme lengths 
are objectionable. The. grate area was thus quite definitely 
limited. The Wootten fire-box, illustrated in Fig. 194, obtained 
a fire-box eight feet v/ide by raising it above the lev6l of the 
drivers, as shown, but this required that the drivers should be 
objectionably small in diameter, except for low-speed engines, 
or that the fire-box would be set objectionably high. The last 
difficulty has been solved by engines of the " Columbia," '' At- 
lantic," " Pacific," " Mikado," and " Santa Fe " types, all of 
which have a pair of trailing wheels, 36 to 45 ins. in diameter, 
set back of the driving wheels and under the fire-box, which may 
thus be widened to 7 or 8 ft., the entire fire-box being placed back 
of the driving wheels. 

405. Superheaters. Inside of a boiler the steam has a tem- 
perature corresponding to its pressure. For example, if the 
pressure is 180 lbs., the temperature is about 379° F. When the 
steam of a locomotive is superheated, the steam is conducted 
from the throttle to the cylinders through pipes which are pur- 



§ 406. ROLLING-STOCK. 439 

posely placed in the path of the flue gases on their way to the 
smokestack. A simple form of superheater is a series of tubes 
and drums located in the smokebox. Here the temperature is 
perhaps 600° F., which is sufficient to heat the steam from 30° 
to 50° above the boiler temperature and to produce substantial 
economies. In another more effective but more costly type a 
considerable number of the ordinary 2|-inch boiler tubes are 
replaced by 5|-inch tubes, inside of each of which is a pipe loop 
extending from the smokebox headers to within a short dis- 
tance of the fire-box, where the temperature approaches the 
fire-box temperature, which is perhaps 2000° F. The live steam 
passes through these loops and is so heated that, even after it 
reaches the cylinder, it has a superheat of 150° to 200° over the 
boiler temperature, but since its pressure is substantially the 
boiler pressure, the quantity (or weight) of steam required to fill 
the cylinder at that temperature and pressure is much less 
than the quantity of steam at the same pressure but lower tem- 
perature. Superheating also has the advantage of making the 
steam more dry and of preventing condensation in the cylinders 
until the steam has lost in temperature at least the amount of its 
superheat. Superheating is chiefly advantageous for use with 
passenger engines, when they must work at high power for long, 
continuous runs. An economy of 15 to 25% in coal consumption 
(and even 30% in some tests), can ordinarily be obtained by the 
use of superheaters, but the economy is somewhat offset by the 
additional cost for installation and for subsequent repairs and 
maintenance. 

406. Reheaters. A reheater is substantially the same as a 
superheater in its general principle of construction. When steam 
has been exhausted from a high-pressure cylinder, the tempera- 
ture and pressure are both considerably lower than their boiler 
values. If the steam is to be again used, an economy is obtained 
and the steam is dried by passing it through a reheater. They 
are generally used on Mallet engines to reheat the steam in its 
passage from the high-pressure to the low-pressure cylinders. 

407. Coal consumption. No form of steam-boiler (except 
a boiler for a steam fire-engine) requires as rapid production of 
steam, considering the size of the boiler and fire-box, as a 
locomotive. The combustion of coal per square foot of grate 
per hour for stationary boilers averages about 15 to 25 lbs. and 
seldom exceeds that amount. An ordinary maximum for a 



440 RAILROAD CONSTRUCTION. § 407. 

locomotive is 125 lbs. of coal per square foot of grate-area per 
hour, and in some recent practice 220 lbs, have been used. Of 
course such excessive amounts are wasteful of coal, because 
a considerable percentage of the coal will be blown out of the 
smoke-stack unconsumed, the draft necessary for such rapid 
consumption being very great. The onty justification of such 
rapid and wasteful coal consumption is the necessity for rapid 
production of steam. The best quality of coal is capable of 
evaporating about 14 lbs. of water per pound of coal, i.e., change 
it from water at 212° to steam at 212°; the heat required to 
change water at ordinary temperatures to steam at ordinary 
working pressure is (roughly) about 20% more. From 6 to 9 lbs. 
of water per pound of coal is the average performance of ordinary 
locomotives, the efficiency being less with the higher rates of 
combustion. Some careful tests of locomotive coal consump- 
tion gave the following figures: when the consumption of coal 
was 50 lbs, per square foot of grate-area per hour, the rate of 
evaporation was 8 lbs. of water per pound of coal. When the 
rate of coal consumption was raised to ISO, the evaporation 
dropped to 5 lbs. of water per pound of coal. It has been 
demonstrated that the efficiency of the boiler is largely increased 
by an increased length of boiler-tubes. The actual consump- 
tion of coal per mile is of course an exceedingly variable quan- 
tity, depending on the size and type of the engine and also on 
the work it is doing — -whether climbing a heavy grade with its 
maximum train4oad or running easily over a level or down 
grade. A test of a 50-ton engine, running without any train at 
about 20 to 25 miles per hour, showed an average consumption 
of 21 lbs. of coal per mile. Statistics of the Pennsylvania Rail 
road show a large increase (as might be expected, considering 
the growth in size of engines and weight of trains) in the aver- 
age number of pounds of coal burned per train-nnle — some of 
the figures being 55 lbs. in 1863, 72 lbs. in 1872, and nearly _, 
84 lbs. in 1883. Figures are published showing an average 
consumption of a,bout 10 lbs. of coal per passenger-car mile, 
and 4 to 5 lbs. per freight-car mile. But these figures are always ' 
obtained by dividing the total consumption per train-mile by 
the number of ears, the coal due to the weight of the engine 
being thrown in. Wellington developed a rule, based on the 
actual pei"forinance of ^a Yery large number of passenger- trains, 
that the number of pounds of coal per mile — 21:14*6.74 times 
the number oa passenger-cars. The amount of coal assigned 



§ 408. ROLLING-STOCK. 441 

to the engine agrees remarkably with the test noted above 
For freight-trains the amount assigned to the engine should 
be much greater (since the engine is much heavier), and that 
assigned to the individual cars much less, although the great 
increase in freight-car weights in recent years has caused an 
increase in the coal required per car* 

There is a physical limit to the amount of coal which can be 
shovelled into a firebox by a fireman. Tests have shown that 
the average fireman can handle about 4000 lbs. of coal per hour 
and keep up such work almost indefinitely. For a short time 
he can shovel coal at the rate of 80 or 90 lbs. per minute, and 
this may be necessary to keep up steam while the train, is going 
over some hump, but it must be followed by some relief which 
will make the average about the same. Automatic stokers have 
been devised for locomotives which can feed as much as 6000 lbs. 
of coal per hour when the grate area is less than 70 square feet 
and up to 8000 lbs. per hour when the grate area is 70 square feet 
or over. These are necessary on some of the most powerful 
locomotives in order to produce steam fast enough to develop 
their maximum capacity. 

408. Oil-burning locomotives. In 1912 over one-sixth of all 
the locomotives west of the Mississippi River used oil as fuel. 
Some of the advantages in using oil are as follows: (1) the 
British thermal units in one pound of oil vary from about 19,000 
to 21,000; those in a pound of coal vary from perhaps 14,000 for 
the vary best down to 5000 for the poorer grades of lignite found 
in the western parts of the United States, and this means a great 
reduction in the cost of carrying and storing fuel, measured in 
heat units; (2) the cost of handling fuel is reduced and that of 
disposing of ashes is eliminated; (3) engine repairs are reduced 
in many respects, although it is said that the increased cost of 
fire-box repairs, due to the intense heat of the oil flame, offsets 
any reduction in other items; (4) the fires can be more easily 
controlled and waste of heat reduced during stoppages or when 
drifting down grade; (5) wayside fires due to sparks are alto- 
gether eliminated; (6) there is a practical limitation (see § 407), 
to the amount of coal that one fireman can feed to a fire; but 
there is no such limitation when using oil; (7) there is an equality 
in cost of heat units when a 42-gallon barrel of oil, weighing 7.3 
lbs. per gallon, costs 60 cents and a ton (2000 lbs.) of coal, having 

* See Chap. XVIII for further discussion of relation of coal consumed 
topower produced. 



442 EAILROAD CONSTRUCTION. - § 409. 

two-thirds as many heat units per pound, costs $2.61, or 4.35 
times as much. The other items of difference almost invariably 
favor the oil and might make it more desirable even when the 
ratio of cost seemed to favor the coal. The extensive use of oil 
west of the Mississippi River is due to the fact that in many 
localities a very suitable quality of crude oil. is plentiful and 
cheap while coal is expensive and of low calorific power. 

409. Heating-surface. The rapid production of steam requires 
that the hot gases shall have a large heating-surface to which 
they can impart their heat. From 50 to 75 square feet of 
heating-surface is usually designed for each square foot of 
grate-area. A more recently used rule is that there should be 
from 60 to 70 square feet of tube heating-surface per square 
foot of grate-area for bituminous coal. 40 or 50 to 1 is more 
desirable for anthracite coal. Almost the whole surface of 
the fire-box has water behind it, and hence constitutes heating- 
surface. Although this surface forms but a small part of the 
total (nominally), it is really the most effective portion, since 
the difference of temperature of the gases of combustion and 
the water is here a maximum, and the flow of heat is therefore 
the most rapid. The heating-surface of the tubes varies from 
85 to 93% of the total, or about 7 to 15 times the heating-surface 
in the fire-box. By dividing the total weight of a well-designed 
engine (exclusive of tender) by the number of square feet of 
heating-surface (fire-box and tubes), we get a quotient which 
varies from 60 to 80 or over. For example, a light engine, weigh- 
ing only 96,450 lbs. had a total heating surface of 1449 square 
feet, or about 67 lbs. per square foot. On the other hand, a 
Mikado engine, weighing 297,500 lbs., had 4359 square feet of 
heating surface, or 68 lbs, per square foot. 

410. Loss of efficiency in steam pressure. The effective 
work done by the piston is never equal to the theoretical energy 
contained in the steam withdrawn from the boiler. This is due 
chiefly to the following causes: 

(a) The steam is " wire-drawn," i.e., the pressure in the 
cylinder is seldom more than 85 to 90% of the boiler pressure. 
This is due largely to the fact that the steam-ports are so small 
that the steam cannot get into the cylinder fast enough to exert 
its full pressure. Partially closing the throttle, so that the 
steam will be used less rapidly, also wire-draws the steam. 

(6) Entrained water. Steam is always drawn from a dome 



§ 411. ROLLING-STOCK. 443 

placed over the boiler so that the steam shall be as far above 
the water-surface as possible, and shall be as dry as possible. 
In spite of this the steam is not perfectly dry and carries with 
it water at a temperature of, say, 361°, and pressure of 140 lbs. 
per square inch. When the pressure falls during the expan- 
sion and exhaust, this hot water turns into steam and absorbs 
the necessary heat from the hot cylinder-walls. This heat is 
then carried out by the exhaust and wasted. 

(c) The back pressure of the exhaust-steam, which depends 
on the form of the exhaust-passages, etc. This amounts to 
from 2 to 20% of the power developed. 

(d) Clearance-spaces. When cutting off at full stroke this 
waste is considerable (7 to 9%), but when the steam is used 
expansively the steam in these clearance-spaces expands and 
so its power is not wholly lost. 

(e) Radiation. In spite of all possible care in jacketing the 
cylinders, some heat is lost by radiation. 

(/) Radiation into the exhaust-steam. This is somewhat 
analogous to (h). Steam enters the cylinder at a temperature 
of, say, 361°; the walls of the cylinder are much cooler, say 250°; 
some heat is used in raising the temperature of the cylinder- 
walls; some steam is vaporized in so doing; Avhen the exhaust 
is opened the temperature and pressure fall; the heat tem- 
porarily absorbed by the cjdinder-walls is reabsorbed by the 
exhaust-steam, re-evaporating the vapor previously formed, 
and thus a certain portion of heat-energy goes through the 
cylinder vithout doing any useful work. With an early cut-off 
the loss due to this cause is very great. 

The sum of all these losses is exceedingly variable. They 
are usually less at lower speeds. The loss in initial j^ressure 
(the difference between boiler pressure and the cylinder pres- 
sure at the beginning of the stroke) is frequently over 20%, 
but this is not all a net loss With an early cut-off the average 
cylinder pressure for the whole stroke is but a small part of 
the boiler pressure, yet the horse -power developed may be as 
great as, or greater than that developed at a lower speed, later 
cut-off, and higher average pressure 

411. Tractive power The work done by the two cylinders 
during a complete revolution of the drivers evident^ = area of 
pistons X average steam pressure X stroke X2X2. The resist- 
ance overcome evidently = tractive force at circumference of 



444 RAILROAD CONSTRUCTION. § 412. 

drivers times distance traveled by drivers (which is the cir- 
cumference of the drivers) Therefore 

( area pistons X a\^erage steam pressure 

^ ^. . } XstrokeX2X2. 

Tractive force = ) -. ^ , , , . 

I circumierence ot drivers 

Dividing numerator and denominator hj n (3.1415), we have 

r (diam piston) '^X average steam 

\ pressure X stroke , 

Tractive force = ) %-. ^-r-. , . (lO.i) 

( diameter oi driver 

which is the usual rule Although the rule is generally stated 
in this form, there are several deductions In the first place 
the net effective area of the piston is less than the nominal on 
account of the area of the piston-rod. The ratio of the areas 
of the piston-rod and piston varies, but the effect of this reduc- 
tion is usually from 1.3 to 1-7%. No allowance has been made 
for friction — of the piston, piston-rod; cross-head, and the 
various bearings This would make a still further reduction 
of several per cent. Nevertheless the above simple rule is 
used, because^, as will be shown, no great accuracy can be 
utilized. 

The maximum draw bar pull is limited by the adhesion between 
the driving wheels and the rails. This is usually about one-, 
fourth of the weight. The use of sand may increase it to one- 
third. But this ratio is important only when starting or at very 
low speeds. The adhesion is always ample for the much lower 
cylinder power which can be developed at higher speeds. This 
is considered more fully in Chapter XVIII. 

RUNNING GEAR. 

412. Equalizing-levers. The ideal condition of track, from 
the standpoint of smooth running of the rolling stock, is that 
the rails should always lie in a plane surface. While this con- 
dition is theoretically possible on tangents, it is unobtainable 
on curves, and especially on the approaches to curves when the 
outer rail is being raised. Even on tangents it is impossible 
to maintain a perfect surface, no matter how perfectly the 
track may" have been laid. In consequence of this, the points 



§412. 



ROLLING-STOCK. 



445 



of contact of the wheels of a locomotive, or even of a four- 
wheeled truck, will not ordinarily lie in one plane. The rougher 
and more defective the track, the worse the condition in this 
respect. Since the frame of a locomotive is practically rigid, 
and the frame rests on the driver-axles through the medium of 
springs at each axle-bearing, the compression, of the springs 
(and hence the pressure of the drivers on the rail) will be varia- 
ble if the bearing-points of the drivers are not in one plane 
This variable pressure affects the tractive power and severely 
strains the frame. Applying the principle that a tripod will 
stand on an uneven surface, a mechanism is employed which 

lb 




Fig. 195. — Action of Eqxjalizing-levers. 

Virtually supports the locomotive on three points, of which one 
is usually the center-bearing of the forward truck. On each 
side the pressure is so distributed among the drivers that even 
it a driver rises or falls with reference to the others, the load 
carried by each driver is unaltered, and that side of the engine 
rises or falls by one nth of the rise or fall of the single driver, 
where n represents the number of wheels. The principle in- 
volved is shown in an exaggerated form in Fig. 195. In the 
diagram, MN represents the normal position of the frame when 
the wheels are on line. The frame is supported by the hangers 
at a^ c, /, and h. ab, de, and gh are horizontal levers vibrating 
about the points H\ K, and L, which are supported by the 
axles. While it is possible with such a system of levers to make 



446 RAILROAD CONSTRUCTION. § 412. 

MN assume a position not parallel with its natural position, 
yet; by an extension of the principle that a beam balance loaded 
with equal weights will always be horizontal, the effect of rais- 
ing or lowering a wheel will be to move MN parallel to itself. 
It only remains to determine how much is the motion of MN 
relative to the rise or drop of the wheel. 

The dotted lines represent the positions of the wheels and 
levers when one wheel drops into a depression. The wheel 
center drops from p to g, a distance m. L drops to U, a 
distance m (s33 Fig. 195, 6); M drops to M', an unknown dis- 
tance X] therefore aa' =x] hb' —x] cc' =x; dd' = 3x = ee''y ff = x', 
.'. gg' = 5x; hh'=x; LU — \{gg' -\-hh') = \{Qx)='m] .'. ic = |m; 
i.e., MN drops, parallel to itself, 1/n as much as the wheel 
drops, where n is the number of wheels. The resultant effect 
caused by the simultaneous motion of two wheels with refer- 
ence to the third is evidently the algebraic sum of the effects 
of each wheel taken, separately. 

The practical benefits of this device are therefore as follows: 

(a) When any driver reaches a rough place in the track, a 
high place or a low place, the stress in all the various hangers 
and levers is unchanged. 

(6) The motion of the frame (represented by the bar MN 
m Fig. 195) is but \/n of the motion of the wheel, and the jar 
and vibration caused by a .roughness in the track is correspond- 
ingly reduced. 

The details of applying these principles are varied, but in 
general it is done as follows: 

(a) American and ten wheeled types. Drivers on each side 
form a system. The center-bearing pilot-truck is the third 
point of support. The method is illustrated in Fig. 196. 

(b) Mogul and consolidation types. The front pair of drivers 
is connected with the two-wheeled pilot-truck (as illustrated 
in Fig. 197) to form one system. The remaining drivers on 
each side are each formed into a system. 

The device of equalizers is an American invention. Until 
recently it has not been used on foreign locomotives. The 
necessity for its use becomes less as the track is maintained 
with greater perfection and is more free from sharp curves. A 
locomotive not equipped > with this device would deteriorate 
very rapidly on the comparatively rough tracks which are 
usually found on hght-traffic roads, It is still an open ques- 



§41^. 



ROLLING-STOCK 



447 





tion to what extent the neglect of this device is responsible for 
the statistical fact thataverage freight-train loads on foreign 



448 RAtLRdAD CONSTRUCfiO]^. §413. 

trains are less in proportion to the weight on the drivers than 
is the case with American practice. The recent increasing use 
of this device on foreign heavy freight locomotives is perhaps 
an acknowledgment of this principle. 

413. Counterbalancing. At very high velocities the cen- 
trifugal force developed by the weight of the rotating parts 
becomes a quantity which cannot be safely neglected. These 
rotating parts include the crank-pin, the crank-pin boss, the 
side rod, and that part of the weight of the connecting-rod 
which may be considered as rotating about the center of the 
crank-driver. As a numerical illustration, a driving-wheel 
62" in diariieter, running 60 miles per hour, will revolve 325 
times per minute. The weights are: 

Crank-pin 110 lbs. 

boss 150 "* 



One-half side rod 240 






Back end of connecting-rod (56%) . . . 190 

Total .' 690 lbs. 

If the stroke is 24", the radius of rotation is 12", or 1 foot. Then 

-gi= 32.2X1X6Q2 =^^^^^ ^^'^ 

which is half as much again as the weight on a driver, 16000 lbs. 
Therefore if no counterbalancing were !iused, the pressure be- 
tween the drivers and the raih would always be less (at any 
v:^locity) when the crank-pin was at its' highest point. At a 
velocity of about 48 miles per hour the pressure would become 
zero, and at higher velocities the wheel would actually be 
thrown from the rail. As an additional objection, when the 
crank-pin was at the lowest point, the rail pressure would be 
increased (velocity 60 miles per hour) from 16000 lbs. to nearly 
41000 lbs., an objectionably high pressure. These injurious 
effects are neutralized by ''counterbalancing." Since all of 
the above-mentioned weights can be considered as concen- 
trated at the center of the crank-pin, if a sufficient weight is so 
placed in the drivers that the center of gravity of the eccentric 
weight is diametrically opposite to the crank-pin, this centrifu- 
gal force can be wholly balanced. This is done by filling up 
a portion of the space between the spokes. If the center of 
■gravity of the counterbalancing weight is 20" from the center, 
then, since the crank-pin radius is 12", the required weight 
would be 690X^1 = 414 lbs. 



§ 413. ROLLING-STOCK. • 449 

/ 

In addition to the effect of these revolving parts there is 

the effect of the sudden acceleration and retardation of the 

reciprocating parts. In the engine above considered the weights 

of these reciprocating parts will be: 

Front end of connecting-rod (44%) . . 150 lbs. 

Cross-head; 174 ' ' 

Piston and piston-rod 300 '' 

Total 624 lbs. 

Assume as before that the reciprocating parts may be con- 
sidered as concentrated at one point, the point P of the dia- 
gram in Fig. 198, Since the 

motion of P is horizontal y'"" ""^,^\A/' 

only, the force required to / /^ i"^P \ 

overcome its inertia at any^ ^. / [ ]/^_jJi-\ 

point will exactly equal p^ " — ^***-i..^\L/''^i^ 

the horizontal component of ^S' ' 

the force required to over- ^^-^^^^ ^^/ 

come the inertia of an equal 

„r^,- 1,+ «+ e ^^,r^Ur;.,^ i^ FiG. 198. — Action of Cox nterbalance. 

weight at o revolvmg m 

a circular path. Then evidently the horizontal component of 
the force required to keep W in the circular path will exactly 
balance the force required to overcome the inertia of P. Of 
course W=P. But a smaller weight TF', whose weight is 
inversel}'^ proportional to its radius of rotation, wdll evidently 
accomplish the same result. In the above numerical case, if 
the center of gravity of the counterweights is 20'' from the 
center, the required weight to completely counterbalance 
the reciprocating parts would be 624 X|^f = 374.4. lbs. This 
counterweight need not be all placed on the driver carrying 
the main crank-pin, but can be (and is) distributed among all 
the drivers. Suppose it were divided between the two drivers 
in the above case. At 60 miles per hour such a counterweight 
would produce an additional pressure of 11211 lbs, when the 
counterweight was down, or a lifting force of the same amount 
when the counterweight was up. Although this is not suffi- 
cient to lift the driver from the rail, it would produce an objec- 
tionably high pressure on the rail (over 27000 lbs,), thus inducing 
just what it was desired to avoid on account of the eccentric 
rotating parts. Therefore a compromise must be made. Only 
a portion (one half to three fourths) of the weight of the recip- 
rocating parts is balanced. Since the effect of the rotating 



450 RAILROAD CONSTRUCTION. § 413. 

weights is to cause variable pressure on the rail, while the effect 
of the reciprocating parts is to cause a horizontal wobbling or 
"nosing" of the locomotive, it is impossible to balance both. 
Enough counterweight is introduced to partially neutralize the 
effect of the reciprocating parts, still leaving some tendency to 
horizontal wobbling, while the counterweights which were 
introduced to reduce the wobbling cause some variation of 
pressure. The vertical or horizontal pressure developed by the 
unbalanced rotating and reciprocating parts is called the 
dynamic augment. 

An additional injurious effect on the track of the dynamic 
augment is due to the fact that the center of gravity of the side 
rod is several inches outside of the vertical plane in which the 
counterweight revolves, and that the center of gravity of the 
main rod, or connecting rod, is still further outside. The dy- 
namic augment will be increased by the ratio of the distance 
between these planes of rotation to the distance between the 
centers of the companion drivers. This ratio averages about 
11% for the side rods and for the part of the pin within the side 
rod; the corresponding figure for the main rod is about 23%. 
The physical effect of the dynamic augment on the stresses 
produced in the track is further discussed in Chapter XXV, 

By using hollow piston-rods of steel, ribbed cross-heads, 
and connecting- and side-rods with an I section, the weight 
of the reciprocating parts may be greatly lessened without 
reducing their strength, and with a decrease in weight the 
effect of the unbalanced reciprocating parts and of the " ex- 
cess balance " (that used to balance the reciprocating parts) is 
largely reduced. 

Current practice is somewhat variable on three features : 

(a) The proportion of the weight of the connecting-rod which 
should be considered as revolving weight. 

(&) The proportion of the total reciprocating weight that 
should be balanced. 

(c) The distribution among the drivers of the counterweight 
to balance the reciprocating parts. 

The principal rules which have been formulated for counter- 
balancing may be stated as follows, although there is consider- 
able variation in the figures used in rules 2 and 3. 

1. Each wheel should be balanced correctly for the revolving 
parts connected with it. 



§414. 



ROLLING-STOCK. 



451 



2. In addition, introduce counterbalance sufficient for 50% 
of the weight of the reciprocating parts for ordinary engines, 
increasing this to 75% when the reciprocating parts are exces- 
sively heavy (as in compound locomotives) or when the engine 
is light and unable to withstand much lateral strain or when 
the wheel-base is short. 

3. Consider the weight of the connecting-rod as ^ revolving 
and ^ reciprocating when it is over 8 feet long; when shorter 
than 8 feet, consider ^^ of the weight as revolving and -^-^ as 
reciprocating. 

4. The part of the weight of the connecting-rod considered 
as revolving should be entirely balanced in the crank-driver wheel. 

5. The "excess balance" should be divided equally among 
the drivers. 

6. Place the counterbalance as near the rim of the wheel 
as possible and also as near the outside 
of the wheel as possible in order that 
the center of gravity shall be as near 
as possible opposite the center of 
gravity of the rods, etc., which are all 
outside of even the plane of the face 
of the wheel. 

In Fig. 199 is shown a section of a 
locomotive driver with the cavities in 
the casting for the accommodation of 
the lead which is used for the counter- 
balance weight. Incidentally several 
other features and dimensions are shown 
in the illustration. 

414. Mutual relations of the boiler power, tractive power, 
and cylinder power for various types. The design of a locomo- 
tive includes three distinct features which are varied in their 
mutual relations according to the work which the engine is 
expected to do. 

(a) The boiler power. This is limited by the rate at which 
steam may be generated in a boiler of admissible size and weight. 
Engines which are designed to haul very fast trains which are 
comparatively light must be equipped with very large grates and 
heating surfaces so that steam may be developed with great 
rapidity in order to keep up with thQ yery rapid consumption. 




Fig. 199. — Section op 
Locomotive-driver. 



452 RAILROAD CONSTRUCTION. §414. 

Engines for very heavy freight work are run at very much 
lower velocity and at a lower piston speed in spite of the fact 
that more strokes are required to cover a givea distance and 
the demand on the boiler for rajdd steam production is not 
as great as with high-speed passenger-engines. The capacity of 
a boiler to produce steam is therefore limited by the limiting 
weight of the general type of engine required. Although im- 
provements may be and have been made in the design of fire- 
boxes so as to increase the steam-producing capacity without 
adding proportionately to the weight, yet there is a more or less 
definite limit to the boiler power of an engine of given weight. 

(b) The tractive power. This is limited by the possible driver 
adhesion. The absolute limit of tractive adhesion between a 
steel- tired wheel and a steel rail is about one-third of the pressure, 
but not more than one-fourth gf the weight on the drivers can 
be depended on for adhesion and wet rails will often reduce 
this to one fifth and even less. The tractive power is therefore 
absolutely limited by the practicable weight of the engine. In 
some designs, when the maximum tractive power is desired, not 
only is the entire weight of the boiler and running gear thrown 
on the drivers, but even the tank and fuel-box are loaded on. 
Such designs are generally employed in switching-engines (or 
on engines designed for use on abnormally heavy mountain 
grades) in which the maximum tractive power is required, but 
in which there is no great tax on the boiler for rajpid steam pro- 
duction (the speed being always very low), and the boiler and 
fire-box, which furnish the great bulk of the weight of an engine, 
are therefore comparatively light, and the requisite weight for 
traction must, therefore, be obtained b}- loading the drivers 
as much as possible. On the other hand, engines of the liighest 
speed cannot possibly produce steam fast enough to maintain ' 
the required speed unless the load be cut down to a compara- 
tively small amount. The tractive power required for this 
comparatively small load will be but a small part of the weight 
of the engine, and therefore engines of this class have but a 
small proportion of their weight on the drivers; generally 
have but two driving-axles and sometimes but one. 
• (c) Cylinder power. The running gear forms a mechanism 
which is simply a means of transforming the energy of the boiler 
into tractive force and its power is unlimited, within the prac- 
tical conditions of the problem. The power of the running 



§414. 



ROLLING-STOCK. 



453 



gear depends on the steam pressure, on the area of the piston, 
on the diameter of the drivers, and. on the ratio of crank-pin 
radius to wheel radius, or of stroke to driver diameter. It 
is always possible to increase one or more of these elements 
by a relatively small increase of expenditure until the cylinders 
are able to make the drivers slip, assuming a sufficiently great 
resistance. Since the power of the engine is limited by the 
power of its weakest feature, and since the running gear is the 
most easily controlled feature, the power of the running gear 
(or the "cylinder power") is always made somewhat excessive 
on all well-designed engines. It indicates a badly designed 
engine if it is stalled and unable to move its drivers, the steam 
pressure being normal. If it is attempted to use a freight- 
engine on fast passenger service, it will probably fail to attain 
the desired speed on account of the steam pressure falling. 
The tractive power and cylinder power are superabundant, but 
the boiler cannot make steam as fast as it is needed for high 
speed, especially when the drivers are small. The practical 
result would be a comparatively low speed kept up with a forced 
fire. If it is attempted to use a high-speed passenger-engine 
on heavy freight service, the logical result is a slipping of the 
drivers until the load is reduced. The boiler power and cylinder 
power are ample, but the weight on the drivers is so small that 
the tractive power is only sufficient to draw a comparatively 
small load. 

These relations between boiler, cylinder, and tractive power 
are illustrated in the following comparative figures referring 
to a fast passenger-engine, a heavy freight-engine, and a switch- 
ing-engine. The weights of the passenger- and freight-engines 
are about the same, but the passenger-engine has only 74% * 





Cylinders. 


Total 
Wght. 


Wt. on 
Driv'rs 


Heat- 
ing 
Sur- 
face, 

sq. ft. 

1831.8 
1498 . 3 
1498.0 


Grate 
area 

sq. ft. 


Steam 
Pres- 
sure in 
Boiler. 


Stroke. 


Kind. 


Diam. 
Driver. 


Fast passenger . 
Heavy freight . 
Switcher 


19"X24" 
20"X24" 
19"X24" 


126700 
128700 
109000 


81500 
112600 
109000 


26.2 
31.5 

22.8 


180 
140 
160 





* Computed from Eq. 103. 



454 KAILROAD CONSTRUCTION. § 415. 

of the tractive power of the freight. But the passenger-engine 
has 22% more heating-surface and can generate steam much 
faster; it makes less than two-thirds as many strokes in cover- 
ing a given distance, but it runs at perhaps twice the speed 
and probably consumes steam much faster. The switch- 
engine is lighter in total weight, but the tractive power is a little 
greater than the freight and much greater than the passenger- 
engine. While the heating-surfaces of the freight- and switch- 
ing engines are practically identical, the grate area of the switcher 
is much less; its speed is always low and there is but little neces- 
sity for rapid steam development. 

While these figures show the general tendency for the relative 
proportions, and in this respect may be considered as typical, 
there are large variations. The recent enormous increase in 
the dead weight of passenger-trains has necessitated greater 
tractive power. This has been provided sometimes by using 
the " Pacific " type, which combines rapid steaming capacity 
and great tractive power. On the other hand, the demand for 
fast-freight service, and the possibility of safely operating such 
trains by the use of air-brakes, has required that heavy freight- 
engines shall be run at comparatively high speeds, and that 
requires the rapid production of steam, large grate areas and 
heating surfaces. But in spite of these variations, the normal 
standard for passenger service is a four-driver engine carrying 
about two-thirds of the weight of the engine on the drivers, 
which are very large; the normal standard for freight work is 
an 8-driver engine with perhaps 90% of the weight on the 
drivers, which are small, but which must have the pony truck 
for such speed as it uses; and finally the normal standard for 
switching service has all the weight on the drivers and has com- 
paratively low steam-producing capacity. 

415. Life of Locomotives, The Ufe of locomotives (as a 
whole) may be taken as about 800000 miles or about 22 to 24 
years. W^hile its life should be and is considered as the period 
between its construction and its final consignment to the scrap 
pile, parts of the locomotive may have been renewed more 
than once. The boiler and fire-box are especially subject to 
renewal. The mileage life is much longer than formerlj^ This 
is due partly to better design and partly to the custom of 
drawing the fires less frequently and thereby avoiding some 
of the destructive strains caused by extreme alternations of. 



§ 416. R0LL1NG-3T0CK. 455 

heat and cold. Recent statistics give the average annual 
mileage on twenty-three leading roads to be 41000 miles. 

CARS. 

416. Capacity and size of cars. The capacity of freight-cars 
has been enormously increased of late years. In 1870 the usual 
live-load capacity for a box-car was about 20000 lbs. In 1916, 
out of 58299 box cars owned by the Pennsylvania R. R., 32923 
or 56% had a capacity of 100000 or over; 49597 or 85% had a 
capacity 70000 or over; only 555, less than 1%, had a capacity 
of less than 60000 lbs., and the most of these were refrigerator 
cars or cars for special service. The Norfolk & Western R. R. 
had (in 1916), 750 gondola drop-bottom coal cars, each with a 
nominal capacity of 180000 lbs.; their length is 46 feet 10| 
inches, and the extreme width 10 feet 4| inches. These cars 
are carried on six-wheel trucks. The usual width of freight- 
cars is about 9 to 10 feet, while parlor-cars and sleepers are 
generally 10 feet wide and sometimes 11 feet. The highest 
point of a train is usually the smokestack of the locomotive, 
which is generally 15 feet above the rails and occasionally over 
16 feet. A sleeping-car usually has the highest point of the 
car about 14 feet above the rails. Box-cars are usually about 
8 feet high (above the sills), with a total height of 13 to 14 feet. 
Some furniture and automobile cars, whose unit live load per 
cubic foot of space is not high, have a total height of over 15 feet. 
The average length of freight cars, as required in the design of 
freight yards, is now considered to be 42 feet; the allowance for 
each car was formerly 40 feet. The P. R. R. standards vary 
between 38 feet 1 inch and 44 feet 6 inches in length. Day 
coaches have an extreme length varying from 45 to 80 feet. An 
80-foot all-steel coach weighs about 118000 lbs. and has a seating 
capacity of 88. Allowing the high average weight of 150 lbs., 
the maximum live load would be 13200 lbs., a little over 11% of 
the dead load, which shows that the tractive force required to 
haul the car will be almost constant, whether the car is full or 
empty. A dining-car may weigh 150000 lbs. and a sleeper even 
more. The weight of the 25 or 30 passengers it maij carry is 

- hardly worth considering in comparison. 

417. Stresses to which car-frames are subjected. A car 
is structurally a truss, supported at points at some distance 
from the ends and subjected to transverse stress. There is, 



456 



RAILROAD CONSTRUCTlJOIsr. 



§418. 



therefore, a change of flexure at two points between the trucks. 
Besides this stress the floor is subjected to compression when, 
the cars are suddenly stopped and to tension when in ordinary j 
motion, the tension being greater as the train resistance is i 
greater and as the car is nearer the engine. The shocks, jars, 
and sudden strains to which the car-frames are subjected are 
very much harder on them than the mere static strains due to 
their maximum loads if the loads were quiescent. Consequently 
any calculations based on the static loads are practically value- 
less, except as a very rough guide, and previous experience 
must be relied on in designing car bodies. As evidence of the 
increasing demand for strength in car-frames, it has been re- 
cently observed that freight-cars, built some years ago and 
built almost entirely of wood, are requiring repairs of wooden 
parts which have been crushed in service, the wood. being per- 
fectly sound as regards decay. ' 
' ' 418. The use of metal. The use of metal in car construction 




Fig. 201. 

is very rapidly increasing. The demand for greater strength 
in car-frames has gro-v\m until the M'^ooden framing has become 
so heavy that it is found possible to make steel frames and 
trucks at a small additional cost, the steel frames being twice 
as strong and yet reducing the dead weight of the car about 
5000 lbs., a consideration of no small value, especially on roads 
having heavy grades. Another reason for the increasing use 
of metal is the great reduction in the price of rolled or pressed 




:10aOOO-LB. Box Car. 




Steel Coal Car. 




Wooden Box Lak, ^JrEEL 1?iiamb.j 
Fiei. 200. — Some Heavt FsEittHT Casb, 
(To face page 456.) 



§ 419. ROLLING-STOCK. 457 

steel, while the cost of wood is possibly higher than before. 
The advocates of the use of steel advise steel floors, sides, etc. 
For box-cars a wooden floor has advantages. For ore and 
coal-cars an all-metal construction has advantages. (Fig. 200.) 
In Germany, where steel frames have been almost exclusively 
in use for many years, they have not yet been able to determine 
the normal age limit of such frames ; none have yet worn out. 
The life is estimated at 50 to 80 years 

Brake beams are also best made of metal rather than wood, 
as was formerly done. Metal brake-beams are generally used on 
cars having air-brakes, as a wooden beam must be excessively 
large and heavy in order to have sufficient rigidity. 

Truck-frames (see Fig. 201), which were formerly made prin- 
cipally of wood, are now largely made of pressed steel. It makes 
a reduction in weight of about 3000 lbs. per car. The increased 
durability is still an uncertain quantity. 

419. Draft gear. The enormous increase in the weight and 
live load capacities of rolling stock have necessitated a corre- 
sponding development in draft gear. Even within recent years, 
"coal- jimmies," carrying a few tons have been made up into 
trains by dropping a chain of three big links over hooks on the 
ends of the cars. But the great stresses due to present loadings 
would tear such hooks from the cars or tear the cars apart if 
such cars were used in the make-up of long heavy trains as now 
operated. The next stage in the development of draft gear was 
the invention of the "spring coupler," by which the energy due 
to a sudden tensile jerk or the impact of compression may be 
absorbed by heavy springs and gradually imparted to the car 
body. Such devices, for which there are many designs, seemed 
to answer the purpose for cars of 25 to 40 tons capacity. The 
use of 100,000-pound steel cars soon proved the inadequacy of 
even spring couplers. The friction-draft gear was then in- 
vented. The general principle of such a gear is that, when 
acting at or near its maximum capacity, it harmlessly trans- 
forms into heat the excessive energy developed by jerks or 
i compression. There are several different designs of such gear, 
but the general principle underlying all of them may be illus- 
trated by a description of the Westinghouse draft gear. The 
gear employs springs which have sufficient stiffness to act as 
ordinary spring-couplers for the ordinary pushing and pulling 
of train operations. Sections of the gear are shown in Fig. 202, 



458 



EAILROAD CONSTRUCTION. 



411 




m 

M 

H 



H 
O 

« 

Q 
w 

m 
& 
O 

W 

iz; 

M 

m 



O 

d 

M 



[§ 420. HOLLING-SfOCK. 459 

while the method of its application to the framing of a car of 
the pressed steel type is shown in Fig. 203, a and h. When 
the draft gear is in tension the coupler, which is rigidly attached 
to B, is drawn to the left, drawing the follower Z with it. Com- 
pression is then exerted through the gear mechanism to the 
follower A which, being restrained by the shoulders RR, against 
which it presses, causes the gear to absorb the compression. 
The coil-spring C forces the eight wedges n against the eight 
corresponding segments E, The great compression of these 
surfaces against the outer shell produces a friction which retards 
the compression of the gear. The total possible movement of 
the gear, as determined by an official test, was 2.42 inches, when 
the maximum stress was 180,000 pounds. The work done in 
producing this stress amounted to 18,399 foot-pounds. Of this 
total energy 16,666 foot-pounds, or over 90%, represents the 
amount of energy absorbed and dissipated as heat by the 
frictional gear. The remaining 10% is given back by the 
recoil. The main release spring K is used for returning the 
segments and friction strips to their normal position after the 
force to close them has been removed. It also gives additional 
capacity to the entire mechanism. The auxiliary spring L 
releases the wedge D, whUe the release pin M releases the pres- 
sure of the auxiliary spring L against the wedge during fric- 
tional operation. If we omit from the above design the fric- 
tional features and consider only the two followers A and Z, 
separated by the springs C and K, acting as one spring, we have 
the essential elements of a spring-draft gear. In fact, this 
gear acts exactly like a spring-draft gear for all ordinary service, 
the frictional device only acting during severe tension and com- 
pression. 

420. Gauge of wheels and form of wheel-tread. — In Fig. 20 1 
is shown the standard adopted by the Master Car Builders* 
Association at their twentieth annual convention. Note the 
normal position of the gauge-line on the wheel-tread. In 
Fig. 118, § 267, the relation of rail to wheel-tread is shown 
on a smaller scale. It should be noted that there is no definite 
position where the wheel-flange is absolutely "chock-a-block" 
against the rail. As the pressure increases the wheel mounts 
a little higher on the rail until a point is soon reached when the 
resistance is too great for it to mount still higher. By this 
means is avoided the shock of unyielding impact when the car 



460 



EAII.BOAD CONSTKtrCTION. 



§420. 




o o o 



) 



_0-Cua_ 



^^ 



u 



□ 



□I 



CO 



LL 



ii 



N 



3^ 



a 



( 



c 

Q 



ca 



o 

to 



b A d'^ 



n 




§421. ROLLING-STOCK. 4G1 

sways from side to side. When the gauge between the inner 
faces of the wheels is greater or less than the limits given in 
the figure, the interchange rules of the Master Car Builders' 
Association authorize a road to refuse to accept a car from 
another road for transportation. At junction points of rail- 
roads inspectors are detailed to see that this rule (as well as 
many others) is complied with in respect to all cars offered 
for transfer. 

TRAIN-BRAKES. 

421. Introduction. Owing to the very general misappre- 
hension that exists regarding the nature and intensity of the 
action of brakes, a complete analysis of the problem is con- 
sidered justifiable, This misapprehension is illustrated by the 
common notion (and even practice) that the effectiveness of 
braking a car is proportional to the brake pressure, and there- 
fore a brakeman is frequently seen using a bar to obtain a 
greater leverage on the brake-wheel and using his utmost 
strength to obtain the maximum pull on the brake-chain while- 
the car is skidding along with locked wheels. 

When a vehicle is moving on a track with a considerable 
velocity, the mass of the vehicle possesses kinetic energy of 
translation and the wheels possess kinetic energy of rotation. 
To stop the vehicle, this energy must be destroyed. The 
rotary kinetic energy will vary from about 4 to 8% of the 
kinetic energy of translation, according to the T3ar loading 
(see § 435). On steam railroads brake action is obtained by 
pressing brake-shoes against car-wheel treads. As the brake- 
shoe pressure increases, the brake-shoes retard with increasing 
force the rotary action of the wheels. As long as the wheels 
do not slip or "skid" on the rails, the adhesion of the rails 
forces them to rotate with a circumferential velocity equal to 
the train velocity. The retarding action of the brake-shoe 
checks first the rotative kinetic energy (which is small), and 
the remainder develops a tendency for the wheel to slip on the 
rail. Since the rotative kinetic energy is such a small per- 
centage of the total, it will hereafter be ignored, except as 
specifically stated, and it will be assumed for simplicity that 
the only work of the brakes is to overcome the kinetic energy 
of translation. The possible effect of grade in assisting or 
preventing retardation, and the effect of all other track resist- 



462 



RAILEOAD CONSTRUCTION. 



§421. 




Fia. 204. — M. C. B. Standard Wheel-tread and Axle, (1918.) 



§ 422. ROLLING-STOCK. 463 

ances, is also ignored. The amount of the developed force 
which retards the train movement is limited to the possible 
adhesion or static friction between the wheel and the rail. 
When the friction between the brake-shoe and the wheel ex- 
ceeds the adhesion between the wheel and the rail, the wheel 
skids, and then the friction between the wheel and the rail 
at once drops to a much less quantity. It nmst therefore be 
remembered at the outset that the retarding action of brake- 
shoes on wheels as a means of stopping a train is absolutely- 
limited by the possible static friction between the braked 
wheels and the rails. 

422. Laws of friction as applied to this problem. Much of 
the misapprehension regarding this problem arises from a very 
common and widespread misstatement of the general laws of 
friction. It is frequently stated that friction is independent 
of the velocity and of the unit of pressure. The first of these 
so-called law^s is not even approximately true. A very exhaus- 
tive series of tests were made by Capt. Douglas Galton on the 
Brighton Railway in England in 1878 and 1879, and by M. 
George Marie on the Paris and Lyons Railway in 1879, with 
trains which were specially fitted with train-brakes and with 
dynagraphs of various kinds to measure the action of the 
brakes. Experience proved that variations in the condition of 
the rails (wet or dry), and numerous irregularities incident to 
measuring the forces acting on a heavy body moving with a 
high velocity, were such as to give somewhat discordant re- 
sults, even when the conditions were made as nearly identical 
as possible. But the tests were carried so far and so persist- 
ently that the general laws stated below were demonstrated 
beyond question, and even the numerical constants were deter- 
mined as closely as they may be practically utilized. These 
laws may be briefly stated as follows: 

(a) The coefficient of friction between cast-iron brake-blocks 
and steel tires is about .3 when the wheels are "just mov- 
ing"; it drops to about .16 when the velocity is about 30 miles 
per hour, and is less than .10 when the velocity is 60 miles per 
hour. These figures fluctua,te considerably with the condition 
of the rails, wet or dry. 

(&) The coefficient of friction is greatest when the brakes 
are first applied; it then reduces very rapidly, decreasing 
nearly one third after the brakes have been applied 10 seconds^ 



4^4 RAILROAD CONSTRUCTION. § 422. 

and dropping to nearly one half in the course of 20 seconds. 
Although the general truth of this law was established beyond 
question, the tests to demonstrate the law of the variation of 
friction with time of application were too few to determine 
accurately the numerical constants. 

(c) The friction of skidded wheels on rails is always very 
much less than the adhesion when the wheel is rolling on the 
rail— sometimes less than one third as much. 

(d) An analysis of the tests all pointed to a law that the 
friction developed does not increase as rapidly as the intensity 
of pressure increases^, but this may hardly be considered as 
an established law. 

(e) The adhesion between the wheel and the rail appears to 
be independent of velocity. The adhesion here means the force 
that must be developed before the wheel will slip on the rail. 

The practical effect of these laws is shown by the following 
observed phenomena: 

(a) When the brakes are first applied (the velocity being 
very high), a brake pressure far in excess of the weight on the 
wheel (even three or four times as much) may be applied with- 
out skidding the wheel. This is partly due to the fact that 
the wheel has a very high rotative kinetic energy (which varies 
as the square of the velocity, and which must be overcome 
first), but it is chiefly due to the fact that the coefficient of 
friction at the higher velocity is very small (at 60 miles per 
hour it is about .07), while the adhesion between the wheel and 
the rail is independent of the velocity. 

(b) As' the velocity decreases the brake pressure must be 
decreased or the wheels will skid. Although the friction de- 
creases with the time required to stop and increases with the 
reduction of speed, and these two effects tend to neutralize 
each other, yet unless the stop is very slow, the increase, in 
friction due to reduction of speed is much greater than the 
decrease due to time, and therefore the brake pressure must 
not be greater than the weight on the wheel, unless momentarily 
while the speed is still very high. 

(c) The adhesion between wheels and rails varies from .20 
to .25 and over when the rail is dry. When wet and slippery 
it may fall to .18 or even .15. The use. of sand will always 
raise it above .20, and on a dry rail, when the sand is not blown 
away by wind, it may raise it to .35 or even .40. 



§423. EOLLING-STOCK. 465 

(<f) Experiments were made with an automatic valve by 
which the brake-shoe pressure against the wheel should be 
reduced as the friction increased, but since (1) the essential 
requirement is that the friction produced by the brake-shoes 
shall not exceed the adhesion between rail and wheel, and 
since (2) the rail-wheel adhesion is a very variable quantity, 
depending on whether the rail is wet or dry, it has been found 
impracticable to use such a valve, and that the best plan is to 
leave it to the engineer to vary the pressure, if necessary, by the 
use of the brake-valve. 

MECHANISM OF BRAKES. 

423. Hand-brakes. The old style of brakes consists of brake- 
shoes of some type which are pressed against the wheel-treads 
by means of a brake-beam, which is operated by means of a 
hand-windlass and chain operating a set of levers. It is desir- 
able that brakes shall not be set so tightly that the wheels 
shall be locked, and then slide over the track, producing 
flat places on them, which are very destructive to the 
rolling-stock and track afterward, on account of the impact 
occasioned at each revolution. With air-brakes the maximum 
pressure of the brake-shoes can be quite carefully regulated, 
and they are so designed that the maximum pressure exerted 
by any pair of brake-shoes on the wheels of any axle shall not 
exceed a certain per cent, of the weight carried by that axle 
when the car is empty, 90% being the figure usually adopted 
for passenger-cars and 70% for freight-cars. Consider the 
case of a freight-car of 100000 lbs. capacity, weighing 33100 lbs., 
or 8275 lbs. on an axle, and equipped with a hand-brake which 
operates the levers and brake-beams, which are sketched in 
Fig. 205. The dead weight on an axle is 8275 lbs.; 70% of 
this is 5792 lbs., which is the maximum allowable pressure 
per brake-beam, or 2896 lbs. per brake-shoe. With the dimen- 
sions shown, such a pressure will be produced by a pull of about 
1158 lbs. on the brake-chain. The power gained by the brake- 
wheel is not equal to the ratio of the brake-wheel diameter 
to the diameter of the shaft, about which the brake-chain 
winds, which is about 16 to 1|. The ratio of the circumfer- 
ence of the brake-wheel to the length of chain wound up by 
one complete turn would be a closer figure. The loss of efh- 



466 



KAILROAD CONSTRUCTION. 



§424. 



ciency in such a clumsy mechanism also reduces the effective 
ratio. Assuming the effective ratio as C : 1 it would require a 
pull of 193 lbs. at the circumference of the brake-wheel to 
exert 1158 lbs. pull on the brake-chain, or 5792 lbs. pressure 
on the wheels at B, and even this will not lock the wheels when 
the car is empty, much less when it is loaded. Note that the 
pressures at A and B are unequal. This is somewhat objec- 
tionable, but it is unavoidable with this simple form of brake- 
beam. More complicated forms to avoid this are sometimes 
used. Hand-brakes are, of course, cheapest in first cost, and 
even with the best of automatic brakes,, additional mechanism 
to operate the brakes by hand in' an emergency is always pro- 
vided, but their slow operation when a quick stop is desired 
makes it exceedingly dangerous to attempt to run a train at 
high speed unless some automatic brake directly under the 
control of the engineer is at hand. The great increase in the 




==3" 5.792 



Fig. 205. — Sketch of Mechanism of Hand-brake. 



average velocity of trains during recent j^ears has only been 
rendered possible by the invention of automatic brakes. 

424. "Straight" air-brakes. The essential constructive fea- 
tures of this form of brake are (1) an air-pump on the engine, 
operated by steam, which compresses air into a reservoir on 
the engine; (2) a "brake-pipe" running from the reservoir 
to the rear of the engine and pipes running under each car, 
the pipes having flexible connections at the ends of the cars 
and engine; (3) a cylinder and piston under each car which 



§ 425. eolling-stock:. 46? 

operates the brakes by a system of levers, the cyhnder being 
connected to the brake-pipe. The reservoir on the engine 
holds compressed air at about 45 lbs. pressure. To operate the 
brakes, a valve on the engine is opened which allows the com- 
pressed air to flow from the reservoir through the brake-pipe 
to each cylinder, moving the piston, wliich thereby moves the 
levers and applies the brakes. The defects of this system are 
many: (1) With a long train, considerable time is required for 
the air to flow from the reservoir on the engine to the rear cars, 
and for an emergency-stop even this delay would often be 
fatal; (2) if the train breaks in two, the rear portion is not 
provided with power for operating the brakes, and a dangerous 
collision would often be the result; (3) if an air-pipe couphng 
bursts under any car, the whole sj^stem becomes absolutely 
helpless, and as such a thing might happen during some emer- 
gency, the accident would then be especially fatal. 

This form of brake has almost, if not entirely, passed out of 
use. It is here briefly described in order to show the logical 
development of the form which is now in almost universal use, 
the automatic. 

425. Automatic air-brakes. The above defects have been 
overcome by a method which may be briefly stated as follows: 
A reservoir for compressed air is placed under each car and the 
tender; whenever the pressure in these reservoirs is reduced 
for any reason, it is automatically replenished from the main 
reservoir on the engine; whenever the pressure in the brake-' 
pipe is reduced for any cause (opening a valve at any point of 
its length, parting of the train, or bursting of a pipe or coupler), 
valves are automatically moved under each car to operate the 
piston and put on the brakes. All the brakes on the train are 
thus applied almost simultaneously. If the train breaks in two, 
both sections will at once have all the brakes applied automati- 
cally ; if a coupling or pipe bursts, the brakes are at once applied 
and attention is thereby attracted to the defect ; if an emer- 
gency should arise, such that the conductor desires to stop 
the train instantly without even taking time to signal to the 
engineer, he can do so by opening a valve placed on each car, 
which admits air to the train-pipe, which will set the brakes 
on the whole train, and the engineer, being able to discover 
instantly what had occurred, would shut off steam and do 
whatever else was necessary to stop the train as quickly as pos- 



4:68 EAtLROAD CONSTRUCTION. § 426. 

sible. The most important and essential detail of this system 
is the "automatic triple valve" placed under each car. Quot- 
ing from the Westinghouse Air-brake Company's Instruction 
Book, "A moderate reduction of air pressure in the train-pipe 
causes the greater pressure remaining stored in the auxiliary 
reservoir to force the piston of the triple valve and its slide- 
valve to a position which will allow the air in the auxiliary 
reservoir to pass directly into the brake-cylinder and apply the 
brake. A sudden or violent reduction of the air in the train- 
pipe produces the same effect, and in addition causes supple- 
mental valves in the triple valve to be opened, permitting the 
pressure from the train-pipe to also enter the brake-cylinder, 
augmenting the pressure derived from the auxiliary reservoir 
about 20%, producing practically instantaneous action of the 
brakes to their highest efficiency throughout the entire train. 
When the pressure in the brake-pipe is again restored to an 
amount in excess of that remaining in the auxiliary reservoir, 
the piston- and slide-valves are forced in the opposite direction , 
to their normal position, opening communication from the train- 
pipe to the auxiliary reservoir, and permitting the air in the 
brake-cylinder to escape to the atmosphere, thus releasing the 
brakes. If the engineer wishes to apply the brake, he moves 
the handle of the engineer's brake-valve to the right, which 
first closes a port, retaining the pressure in the main reservoir, 
.and then permits a portion of the air in the train-pipe to escape. 
To release the brakes, he moves the handle to the extreme 
left, which allows the air in the main reservoir to flow freely 
into the brake-pipe, restoring the pressure therein." 

426. Tests to measure the efficiency of brakes. Let v repre- 
sent the velocity of a train in feet per second; W, its weight; 
F, the retarding force due to the brakes; d, the distance in feet 
required to make a stop; and g, the acceleration of gravity 
(32.16 feet per square second); then the kinetic energy pos- 
sessed by the train (disregarding for the present the rotative 

kinetic energy of the wheels) =-^ — . The work done in stop- 

ping the train =Frf, .'.Fd = -^. The ratio of the retarding 

force to the weight, 

F v^ v^ 



§427. ROLLING-STOCK. 469 

In order to compare tests made under varying conditions, the 
ratio F-i-W should be corrected for the effect of grade ( + or — ), 
if any, and also for the proportion of the weight of the train 
which is on braked wheels. For example, a train weighed 
146076 lbs., the proportion on braked wheels was 67%, speed 
60 feet per second, length of stop 450 feet, track level. Sub- 
stituting these values in the above formula, we find (F-^W) 
= .124. This value is really unduly favorable, since the ordi- 
nary track resistance helps to stop the train. This has a value 
of from 6 to 20 lbs. per ton, averaging say 10 lbs. per ton dur- 
ing the stop, or .005 of the weight. Since the effect of this is 
small and is nearly constant for all trains, it may be ignored 
in comparative tests. The grade in this case was level, and 
therefore grade had no effect. But since only 67% of the 
weight was on braked wheels, the ratio, on the basis of all the 
wheels braked, or of the weight reduced to that actually on 
the braked wheels, is 0.124 -r- .67 = 0.185. This was called 
a " good " stop, although as high a ratio as 0.200 has been 
obtained. 

427. Brake-shoes. Brake-shoes were formerly made of 
wrought iron, but when it was discovered that cast-iron shoes 
would answer the purpose, the use of wrought-iron shoes was 
alSandoned, since the cast-iron shoes are so much cheaper. A 
cheap practice is to form the brake-shoe and its head in one 
piece, which is cheaper in first cost, but when the wearing-sur- 
face is too far gone for further use, the whole casting must be 
renewed. The ''Christie" shoe, adopted by the Master Car 
.Builders' Association as standard, has a separate shoe which 
is fastened to the head by means of a wrought-iron key. The 
shoe is beveled i'' in a width of 3f " to fit the coned wheel. 
This is a greater bevel than the standard coning of a car-wheel. 
It is perhaps done to allow for some bending of the brake- 
beam and also so that the maximum pressure (and wear) should 
come on the outside of the tread, rather than next to the 'flange, 
where it might tend to produce sharp flanges. ■ By concen- 
trating the brake-shoe wear on the outer side of the tread, the 
wear on the tread is more nearly equalized, since the rail wears 
the wheel-tread chiefly near the flange. This same idea is 
developed still further in the "flange-shoes," which have a 
curved form to fit the wheel-flange and which bear on the 
wheel on the flange and on the outside of the tread. It is 



470 KAILROAD CONSTRUCTION. §427. 

claimed that by this means the standard form of the tread is 
better preserved than when the wear is entirely on the tread. 
The Congdon brake-shoe is one of a type in which wrought- 
iron pieces are inserted in the face of a cast-iron shoe. It is 
claimed that these increase the life of the shoe. 



CHAPTER XVI. 
TRAIN RESISTANCE. 

428. Classification of the various forms. The various resist- 
ances which must be overcome by the power of the locomotive 
may be classified as follows : 

(a) Resistances internal to the locomotive, which include fric- 
tion of the valve-gear, piston- and connecting-rods, journal 
friction of the drivers; also all the loss due to radiation, con- 
densation, friction of the steam in the passages, etc. In short, 
these resistances are the sum-total of the losses by which the 
power at the circumference of the drivers is less than the power 
developed by the boiler. 

(6) Velocity resistances, which include the atmospheric resist- 
ances on the ends and sides; oscillation and concussion resist- 
ances, due to uneven track, etc. 

(c) Wheel resistances, which include the rolling friction be- 
tween the wheels and the rails of all the wheels (including the 
drivers) ; also the journal friction of all the axles, except those 
of the drivers. 

(d) Grade and curve resistances, which include those resist- 
ances which are due to grade and to curves, and which are not 
found on a straight and level track. 

(e) Brake resistances. As shown later, brakes consume 
power and to the extent of their use increase the energy to 
be developed by the locomotive. 

(/) Inertia resistances. The resistance due to inertia is not 
generally considered as a train resistance because the energy 
which is stored up in the train as kinetic energy may be util- 
ized in overcoming future resistances. But in a discussion 
of the demands on the tractive power of the engine, one of the 
chief items is the energy required to rapidly give to a starting 
train its normal velocity. This is especially true of suburban 
trains, which must acquire speed very quickly in order that 

471 



472 RAILROAD CONSTRUCTION. § 429. 

their general average speed between termini may be even reason- 
ably fast. 

429. Resistance internal to the locomotive. These are re- 
sistances which do not tax the adhesion of the drivers to the. 
rails, and hence are frequently considered as not being a part 
of the train resistance properly so called. If the engine were 
considered as lifted from the rails and made to drive a belt 
placed around the drivers, then all the power that reached the 
belt would be the power that is ordinarily available for adhe- 
sion, while the remainder would be that consumed internally 
by the engine. The power developed by an engine may be 
obtained by taking indicator diagrams which show the actual 
steam pressure in a cylinder at any part of a stroke. From 
guch a diagram the average steam pressure is easily obtained, 
and this average pressure, multiplied by the length of the stroke 
and by the net area of the piston, gives the energy developed 
by one hal f-stroke of one piston. Four times this product 
divided by 550 times the time in seconds required for one stroke 
gives the "indicated horse-power" Even this calculation 
gives merely the power behind the piston, which is several per 
cent, greater than the power which reaches the circumference 
of the drivers, owing to the friction of the piston, piston-rod, 
cross-head, connecting-rod bearings, and driving-wheel jour- 
nals. (See § 411, Chapter XV.) By measuring the amount 
of water used and turned into steam, and by noting the boiler 
pressure, the energy possessed by the steam used is readily 
computed. The indicator diagrams will show the amount of 
steam that has been effective in producing power at the cylin- 
ders. The steam accounted for by the diagrams will ordinarily 
amount to 80 or 85% of the steam developed by the boiler, 
and the other 15 or 20% represents the loss of energy due to 
radiation, condensation, etc. 

Locomotive resistance has been estimated and tabulated by a 
Committee of the Amer. Rwy. Eng. Assoc, and the results are 
given in Table XXIX, which is taken from the Manual of that 
Association. As a numerical illustration, what is the computed 
resistance for a Mikado locomotive of which the total weight of 
engine and tender is 315,000 lbs. of which 153,200 lbs. is carried 
on the drivers, at a velocity of 6 miles per hour ? In this case, 
Item A = (18.7X76.6) + (80X4) = 1432 lbs. The weight carried 
on the engine and tender trucks = 315,000 -153,200 = 161,800 



§430. 



TRAIN RESISTANCE. 



473 



=80.9 tons. Item B = (2.6X80.9) + (20X6) =330 lbs. Item C 
is comparatively insignificant at this low velocity. From the 
table, we read 9 lbs. Then the sum of A, B, and C = 1771 lbs,, 
which must be subtracted from a computed tractive effort to 
obtain the estimated draw-baf- pull. 

TABLE XXIX. LOCOMOTIVE HfeSlSTANCES. 

Total Locomotive Resistance =A -\-B-\rC, in whieh 

A = resistance between cylinder and rim of drivers, and in pounds 

= 18.7r+80iV 

in which T =tons weight on drivers, and 
N = number of dWvirig axies; 

B =r'esistance of engine and tender trucks, and in pounds 
=2.%T+2QN 

in which T = tons Weight on enginfe aad t'ender trucks 
and' N = number of truck axles; 

C=head end' or " air " resistance, and in pounds 
= .002F2j.' 

in which V = velocity in miles per hour, and 
A =end area of locomotive. 

On the basis that the end area averages 125 square feet, the formula 
becomes C=0.25F2_ The number of pounds air resistance for varioiiS ve- 
locities is as given below. 



Vel. 


Res. 


1 


0.25 


2 


1.00 


3 


2.25 


4 


4.00 


5 


6.25[ 


6 


9. 00-' 


7 


12.25| 



Vel. Res. 



8 1 16.00 

9 120.25 



10 
11 
12 
13 
14 



25 . 00 

30 

36 

42 

49 



Vel. 


Res; 


vei; 


Res. 


Vei: 


Res". 


tel. 


15 


56 


22 


121 


29 


210 


36 


16 


64 


2S 


132 


30 


225' 


3^ 


i 17 


72 


24 


144 


31 


240 


3?, 


18 


81 


25 


156 


32 


256 


39 


19 


90 


26 


169 


33 


272 


40 


20 


100 


27 


182 


34 


289 


50 


21 


110 


28 


196 


35 


306 


60 



^ 



,es. 



324 

342 

361 

38b' 

400 

625 

900' 



Draw-bar pull on level tangent equals the cylinder tratciiVe'pd'wef' lfe'# 
the suift of the engine' resistances. 

At low speeds, the adhesion of the drivers should be considered and avail- 
able draw-bar pull should never be estimated greater than 30% of weight 
on drivers at starting with use of sand, 25% of weight on drivers at' ruiining 
speeds. 



Taken from Table 7 in " Economics 
Rwy. Eng. Assoc, 1915 edition. 



section of Mahu&l of ' tbe' Adf^f*. 



430. Velocity resistance, (a) Atmospheric. This consists of 
the head and tail resistances and the side riesistance. Thtei'head 



/ 



474 RAILROAD CONSTRUCTION. §431. 

and tail resistances are nearly constant for all trains of given 
velocity, varying but slightly with the varying cross-sections 
of engines and cars. The side resistance varies with the length 
of the train and the character of the cars, box-cars or flats, etc. 
Vestibuling cars has a considerable effect in reducing this side i 
resistance by preventing much of the eddying of air-currents 
between the cars, although this is one of the least of the advan- 
tages of vestibuling. Atmospheric resistance is generally 
assumed to vary as the square of the velocity, and although 
this may be nearly true, it has been experimentally demon- 
strated to be at least inaccurate. Values for head resistance 
are given in Table XXIX, which are probably accurate enough 
for all practical purposes, especially at ordinary freight train 
velocities. A freight-train composed partly of flat-cars and 
partly of box-cars will encounter considerably more atmospheric 
resistance than one made exclusively of either kind, other things 
being equal. The definite information on this subject is very 
unsatisfactory, but this is possibly due to the fact that it is of 
little practical importance to know just how much such resistance 
amounts to. 

(6) Oscillatory and concussive. These resistances are con- 
sidered to vary as the square of the velocity. Probably this 
is nearly, if not quite, correct on the general principle that such 
resistances are a succession of impacts and the force of impacts 
varies as the square of the velocity. These impacts are due to 
the defects of the track, and even though it were possible to 
make a precise determination of the amount of this resistance 
in any particular case, the value obtained would only be true 
for that particular piece of track and for the particular degree 
of excellence or defect which the track then possessed. The 
general improvement of track maintenance during late years 
has had a large influence in increasing the possible train-load 
by decreasing the train resistance. The expenditure of money 
to improve track will give a road a large advantage over a 
competing road with a poorer track, by reducing train resist- 
ance, and thus reducing the cost of handling traffic. 

431. Wheel resistances, (a) Rolling friction of the wheels. 
To determine experimentally the rolling friction of wheels, 
apart from all journal friction, is a very difficult matter and 
has never been satisfactorily accomplished. Theory as well 
as practice shows that the higher and the more perfect the 



§ 431. TRAIN RESISTANCE. 475 

elasticity of the wheel and the surface, the less will be the roll- 
ing friction. But the determination, if made, would be of 
theoretical interest only. 

The combined effect of rolling friction and journal friction 
is determinable with comparative ease. From the nature of 
the case no great reduction of the rolling friction by any device 
is possible. It is only a very insignificant part of the total 
train resistance. 

(b) Journal friction of the axles. This form of resistance has 
been studied quite extensively by means of the measurement 
of the force required to turn an axle in its bearings under 
various conditions of pressure, speed, extent of lubrication, 
and temperature. The following laws have been fairly well 
established: (1) The coefficient of friction increases as the pres- 
sure diminishes; (2) it is higher at very slow speeds, gradually 
diminishing to a minimum at a speed corresponding to a train 
velocity of about 10 miles per hour, then slowly increasing 
with the speed; it is very dependent on the perfection of the 
lubrication, it being reduced to one sixth or one tenth, when the 
axle is lubricated by a bath of oil rather than by a mere pad 
or wad of waste on one side of the journal; (3) it is much lower 
at higher temperature, and vice versa. The practical effect of 
these laws is shown by the observed facts that (1) loaded cars 
have a less resistance per ton than unloaded cars, the figures 
being, for speeds of about 10 miles per hour, approximately: 

For passenger- and loaded freight-cars . . 4 lbs. per ton 

" empty freight-cars 8 " " " 

" street-cars.... 10 " " " 

" freight-trucks without load 14 " " " 

(2) When starting a train, the resistances are about 20 lbs. 
per ton, notwithstanding the fact that the velocity resistances 
are practically zero; at about 2 miles per hour it will drop to 
10 lbs. per ton and above 10 miles per hour it may drop to 
4 lbs. per ton if the cars are in good condition. (3) The re- 
sistance could probably be materially lowered if some practicable 
form of journal-box could be devised which would give a more 
perfect lubrication. (4) It is observed that freight-train loads 
must be cut down in winter by about 10 or 15% of the loads 
that the same engine can haul over the same track in summer. 
This is due partly to the extra roughness and inelasticity of the 



476 



RAILROAD CONSTRUCTION. 



§432. 



track in winter, and partly to increased radiation from the 
engine wasting some energy, but this will not account for all 
of the loss, and the effect, which is probably due largely to the 
lower temperature of the journal-boxes, is very marked and 
costly. It has been suggested that a jacketing of the journal- 
boxes, which would prevent rapid radiation of heat and enable 
them to retain some of the heat developed by friction, would 
result in a saving amply repaying the cost of the device. 

Roller journals for cars have been frequently suggested, amd 
experiments have been made with them. It is found that they 
are very effective at low velocities, greatly reducing the start- 
ing resistance, which is very high with the ordinary forms of 
journals. But the advantages disappear as the velocity in- 
creases. The advantages also decrease as the load is increased, 
so that with heavily loaded cars the gain is small. The excess 
of cost for construction and maintenance has been found to be 
more than the gain from power saved. 

432. Grade resistance. The amount of this may be com- 
puted with mathematical exactness. Assume that the bail 
or cylinder (see Fig. 206) is being drawn up the plane. If W 




Fig. 206. 
is the weight, N the normal pressure against the rail, and O 
the force required to hold it or to draw it up the plane with 
uniform velocity, the rolling resistances being considered zero 
or considered as provided for by other forces, then 

G:W::h:d, or G = ^; 

d 

but for all ordinary railroad grades, d=c io within a tenth of 

Wh 
1%, i.e., G = = IT X rate of grade. In order that the student 

may appreciate the exact amount of this approximation the per- 
centage of slope distance to its horizontal projection is given in 
the following tabular farm: 



§432. 



TRAIN RESISTANCE. 



477 



Grade in per cent. 


1 


2 


3 


4 


5 


Slope dist.^^QQ 

hor. dist. 


100.005 


100.020 


100.045 


100.080 


100.125 



Grade in per cent. 


6 


7 


8 


9 


10 


^''^^'^'fxioo 

nor. dist. 


100.180 


100.245 


100.319 


100.404 


100.499 



This shows also the error on various grades of measuring with 
the tape on the ground rather than held horizontally. Since 
almost all railroad grades are less than 2% (where the error 
is but .02 of 1%), and anything in excess of 4% is unheard 
of for normal construction, the error in the approximation 
is generally too small for practical consideration. 

If the rate of grade is 1 : 100, G=W XjUj i-e-> ^^20 lbs. 
per ton ; .*. for any per cent, of grade, G = (20 X per cent, of grade) 
pounds per ton. .When moving up a grade this force G is to 
be overcome in addition to all the other resistances. When 
moving down a grade, the force G assists the motion and may 
be more than sufficient to move the train at its highest allow- 
able velocity. The force required to move a train on a level 
track at ordinary freight-train speeds (say 20 miles per hour) 
is about 7 lbs. per ton. A down grade of /^ of 1% will fur- 
nish the same power; therefore on a down grade of 0.35%, a 
freight-train would move indefinitely at about 20 miles per hour. 
If the grade were higher and the train were allowed to gain 
speed freely, the speed would increase until the resistance at 
that speed would equal W times the rate of grade, when the 
velocity would become uniform and remain so as long as the 
conditions were constant. If this speed was higher than a 
safe permissible speed, brakes must be applied and power 
wasted. The fact that one terminal of a road is considerably 
higher than the other does not necessarily imply that the extra 
power needed to overcome the difference of elevation is a 
total waste of energy, especially if the maximum grades are 
so low that brakes will never need to be applied to reduce a 
dangerously high velocity, for although more power must be 



478 RAILROAD CONSTRUCTION. § 433. 

used in ascending the grades, there is a considerable saving of 
power in descending the grades. The amount of this saving 
will be discussed more fully in Chapter XXIII. 

433. Curve resistance. Some of the principal laws wHl be 
here given -u-ithout elaboration. A more detailed discussion 
will be given in Chapter XXTI. 

(a) While the total curve resistance increases as the degree 
of curve increases, the resistance per degree of curve is much 
greater for easy curves than for sharp curves; e.g., the resist- 
ance on the excessi\el3'- sharp curves (radius 90 feet) of the 
elevated roads of New York City is verj'- much less per degree 
of curve than that on curves of 1° to 5°. (b) Curve resistance 
increases vnth the velocity, (c) The total resistance on a 
curve depends on the central angle rather than on the radius; 
I.e., two curves of the same central angle but of different radius 
would cause about the same total curve resistance. This is 
partly explained by the fact that the longitudinal slipping will 
be the same in each case. (See § 395, Chapter XV.) In each 
case also the trucks must be twisted around and the wheels 
slipped laterally on the rails by the same amount A°. (See 
§396, Chapter XV.) _ -i 

434. Brake resistances. If a down grade is excessively steep 
so that brakes must be applied to prevent the train acquiring 
a dangerous velocity, the energj'- consumed is hopelessly lost 
without any compensation. When trains are required to make 
frequent stops and yet maintain a high average speed, consid- 
erable power is consumed b}'- the application of brakes in stop- 
ping. AH the energy which is thus turned into heat is hope- 
lessly lost, and in addition a very considerable amount of steam 
is drawn from the boiler to operate the air-brakes, which con- 
sume the power already developed. It can be easily demonstrated 
that engines drawing trains in suburban service, making fre- 
quent stops, and yet developing high speed between stops, will 
consume a very large proportion of the total power developed 
by the use of brakes. Note the double loss. The brakes con- 
sume power already developed and stored in the train as kinetic 
or potential energy, while the operation of the brakes requires 
additional steam power from the engine. 

435. Inertia resistance. The two forms of train resistance 
which under some circumstances are the greatest resistances 
to be overcome by the engine are the grade and inertia resist* 



§ 435. TEAIN RESISTANCE. 479 

ances, and fortunately both of these resistances may be com- 
puted with mathematical precision. The problem may be 
stated as follows: What constant force P (in addition to the 
forces required to overcome the various frictional resistances, 
etc.) will be required to impart to a body a velocity of v feet 
per second in a distance of s feet? The required number of 
foot-pounds of energy is evidently Ps. But this work imparts 

a kinetic energy which may be expressed by -^— . Equating 



these values, we have Ps=-^r—, or 

2g 



2fl^ 



Wv^ 



2gs 



(104) 



The force required to increase the velocity from v^ to Vj may 

W 
likewise be stated as P=^ — (^'2^— ^i^)- Substituting in the 

2gs ' ^ ' <=> 

formula the values Tr=2000 lbs. (one ton), g- =32.16, and s = 
5280 feet (one mile), we have 

P = .00588(i;2^-V). 

Multiplying by (5280-^3600)^ to change the unit of velocity 
to miles per hour, we have 

P = .01267(72'-'f^i')' 

But this formula must be modified on account of the rotative 
kinetic energy which must be imparted to the wheels of the cars. 
The precise additional percentage depends on the particular 
design of the cars and their loading and also on the design of 
the locomotive. Consider as an example a box-car, 60000 lbs, 
capacity, weighing 33000 lbs. The wheels have a diameter 
of 36" and their radius of gyration is about 13''. Each wheel 
weighs 700 lbs. The rotative kinetic energy of each wheel is 
4877 ft. -lbs. when the velocity is 20 miles per hour, and for 
the eight wheels it is 39016 ft.-lbs. For greater precision 
(really needless) we may add 192 ft.-lbs. as the rotative kinetic, 
energy of the axles. When the car is fully loaded (weight 
93000 lbs.) the kinetic energy of translation is 1,244,340 ft.-lbs.; 
when empty (weight 33000 lbs.) the energy is 441540 ft.-lbs. 
The rotative kinetic energy thus adds (for this particular 
car) 3.15% (when the car is loaded) and 8.9% (when the car 
is empty) to the kinetic energy of translation. The kinetic 



480 RAILROAD CONSTRUCTION. § 435. 

energy which is similarly added, owing to the rotation of the 
v/heels and axles of the locomotive, might be sijnilarly com- 
puted. For one type of locomotive it has been figured at about 
8%. The variations in design, and particularly the fluctua- 
tions of loading, render useless any great precision in these 
computations. For a train of "empties" the figure would be 
high, probably 8 to 9%; for a fully loaded train it will not 
much exceed 3%. Wellington considered that 6% is a good 
average value to use (actually used G.14% for "ease of compu- 
tation"), but considering (a) the increasing proportion of live 
load to dead load in modern car design, (h) the greater care 
now used to make up full train-loads, and (c) the fact that 
ftdl train-loads are the critical loads, it would appear that 5% 
is a better average for the conditions of modern practice. Even 
this figure allows something for the higher percentage for the 
locomotive and something for a few empties in the train. There- 
fore, adding 5% to the coefficient in the above equation, we 
have the true equation 

P = . 0133(72^-^12), . . . . . (105) 

in which V2 and Vi are the higher and lower velocities respec- 
tively, in miles per hour, and P is the force required per ton to 
impart that difference of velocity in a distance of one mile. If 
more convenient, the formula may be used thus : * 

Pi=^(F2^-7i2), (106) 

o 

in which s is the distance in feet and Pi is the corresponding 
force. 

As a numerical illustration, the force required per ton to 
impart a kinetic energy due to a velocity of 20 miles per hour 
in a distance of 1000 feet will equal 

70(400-0) 
^'- 1000 -28 lbs., 

which is the equivalent (see § 432) of a 1.4% grade. Since the 
velocity enters the formula as V^, while the distance enters 
only in the first power, it follows that it will require four times 

* The slight approximation involved in the transformation from Eq. 105 
to 106, by using the even number 70, is covered by allowing 4.6%, instead 
of 5% for rotary kinetic energy. 



§ 436. TRAIN RESISTANCE. 481 

the forc3 to produce twice the velocity in the same distance, or 
that with the sarne force it will require four times the distance 
to attain twice the velocity. 

As another numerical illustration, if a train is to increase its 
speed from 15 miles per hour to 60 miles per hour in a distance 
of 2000 feet, the force required (in addition to all the other 
resistances) will be 

^, 70(3600-225) ^^_,. 

t\ = — ^ — 9r^ =118 lbs. per ton. 

This is equivalent to a 5.9% grade and shows at once that it 
would be impossible unless there were a very heavy down grade, 
or that the train was very light and the engine very powerful. 

436. Dynamometer tests. These are made by putting a 
" dynamometer-car " between the engine and the cars to be 
tested. Suitable mechanism makes an automatic record of 
the force which is transmitted through the dynamometer at 
any instant, and also a record of the velocity at any instant. 
One of the practical difficulties is the accurate determination 
of the velocity at any instant when the velocity is fluctuating. 
When the velocity is decreasing, the kinetic energy of the train 
is being turned into work and the force transmitted through the 
dynamometer is less than the amount of the resistance which 
is actually being overcome. On the other hand, when the velocity 
is increasing, the dynamometer indicates a larger force than 
that required to overcome the resistances, but the excess force 
is being stored up in the train as kinetic energy. Grade has 
a similar effect, and the force indicated by the dynamometer 
may be greater or less than that required at the given velocity 
on a level by the force which is derived from, or is turned into^ 
potential energy. The effect of curvature should be eliminated 
by subtracting from the dynamometer record 0.6 to 0.8 pound 
per ton per degree of curve, according to the rules for compen- 
sation of curvature as developed in § 511. Correct for grade 
by subtracting from the dynamometer record twenty pounds 
per ton for each percent of grade, assuming that the test train 
is moving up a grade; if the train is moving down grade, add 
a similar amount. Add (or subtract) the effect of change in 
velocity, as computed in § 435. Usually each dynamometer 
observation will need to be corrected by one or all of these cor- 
rections in order to determine what would have been the resist- 
ance on a straight, level track, at some definite uniform velocity. 



482 



RAILROAD CONSTRUCTION. 



§436. 




ci 

■g 3 
& 



^0 30 40 50 60 
Average veeighfe per car-tons 



70 



-'ioi 1908-09 the Railway Eng. Dep't of the Univ. of Illinois 
conducted a series of tests, under the direction of Prof. E. C. 
Schmidt,* which were so elaborate and thorough that they 
definitely demonstrated that (a) the resistance per ton of any 
car depends very considerably upon the weight of the car, which 
is graphically shown in Fig. 206 a, and (b) the actual resist- 
ance per ton is variable and uncertain, and therefore no formula 

or resistance curve can assume 
to represent such resistance 
with a close percentage of ac- 
curacy. This uncertainty is 
illustrated by the fact that, in 
spite of the most elaborate 
tare to eliminate all observa- 
tional error and obtain uniform 
results, one typical group of 
plotted points had an average 
deviation of about 8% from 
the curve of average resistance 
and there was one instance of 
a 23% deviation. The varia- 
tion! in results is probably due to variable condition of the 
track (see § 4306) and shows that no one formula or curve, or 
set of them, is closely applicable to the variable track conditions 
found in the country or even to the variations found on any 
one ^oad. The chief object in observing train resistance is to 
determine the tractive power required to haul a deiinite amount 
of tmffic under certain known conditions, but these tests have 
confirmed what operating experience had already pointed out, 
that actual train resistance is so variable that there must be 
a considerable margin of tractive power in the locomotive or 
trains will be frequently stalled. Nevertheless, resistance for- 
mulae can be and are utilized for comparing proposed track loca- 
tions and for computing, with a proper margin, the train load 
which may be attached to a locomotive of known tractive power. 
The net result of these tests on 32 freight trains of various 
weights have been plotted in Fig. 2066, which shows ten curves, 
each for a different average car weight. For each curve, the 
resistance per ton increases with the velocity, being about 80% 
more for a velocity of 40 miles per hour than for a velocity of 



Fig. 206a. — Relation between Re- 
sistance AND Average Car Weight, 
AT Various Speeds. 

(Reduced from Fig. 10, Schmidt, 
Freight Car Resistance.) 



* Univ. of III, Bull. 43, Freight Train Resistance, by Edward C. Schmidt. 



§437. 



TRAIN RESISTANCE. 



483 



5 miles per hour. Note that the upper curve (15 tons per car) 
is only applicable to a train of empties and the lower curve (75 
tons per car) would mean a train of fully loaded cars. It should 
be fully realized that, in order to practically utilize these or other 
similar curves as a measure of the tractive power demanded of 
locomotive, due allowance should be made for grade, for curv- 
ature, and for the inertia effect of change in velocity; also that 
such figures only claim to measure the resistance behind the 
tender, and that it does not apply if brakes have been used. 



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Fia. 2066. — Relation between Resistance and Speed, for Various 
Average Weights per Freight Car. 

(Reduced from Fig. 11, Schmidt, Freight Car Resistance.) 

437. Gravity or " drop " tests. A drop test utilizes the force 
of gravity which may be measured with mathematical accuracy. 
The general method is to select a stretch of track which has a 
uniform grade of about 0.7% and which is preferably straight 
for 2 or 3 miles. On such a grade cars with running gear 
in good condition may be started by a push. The velocity 
will gradually increase until at some velocity, depending on 
the resistances encountered, the cars will move uniformly. The 
only work requiring extreme care with this method is the deter- 
mination of the velocity. If the velocity is fluctuating, as it 
is during the time when it is of the greatest importance to know 
the velocity, it is not sufficient to determine the time required 
to run some long measured distance, for the average velocity 



484 



RAILROAD CONSTRUCTION. 



§437. 



thus obtained would probably differ considerably from the 
velocity at the beginning and end of that space. If the train 
consists of five cars or more, the velocity may be determined 
electrically (as described by Wellington in his " Economic 
Location," etc., p. 793 et seq.) from the automatic record made 
on a chronograph of the passage of theiirst wheel and the last, 
the chronograph also recording automatically the ticks of a 
clock beating seconds. From this the exact time of the passage 
of the first and last wheels of the train of cars may be determined 
to the tenth or twentieth of a second. 

Velocity-head. From theoretical mechanics we know that if 
a body descends through any path by the action of gravity, and 



4 




Fig. 207. — Loss in Velocity-head. 



is unaffected by friction, its velocity at any point in the direc- 
tion of the path of motion is V = V2gh' If the body is retarded 
by resistances, its velocity at any point will be less than this. 
If AM, Fig. 207, represents any grade (exaggerated of course), 
then BJ, C K, etc., represent the actual fall at any point. Let 

BF represent the fall hi, determined from hi = ^, in which vi is 

the actual observed velocity at /. Then /i^=the velocity- 
head consumed by the resistances between A and /. If the 
train continues to K, the corresponding /?,2 is CG; the remaining 
fall GK consists of GN {—JF, which is the velocity-head lost 
back of J) and N K, the velocity-bead lost between J and K. 
At some velocity (F„) on any grade, the velocity will not further 
increase and the line AFGHl will then be horizontal and at 



§ 438. TRAIN RESISTANCE. 485 

a distance (hn) = EI below A . . . E. The grade AM is the 
"grade of repose " for that velocity (F,.,); i.e., it is the grade 
that would just permit the train to move indefinitely at the 
velocity Vn. The broken line AFGHI should really be a curve, 
and the grade of repose at any point is the angle between AM 
and the tangent to thai curve at the given point. The ," grade 
of repose " by its definition gives the total resistance of the train 
at the particular velocity, or multiplying the grade of repose 
in per cent by 20 gives the pounds per ton of resistance. Thus 
being able to determine the total resistance in pounds per ton 
at any velocity, the variation of total resistance with velocity 
may be determined, and then by varying the resistances, using 
different kinds of cars, empty and loaded, box-cars and flats, 
the resistances of the different kinds at various velocities may 
be determined. Many tests have been made, on the above 
general plan, to determine track resistance, but, since it is imprac- 
ticable and even dangerous to use this method for high velocities, 
the dynamometer-car method has been used for the most recent 
and reliable tests. 

438. Resistance of cars through switches. It has always 
been realized that cars encounter greater resistance while passing 
through switches than on a straight unbroken track. This 
additional resistance would have a vital importance in case a 
passing siding were located on a ruling grade. The additional 
tractive force required to haul a train from a siding through 
a switch on to a main track would limit the length of train which 
might otherwise be hauled. Whenever a passing siding is essen- 
tial on a ruling grade, the grade should be compensated, but the 
rate of compensation is still an uncertain quantity. An anal- 
ogous problem is the rate of grade of a ladder track in a classi- 
fication yard (see Chapter XIII, § 379) in order that, when 
switching cars by gravity from a hump, the added resistance, 
due to passing over the various frogs and switch rails on the 
ladder track, will not so exhaust the inertia due to the initial 
velocity that the cars cannot reach the desired locations on the 
classification tracks. Tests to determine such resistance were 
made in 1913-14, under the direction of Prof. C. L. Eddy, of 
the Case School of Applied Science.* The cars, usually singly 
but occasionally two, three or four together, were dropped from 
the top of a hump down a short 4% grade, by which they 

* Bull, 175, Amer. Rwy. Eng. Assoc, March, 1915, 



486 RAILROAD CONSTRUCTION. § 439. 

acquired a velocity varying from 14 to 21 miles per hour at the 
beginning of the ladder track, which had a downward grade 
of 1.175%. Velocities were observed at two places on the ladder 
track, by setting up at each place a pair of " contact points," 
usually 60 feet apart, by which the time of travelling the 60 feet 
was automatically recorded on a chronograph, which also 
recorded half seconds. The mean distance apart of the two 
pairs of contact points was at first 375 feet; then for other tests 
400 feet and then 421.5 feet. Sometimes the velocity of the 
cars decreased while passing over this measured distance, and 
sometimes it increased. In any case the impelling force was 
the constant gravity force of 20X1.175% = 23 pounds per ton, 
plus the inertia force due to the initial velocity. This net force, 
less the inertia force represented by the final velocity, equals 
the resisting force, in pounds per ton. As usual in such tests, 
the results were very variable, varying in 163 observations 
from a minimum of 4.5 to a maximum of 41.8 pounds per ton. 
The general average was about 22 pounds per ton, which is very 
nearly the gravity force (23.5 lbs.) of the ladder track used in 
this test. Note the increase in the average figure (22) above 
the average resistance per ton for whole trains of cars on a 
straight unbroken track, at the same average velocity of 15 to 
20 m.p.h., which would vary from 3.5 to 9.5 pounds per ton — 
see Fig. 2086. A very small part of the increase is due to the 
extra atmospheric resistance per ton of one car over that of a 
train of cars, but the largest part of the excess resistance is that 
due to the frogs and switch points in the track, which, by their 
variable surface, variable elasticity and uneven support, cause 
shock resistances which average three or four times the normal 
resistance on an unbroken track. The above tests demonstrate 
(a) the very great increase of resistance on switches, and (6) 
that the resistance varies so greatly that no precise calculations 
can be made with respect to it. Although the average resistance 
was about 22 lbs. per ton, an allowance of 30 lbs. per ton would 
only cover 91% of the trials in the above test. It should also 
be noted that the switch work, made up of No. 8 frogs and split 
switches, was on the New York Central system, and was declared 
to be " in good order." It cannot therefore be claimed that 
this switch resistance was abnormally high. 

439. American Railway Engineering Association Formula. In 
1910, the Association Committee on. Economics of Location 



§ 439. TBAIN KESISTANCE. 487 

developed a formula with the special idea of its utilization in the 
comparative study of alternate locations of a railroad line, or 
in the operation of trains. An elaborate study of the very 
numerous formulae which had been published convinced the 
committee that all such formulae were either intrinsically 
worthless or that they were inapplicable to present conditions 
of track and rolling stock. After an exhaustive study of the 
results of recent dynamometer tests on the resistance of freight 
trains, with velocities varying from 5 to 35 m.p.h., it was declared 
that a' formula which is sufficiently accurate for practical pur- 
poses can be put into the form 

R=at-\-hn 

in which t is the total weight of the train, in tons of 2000 lbs. 
and n is the number of cars; a and 6 are constants to be deter- 
mined by tests. The values 2.78 and 113.9 for a and 6 respect- 
ively were first used on the basis of certain tests. Later, on 
the basis of an accumulation of additional tests, these constants 
were modified so as to have varying values according to the 
temperature and the following group of four formulae was 
recommended. 

A rating, temp. =35° F. or above; R = 2.2 t+122 n 
B rating, temp. =20° to 35° F.; i2 = 3.0 f +137 n 
C rating, temp. = 0° to 20° F. ; i2 = 4.0 ^ + 153 n 

D rating, temp. =below 0° F.; i2 = 5.4 ^+171 n 



(107) 



These formulae apply only to level grade. When using them, 
suitable corrections for actual rate of grade and curvature, 
and a proper allowance for inertia, in accordance with the as- 
sumed method of operation, should be added to the resistance 
computed from Eq. 107. 

Comparing these formulae with the results of the tests by 
Schmidt, we should use only the formula for A rating, since 
Schmidt's tests were all made at temperatures above 35° F. 
Assume a train of 53 empties, each weighing 18 tons, or a total 
of 954 tons, which is the value oi t) n=53; then the draw bar 
pull behind the tender equals 

22=2.2X954+122X53 = 2099+6466 = 8565 pounds. 

The mean resistance per ton would be 8565 -r- 954 = 8.97 pounds 
per ton. By Schmidt's curves (Fig. 2066) the resistance would 



488 



RAILROAD CONSTRUCTION. 



§ 439a. 



vary from about 7 lbs. per ton for a velocity of 5 m.p.h. to 
11.4 lbs. per ton at 35 m.p.h., or a total of 6678 to 10876 lbs. 
resistance, depending on velocity. At a velocity of slightly 
over 20 m.p.h. the Schmidt curves show the same average resist- 
ance (8.97 lbs. per ton) for 18-ton ears. 

A similar computation for a train of 30 cars weighing 70 tons 
each, or a total of 2100 tons, indicates a total resistance, by 
Eq. 107, of 8280 lbs. or 3.94 lbs. per ton. This again is the 
resistance per ton indicated by the Schmidt curves for 70-ton 
ears when the velocity is a little over 20 m.p.h. 

The student should note that although the A.R.E.A. formula 
is independent of velocity, while the Schmidt curves indicate 
resistances varying as a function of the first power and also 
of the square of the velocity, the results at a velocity of aboiit 
20 m.p.h. are identical. Secondly both agree (up to 25 m.p.h.) 
that, although the loaded train weighs considerably more than 
twice as much as the train of empties, th« pull on the draw bar 
is actually less, which forcibly illustrates the economy of oper- 
ating full and heavily loaded cars. 

The application of Eq. 107 to the operation of trains, or to 
train rating, is explained in Chapter XVIII, § 467. 



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10 20 30 40 50 

Speed-Miles per Hour 



60 



■70 



207a. — Relation between Resistance and Speed, for Various 
Average Weights per Passenger Car. 
(Reduced from Fig. 6, Schmidt, Passenger Train Resistance.) 



439a. Passenger-car resistance. In 1916, Prof. E. C. Schmidt 
made some tests on passenger-car resistance by the same general 



§ 439a. TRAIN RESISTANCE. 489 

methods used in freight-car tests, as described in § 436.* Tests 
were made on eighteen trains, of which the average car weight 
varied from 48.7 to 71.1 tons. 83% of the cars had six-wheeled 
trucks. 

The curves plotted frorn these tests are shown on a reduced 
scale in Fig. 207a, which shows the same general form of curves 
as those of Fig. 2066. It should also be noted that the tests 
showed that the heavier cars have less resistance per ton than 
lighter cars, the same as for freight ears. Comparing the curves, 
where identical conditions make such comparisons possible, it 
may be noted that, in general, freight cars showed a less resist- 
ance per ton than passenger cars for the same velocity and weight 
of car. Many years ago a committee of the Am. Rwy. Master 
Mechanics Assoc, reported that " six-wheel trucks are found 
to produce greater resistance, and as a consequence absorb 
more hauling power than four-wheel trucks carrying the same 
weight of car." Six-wheeled trucks are considered essential 
for carrying especially heavy cars at high passenger-train^ speed, 
in spite of the proved added per4on resistance. Since nearly 
all trains in the above tests included cars with both six-wheeled 
and four-wheeled trucks, it was impracticable to differentiate 
the results on this basis, but the fact that about 83% of the oars 
had six-wheeled trucks probably explains the higher per-ton 
results. When the passenger-car results are reduced to per- 
ton-per-axle, the freight-car and passenger-car results are more" 
nearly uniform. Whenever these curves are used, it should be 
kept in mind that the effect of grade, curvature and inertia 
resistance have all been eliminated from these results. The 
tests were made in pleasant weather, during the summer. It 
should therefore be expected that the resistance in cold and 
windy weather would be materially greater. 

It is interestizig to note that the careful calculations made of 
the weight of the live load (passengers, baggage, mail and express) 
showed that the maximum load weighed only 5.2% of the gross 
train load, and therefore the cost of running a passenger traiii 
is measurably the same whether it runs full or absolutely empty. 

* Proceedings, Artier. Rwy. Eng. Assoc, Vol. 18, p. 6^9. 



CHAPTER XVII 

COST OF RAILROADS. 

440. General considerations. Although there are many ele- 
ments in the cost of railroads which are roughly constant per 
mile of road, yet the published reports of the cost of railroads 
dilfer very widely. The variation in the figures is due to several 
tauses. (a) Economy requires that a road shall be operated 
and placed on an earning basis as soon as possible. Therefore 
the reported cost of a road during the first few years of its 
existence is somewhat less than that reported later. This is 
well illustrated when a long series of consecutive reports from 
an old-established road is available; nearly every year there 
will be shown an addition to the previous figures. And this 
is as it should be. The magnificent road-beds of some old 
roads cannot be the creation of a single season. It takes many 
years to produce such settled perfect structures. (6) A large 
part of the variation is due to a neglect to charge up " permanent 
improvements" as additions to the cost of the road. For the 
first few years of the life of a road a great deal of work is done 
which is in reality a completion of the work of construction, 
and yet the cost of it is buried under the item "maintenance 
of Avay." For example, a long wooden trestle is replaced by 
an earth embankment and a culvert. Since the original trestle 
is to be considered a temporary structure, the excess of the 
cost of the permanent structure over that of the temporar}'^ 
structure should evidently be considered as an addition to the 
cost of the road. But if the filling-in was done slowh'', a few 
train-loads at a time, and the work scattered over man}?- years, 
the cost of operating the "miud-train" has perhaps been buried 
under "maintenance" charges, (c) The reports from which 
many of the following figures were taken have not always 
analyzed the items of cost with the same detail as has been 
here attempted, and to that is probably due many of the varia- 
tions and apparent discrepancies. 

490 



§ 441. COST OF EAILROADS. 491 

The various items of cost will be classified as follows: 

1. Preliminary financiering. 

2. Surveys and engineering expenses. 

3. Land and land damages. 

4. Clearing and grubbing. 

5. Earthwork, including rockwork; tunneling. 

6. Bridges, trestles, and culverts. 

7. Trackwork, material and track labor. 

^ 8. Buildings and miscellaneous structures. 
9. Interest on construction. 

10. Rolling stock. 

441. Item i. PRELIMINARY FINANCIERING. The cost of this 
preliminary work is exceedingly variable. The work includes 
the clerical and legal work of organization, printing, engraving 
of stocks and bonds, and (sometimes the most expensive of all) 
the securing of a charter. This sometimes requires special 
legislative enactments, or may sometimes be secured from a 
State railroad commission. It has been estimated that about 
2% of the railway capital of Great Britain has been spent in 
Parliamentary expenses over the charters. These expenses 
are usually but a small percentage of the total cost of the enter- 
prise, but for important lines the gross cost is large, while the 
amount of money thus spent by organizations which have 
never succeeded in constructing their roads is, in the aggregate, 
an enormous amount, although it is of course not ascertainable 
by any investigator. 

Another occasional feature of the financing of a road must be 
kept in mind. The promoters of a railroad enterprise frequently 
endeavor to limit their own personal expenditures to the purely 
preliminary expenses as mentioned above. The project, after 
having been surveyed, mapped, and written up in a glowing 
"prospectus," is submitted to capitalists, in the endeavor to 
have them furnish money for construction, the money to be 
secured by bonds. If the project will stand it, the amount of 
the bond issue is made sufficient to pay the entire cost of the 
road, even with a discount of perhaps 15%. The bond issue 
may also provide for a very generous commission to the broker 
who is the intermediary between the promoters and the capi- 
talists. The bond issue may even provide for repaying the 
promot'^rs for their preliminary expenses. Frequently a con- 
siderable proportion of the capital stock goes to the capitalists 



492 RAlLROAt) CONSTRUCTION. § 442. 

who take the bonds, the promoters retaining only such propor- 
tion as may be agreed upon. In such a case, the capital stock 
is "pure velvet," and costs nothing. Its future value, whatever 
it may be, is so much clear profit. The effect of such a financial 
policy is to burden the project with a capitalization which is 
far in excess of the actual cost of constructing the road. Com- 
paratively few projects will stand such over-capitalization. 
The apparent financial failure of many railroads, which have 
gone into the hands of receivers is due to their inability to 
make returns on an over-capitalization rather than because 
they could not earn enough to pay the legitimate cost of their 
construction. These features of financiering are really foreign 
to the engineer's work, but he should know that many projects 
which would return a handsome profit on an investment amount- 
ing only to the legitimate cost, will be rejected by capitalists 
because it is apparent that there is not enough "velvet" 
in it. 

442. Item 2. Surveys and Engineering Expenses. The 
comparison of a large number of itemized reports on the cost 
of construction shows that the cost of the "engineering"' will 
average about 2% of the total cost of construction. This in- 
cludes the cost of surveys and the cost of laying out and super- 
intending the constructive work. The cost of mere surveying 
up to the time when construction actually commences has 
been variously quoted at $60, $75, and even $300 per mile. 
The lower figures generally refer to the hasty, ill-considered work 
which was formerly common and which has resulted in so much 
badly located road, much of which has been reconstructed, 
when improvements are practicable. See the introductory par- 
agraphs of Chapter I. Except when the topography limits the 
location to one very obvious route, a thorough survey may cost 
about $300 per mile. In the estimate given at the end of this 
chapter the cost of " engineering and office expenses " is given 
at 5% of the cost of the construction work. The item then 
includes the cost of the very considerable amount of clerical 
work and superintendence incident to the expenditure of such a 
large sum of money. 

443. Item 3. Land and Land Damages. The cost of this 
item varies from the extreme, in which not only the land foi* 
right-of-way but also grants of public land adjoining the road 
are given to the corporation as a subsidy, to the other extreme 



J 444. COSl^ OF RAILROADS. 493 

where the right-of-way can only be obtained at exorbitant 
prices. The width required is variable, depending on the 
jWidth that may be needed for deep cuts or high fills, or the 
extra land required for yards, stations, etc. A strip of land 
mile long and 8.25 feet Avide contains precisely 1 acre. An 
average width of 4 rods (66 feet), therefore, requires 8 acres per 
mile. On the Boston & Albany Railroad the expenditure 
assigned to ''land and land damages" averages over $25000 
per mile. Of course this includes some especially expensive 
land for terminals and stations in large cities. Less than $300 
per mile was assigned to this item by an unimportant 18-mile 
road. 

444. Item 4. Clearing and Grubbing. The cost of thia 
may vary from zero to 100% for miles at a time, but as an 
average figure it may be taken as about 3 acres per mile at a 
cost of say $50 per acre. The possibility of obtaining valuable 
timber, which may be utilized for trestles, ties, or other\Aise, 
and the value of which may not only repay the cost of clearing 
and grubbing, but also some of the cost of the land, should not 
be forgotten. 

445. Item 5. EARTHWORK. This item also includes rock- 
work. The methods of estimating the cost of earthwork end 
rockwork have been discussed in Chapter III. The percentage 
of this item to the total cost is very variable. On a westerp 
prairie it might not be more than 5 to 10%. On a road through 
the mountains it will run up to 20 or 25%,, and even more. 
The item also includes tunneling, which on some roads is a 
heavy item, 

446. Item 6. Bridges, trestles, and Culverts. This item 
will usually amount to 5 or 6% of the total cost of the road. 
In special cases, where extensive trestling is necessary, or 
several large bridges are required, the percentage will be much 
higher. On the Other hand, a road whose route avoids the 
watercourses may have verj*- little except minor culverts. On 
the Boston & Albany the cost is given as $5860 per mile; on 
the Adirondack Railroad, $2845 per mile. Considering their 
relative character (double and single track), these figures are 
relatively what we might expect. 

447. Item 7. Trackwork. This item will be considered as 
including everything above subgrade, except as otherwise 
itemized. 



494 



RAILROAD CONSTRUCTION. 



§447. 



(a) Ballast. As already elaborated in Chapter VII, Ballast, 
the standards for depth of ballast, in order to produce a uniform 
pressure on sub-grade, have so increased that former estimates; 
are inapplicable. The increased depth now called for is usually 
provided by using a layer of sub-ballast made of comparatively, 
inexpensive material, such as cinders, which, being a by-product, 
has only a nominal cost. The unit cost of ballast per cubic 
yard varies from merely nominal to the cost of broken stone, 
which may cost $1.50 or even $2.00 per cubic yard. 

(b) Ties. Ties cost anywhere from $1.40 down to 50 c. andi 
even less. At an average figure of 80c., 2640 ties per mile will I 
cost $2112 per mile of single track. The cheaper ties are usually ' 
smaller and more must be used per mile, and this tends to com- 
pensate the difference in cost. 

The following tabular form is convenient for reference: 



TABLE XXX. NtTMBEK OF CROSS-TIES PER MILE OF TRACK. 



Number per 


Average spacing 


Number 


33' rail. 


center to center. 


per mile. 


22 


18.0 inches 


3520 


21 


18.9 " 


3360 


20 


19.8 " 


3200 


19 


20.9 " 


3040 


18 


22 . " 


2880 


17 


23.3 " 


2720 


IQ 


24.75 " 


2560 


15 


26.4 " 


2400 


14 


28 . 3 " 


2240 


13 


30.5 ♦• 


2080 



(c) Rails. The total weight of the rails used per mile may 
best be seen by the tabular form. 

A convenient and useful rule to remember is that the number 
of long tons (2240 lbs.) per mile of single track equals the weight 
of the rail per yard times -y^-. The rule is exact. For example, 
there are 3520 yards of rail in a mile of single track; at 70 lbs. 
per yard this equals 246,400 lbs., or 110 long tons (exactly); 
but70X-V^ = 110. 

Any calculation of the required weight of rail for a given 
weight of rolling-stock necessarily depends on the assumptions 
which are made regarding the support which the rails receive 
from the ties. This depends not only on the width and spacing 
of the ties (which are determinable), but also on the support 



§447. 



COST OF RAILROADS. 



495 



TABLE XXXI. — TONS PER MILE OF RAILS OF VARIOUS WEIGHTS. 





Tons 




Tons 




Tons 




Tons 


Weight 


(22401b.) 


Weight 


(22401b.) 


Weight 


(22401b.) 


Weight 


(22401b.) 


in lbs. 


per mile 


in lbs. 


per mile 


in lbs. 


per mile 


in lbs. 


per mile 


per yd. 


of single 


per yd. 


of single 


per yd. 


of single 


per yd. 


of single 




track. 




track. 




track. 




track. 


8 


12.571 


25 


39.286 


55 


86.429 


85 


133.571 


10 


15.714- 


30 


47.143 


60 


94.286 


90 


141.429 


12 


18.857 


35 


55.000 


65 


102.143 


95 


149 . 286 


14 


22.000 


40 


62.857 


70 


110.000 


100 


157.143 


16 


25.143 


45 


70.714 


75 


117.857 


110 


172.857 


20 


31.429 


50 


78.571 


80 


125.714 


120 


188.571 



About two per cent (2%) extra should be allowed for waste in cutting, 

which the ties receive from the ballast, which is not only very- 
uncertain but variable. No general rule can therefore claim 
any degree of precision, but the following is given by the Bald- 
win Locomotive Works: The weight per wheel which can be 
safely carried for each pound weight of rail per yard is approxi- 
mately as follows: 

Light rails; 60 lbs. and less per yard; 250 lbs.; 

Medium rails; 60 lbs. to 90 lbs. per yard; 300 lbs.; 

Heavy rails; 90 lbs. and over per yard; 350 lbs. 

This assumes that the rails are properly supported by cross ties, 
not less than 14 per 30-ft. rail. For example, a Mikado loco- 
motive with 153,200 lbs. on 8 drivers has a load of 19,150 lbs. 
per wheel. This divided by 300 gives 63.8. According to the 
rule, the rails for such a locomotive should weigh at least 63.8 
lbs. per yard. But it should be noted that railroads which 
use Mikado locomotives will also have their track laid with 
heavier than 63.8 (or 65) pound rails. The rule should there- 
fore be considered as the minimum permissible. A road with 
even one high-speed train, or a Class A road (§ 234), should 
use 80 to 90 lb. rails, even if not required by the above rule. 

On the basis of 33-foot lengths, and 10% shorter lengths, 
varying by even feet down to 25 feet (see § 273 e), the average 
length, assuming an equal number each of the shorter length 
rails, would be 32.55 feet. Calculating similarly for 30-ft. rails, 
with 10% shorts to 24 feet, the average length would be 29.65 
feet. 60-ft. rails, used extensively for electric roads, with 10% 
shorts to 40 feet, will have average length of 58.95 feet. 

(d) Splice-bars, track-bolts, and spikes. These are usually 
sold by the pound, except the patented forms of rail- joints, 



496 



EAILROAD CONSTRUCTION. 



§447. 



which are sold by the pair. In any case they are subject to 
market fluctuations in price. As an approximate value the 
following prices are quoted: Splice-bars, 2.50 cents per pound; 
track-bolts, 4.0 cents; spikes, 3.25 cents. The weight of the 
splice-bars will depend on the precise pattern adopted — its 
cross-section and length. 

In Table XXXII are quoted, from a catalogue of the Illinois 
Steel Co., the weights per foot of sections of angle-bars which 
they recommend for various weights of rail and which are de- 
signed to fit standard A. S. C. E. rail sections of those weights. 
The net weight of the angle-bars may be approximated by 
subtracting about 2.5% to 4% from the gross weight to allow 
for the bolt-holes. A deduction of 2.5% is usually about right 
for the heavier sections. Their recommendations regarding 
lengths of angle-bars do not include those for rails heavier than 
50 pounds per yard. On the basis of a length of 24 inches for 
four-hole splices and of 32 inches for six-hole splicas, the weights 
of splice-bars have been computed for the several styles of splices 
for heavier rails, allowing 2.5% for the holes. The lengths 
recommended for track bolts are those which will allow about 
f inch for the nutlock and for margin, except for the lighter rails. 

TABLE XXXII. — SPWCE-BARS FOR VARIOUS WEIGHTS OF RAILS. 



Weight I 


jength 


Weight 


Weight 


Proper 


Proper size 


of 


of 


per 


of 


size of 


of spikes. 


rail. an 


gle-bar. 


foot. 


pair. 


track-bolt. 




30 


21" 


4.49 


15.1 


21" Xf" 


4 "Xi" 


35 


21" 


4.7 


15.9 


2|"X|" 


41" Xf" 


40 


21" 1 


5,54 


18.8 


3 "Xf" 


5 "xr 


45 


21" 


0.3 


21.5 


3 "XI" 


51" X^" 


50 


21" 


0.97 


23.4 


31" XI" 


51" X^" 


65 


24" 


7.5 


29.2 


31" XI" 


"3 ''N 16 


60 


24" 


8.4 


32.8 


31" Xf" 


c 1// v/ 9 // 

•J 2 /^ 1 6 


65 


f 24" 


9.2 


35.9 


4 "XI" 


C 1// S/ 9 // 
Oz Xl6 


i32" 


9.6 


49.9 


H"xr' 


niff s/ 9 // 

•^2 AT6 


70 


f24" 


9.0 


35.1 


4 "XI" 


5r'xA" 


,32" 


10.0 


52.0 


4 "XI" 


5i"XA" 


75 . 


f24" 


10.68 


42.6 


4i"Xf" 


5rx^«" 


,32" 


11.9 


61.9 


4 "Xl" 


5r'xA" 


80 


'24" 


10.61 


42.3 


4i"xr 


5/'X^" 


S2" 


14 . 65 


76.2 


4r'X|" 


5i"Xi^" 


85 


32" 


12.4 


64.5 


41" Xl" 


54"Xi^"orf" 


90 


32" 


13.5 


70.2 


4rxr 


5|"Xt^"or 1" 


95 


32" 


14.7 


76.4 


41" XI" 


5r'X^"or t" 


100 


32" 


15.78 


82. 1 


41" xr' 


5r'Xi^"orr 



(e) Track-laying. Much depends on the force of men em- 
ployed and the use of systematic methods ; $528 per mile was the 



§447. 



COST OF RAILROADS. 



497 



TABLE XXXIII. — EAILROAD SPIKES. 







Ties 24" between cen- 






Average 


ters, 4 spikes per tie, 


Suitable 


Size meas- 


number 


number 


per mile. 


weight of 


ured under 


per keg of 






rail. 


head. 


200 pounds 


Pounds. 


Kegs. 




54" Xf" 


275 


7680 


38.40 


90 to 100 


54" X 3^" 


375 


5632 


28.16 


45 " 100 


5" Xt^o" 


400 


5280 


26.40 


40 " 56 


5" X4'- 


450 


4692 


23.46 


40 


44" X 4" 


530 


3984 


19.92 


35 


4" X4" 


600 


3520 


17.60 


30 


44" X A" 


680 


3104 


15.52 


25 to 30 



TABLE XXXIV. — TRACK-BOLTS. 
Average number in a keg of 200 pounds. 



Size of 


Square 


Hexagonal 


Suitable 


bolt. 


nut. 


nut. 


rail. 


3" xr 


366 


395 


40 pound 


3" Xf" 
3i"x|" 


250 


270 




243 


261 




34" Xf" 


236 


253 


50 


3rXf" 


229 


244 


55 to 60 


4" Xf" 


222 


236 


65 •• 70 


41" Xf" 
34" Xl" 


215 ■ 


228 


75 


170 


180 




3f"xr 


165 


175 




4" Xi" 


161 


170 




41" xr 


157 


165 


80 


44" Xf" 


153 


160 


85 


4f"xr 


149 


156 


90 



TABLE XXXV.- 



-RAIL-JOINTS AND TRACK-BOLTS. 
OF TRACK. 



NUMBER PER MILE 



Length of 
rail. 
Feet. 


Average 

length of 

rail. 

Feet. 


Number of 

rails or 

complete 

joints. 


Number of bolts. 


4-bolt. 


6-bolt. 


All 30 
30-24 
All 33 
33-27 
All 60 
60-40 


30 

29.65. 

33 

32.65 

60 

58.95 


352 

356.2 

320 

323.4 

176 

179.1 


. 1408 

1425 

1280 

1294 

704 

717 


2112 
2137 
1920 
1941 
1056 
1075 



498 RAILROAD CONSTRUCTION. § 448. 

j 

estimate formerly employed by the Pennsylvania Railroad. 
$500 per mile is the estimate given in § 451. See note at 
bottom of p. 536. 

448. Item 8. Buildings and Miscellaneous Structures. Ex- 
cept for rough and preliminary estimates, these items must be 
individually estimated according to the circumstances. The 
subitems include depots, engine-houses, repair-shops, water- 
stations, section- and tool-houses, besides a large variety of 
smaller buildings. The structures include turn-tables, cattle- 
guards, fencing, road-crossings, overhead bridges, telegraph line, 
etc. The detailed estimate, given in § 451, illustrates the cost 
of these smaller items. 

449. Item 9. Interest on Construction. The amount of 
capital that must be spent on a railroad before it has begun 
to earn anything is so very large that the interest on the cost 
during the period of construction is a very considerable item. 
The amount that must be charged to this head depends on the 
current rate of money on the time required for construction and 
on the ability of the capitalists to retain their capital where 
it will be earning something until it is actually needed to pay 
the company's obligations. Of course, it is not necessary to 
have the entire capital needed for construction on hand when 
construction commences. Assuming money to be worth 6%, 
that the work of construction will require one year, that the 
money may be retained where it will earn something for an 
average period of six months after construction commences, 
or, in other words, it will be out of circulation six months before 
the road is opened for traffic and begins to earn its way, then 
we may charge 3% on the total cost of construction. 

450. Item 10. Rolling Stock. The cost depends on the traffic 
to be handled and bears very little relation to the total or the 
mileage cost of the roadbed and track. In each case the cost, 
at proper unit prices, of the locomotives and cars necessary to 
handle the estimated traffic must be computed. 

451. Detailed estimate of the cost of a line of road. The 
following estimate was given in the Engineering News of Dec. 27, 
1900, of the cost of the Duluth, St. Cloud, Glencoe & Mankato 
Railroad, 157.2 miles long. 

The estimate is exactly as copied from the Engineering News. 
There are some numerical discrepancies. Item 26 should evi- 
dently be based on the sum of the first 25 items, and item 27 



«1. 



§ 451. COST OF RAILKOADS. 499 

on the sum of the first 26. The figures in parentheses ( ) are 
deduced from the figures given. 

1. Right-of-way; 1905.3 acres (12.12 acres per mile) @ $100 per 

acre $190530 

2. Clearing and grubbing. 144 acres (0.916 acre per mile) @ $50 

per acre 7200 

3. Earth excavation. 1907590 cu. yds. (12135 cu. yds. per mile) 

@ 15 c 286138 

4. Hock excavation. 5100 cu. yds. (32.44 cu. yds. per mile) @ 80 c. 4080 
j Wooden-box culverts. 508300 ft. B.M. @ $30 per M. . $15249 

^' i Iron-pipe culverts. 879840 lbs. @ 3c. per lb 26395 41644 

\ Pile trestling. 4600 lin. ft. @ 35 c. per lin. ft .....' 1610 

(Timber trestUng. 509300 ft. B.M. @ $30 per M 15279 16889 

\ Bridge masonry: 5520 cu. yds. @ $8 per cu. yd 44160 

■ ( Bridges, iron, 100 spans. 2000000 lbs. @ 4 c. per lb.. . 80000 124160 

8. Cattle-guards ' 8750 

9. Ties (2640 per mile). 419813 (159.02 miles) @ 35 c. . . . . .. . . 146935 

10. Rails (70 lbs. per yd.); 110 tons per mile. 17492.2 tons (159.02 

miles @$26 384707 

11. Rail sidings (70 lbs. per yd.) : 110 tons per mile, 3300 tons 

(30 miles® $26 85800 

12. Switch umbers and ties 3300 

13. Spikes: 5920 lbs. per mile, 1107040 (187 m.) @ 1.75. c. per lb. 19373 

14. Splice-bars. 2635776 lbs. @ 1.35 c. per lb 35583 

15. Track-bolts (2 to joint (?)): 188458.3 lbs. @ 2.4 c. per lb 4520 

16. Track-laying 187.2 miles @ $500 per mile 93600 

17. Ballasting: 2152 cu. yds. per mile, 402854 (187.2 m.) @ 60 c. . 241712 

18. Turn-out and switch furnishings 6450 

19. Road-crossings, 68040 ft. B.M. @ $30 per M 2041 

20. Section and tool-houses, 16 @ $800 12800 

21. Water-stations 15000 

22. Turn-tables, 6 @ $800 4800 

23. Depots, grounds, and repair-shops 78000 

24. Terminal grounds and special land damages 150000 

25. Fencing, 314 miles ($150 per mile) 47100 

26. Engineering and office expenses (5% of $1984458) 99222 

27. Interest on construction (3% &f $2083680) 62510 

28. Rolling-stock ($5000 per mile). 786000 

29. Telegraph line: 157 miles @ $200 per mile 31400 

$3060340 
Average cost per mile ready for operation, $19467. 
Approximate cost of 130 miles from St. Cloud to Duluth, estimated at 

$23000 per mile. 
Approximate cost of entire line from Albert Lea to Duluth, 287.2 roiles, 

$6050340 ($21060 per mile). 

Although the above estimate is now (1921) so old that the 
prices are obsolete, the list is retained since it is a typical analysis 
and may be utilized by making the proper changes in unit prices, 
which is always more or less necessary. 



CHAPTER XVIII. 

THE POWER OF A LOCOMOTIVE. 

452. Pounds of steam produced. The power that can be 
developed by a locomotive depends very greatly on the quality 
of the coal burned and the design of the locomotive must corre- 
spond to the general kind or quality of coal to be used. A 
British thermal unit (symbolized as B.t.u.), is the quantity of 
heat required to raise the temperature of 1 lb. of pure water 
1° F., when the water is at or near its maximum density at 39.1° 
F. When it is said that a certain grade of coal has 14000 
B.t.u. it means that the heat in 1 lb. of that coal will raise the 
temperature of 14000 lbs. of water 1°, or, approximately, 100 lbs. 
of water 140°. But, although it only requires 180.9 heat units 
to heat water from 32° to 212°, it requires 965.7 more heat units 
to change it from water at 212° to steam at 212°. It requires 
only 53.6 more heat units to change it from steam at 212° to 
steam at 387.6° or with a pressure of 200 lbs. per square inch. 

A study of locomotive tests made at the St. Louis Exposition 
resulted in the compilation of Table XXXVI, which is copied 
from the Proceedings of the American Railway Engineering 
Association, and is now included as Table I, in the " Economics '' 
section of their Manual. It was found that the steam produced 
per square foot of heating surface is very nearly proportional to 
the coal burned per square foot of heating surface. The results 
are purposely made about 5% below the results obtained in the 
St. Louis tests to allow for ordinary working conditions. 

453. Numerical example. The theory developed in this 
chapter will be illustrated numerically by applying it to a Mikado 
type of locomotive whose dimensions are as follows: 



Cylinder diam. 22" 

Cylinder stroke 28" 

Driving wheel diam. 57" 

Boiler pressure 185 lbs. 

Fire-box length 102|" 

Fire-box. width 65|" 

Grate area 46 . 8 sq. ft. 



Weight, driving wheels. 153,200 lbs. 

engine alone 196,100 lbs. 

engine and tender. . . . 315,000 lbs. 
Heating surface, fire-box 

and tubes 2565 sq. ft. 

superheating surface . 550 sq. ft. 

500 



§453. 



THE POWER OF A LOCOMOTIVE. 



501 



TABLE XXXVI. — ^AVERAQE EVAPORATION IN LOCOMOTIVE BOILERS 
BURNING BITUMINOUS AND SIMILAR COALS OF VARIOUS QUAL- 
ITIES, AND FOR VARIOUS QUANTITIES CONSUMED PER SQUARE 
Ii^OT OF HEATING SURFACE PER HOUR. 
(Based on feed water at 60° Fahrenheit, and boiler pressure 200 pounds) 





Steam per pour 


id of coal of given thermal value 


Coal per square 






(lb.) 






foot of heating 














surface per hour 














(lb.) 


15,000 


14,000 


13,000 


12,000 


11,000 


10,000 




B.t.u. 


B.t.u. 


B.t.u. 


B.t.u. 


B.t.u. 


B.t.u. 


0.8 


7.86 


7.34 


6.81 


6.29 


5.76 


5.24 


0.9 


7.58 


7.07 


6.57 


6.06 


5.56 


5.05 


1.0 


7.31 


6.82 


6.34 


5.85 


5.36 


4.87 


1.1 


7.06 


6.59 


6.12 


5.65 


5.18 


4.71 


1.2 


6.82 


6.37 


5.91 


5.46 


5.00 


4.55 


1.3 


6.59 


6.15 


5.71 


5.27 


4.83 


4.39 


1.4 


6.37 


5.95 


5.52 


5.10 


4.67 


4.25 


i.5 


6.17 


5.76 


5.35 


4.94 


4.52 


4.11 


1.6 


5.97 


5.57 


5.18 


4.78 


4.38 


3.98 


1.7 


5.79 


5.40 


5.02 


4.63 


4.25 


3.86 


1.8 


5.61 


5.24 


4.86 


4.49 


4.12 


3.74 


1.9 


5.44 


5.08 


4.71 


4.35 


3.99 


3.63 


2.0 


5.27 


4.92 


4.57 


4.22 


3.86 


3.51 


2.1 


5.12 


4.78 


4.44 


4.10 


3.75 


3.41 


2.2 


4.97 


4.64 


4.31 


3.98 


3.64 


3.3l 


2.3 


4.83 


4.51 


4.19 


3.86 


3.54 


3.22 


2.4 


4.69 


4.38 


4.07 


3.75 


3.44 


3.13 


2.5 


4.56 


4.26 


3.95 


3.65 


3.34 


3.04 


2.6 


4.44 


4.14 


3.84 


3.55 


3.25 


2.96 


2.7 


4.32 


4.03 


3.74 


3.46 


3.17 


2.88 


2.8 


4.21 


3.93 


3.64 


3.37 


3.09 


2.80 


2.9 


4.10 


3.83 


3.55 


3.28 


3.01 


2.73 


3.0 


3.99 


3.73 


3.46 


3.19 


2.93 


2.66 



The quantity of steam evaporated for intermediate quantities or qualities 
of coal can be found by interpolation. 

On bad-water districts deduct the following from tabular quantities: 

For each re inch of accumulated scale 10 per cent 

For each grain per U. S. gallon of foaming salts 

in the average feed water 1 per cent 

Assume thatt this locomotive is using coal whose air-dried 
mine samples tested 13000 B.t.u.; then the average run-of-car 
coal would have about 90% of this or 11700 B.t.u. On the 
basis that a fireman can handle 4000 lbs. of coal per hour and 
maintain such work throughout his run, the coal may be fed at 
the rate of (4000 -r- 2565) =1.56 lbs. per hour per square foot of 
heating surface. Interpolating in Table XXXVI for 1.56 and 
11700 we find that the pounds of steam per pound of coal would 
be 4.72. The tests at St. Louis showed that a reduction ill 



502 RAILROAD CONSTRUCTION. §454. 

boiler pressure increased very slightly the amount of steam pro- 
duced, but that this amount was only 0.5% greater when the 
pressure was 160 lbs. instead of 200 lbs. The effect of variation 
of pressure can therefore be ordinarily ignored. In this case 
it might add 0.2% or make the figure 4.73. Considering that 
a superheater adds from 15 to 25% to the efficiency, we will 
assume the average of 20% and say that 0.80 lb. of the super- 
heated steam produced may be considered as having the same 
volume and pressure as 1 lb. of saturated steam. Then the 
amount of steam developed by 1 lb. of coal would be the equiva- 
lent of 4.73-^0.80 = 5.91 lbs. Then the equivalent amount of 
steam developed per hour equals 5.91 X4000 = 23640 lbs. 

454. Weight of steam per stroke at full cut-off. This may be 
computed most easily by utilizing Table XXXVII, which is 
also taken (but somewhat amplified), from the Proceedings of 
the American Railway Engineering Association, and is now 
included as Table 2 in the ''Economics" section of their Manual. 
The weight of steam per foot of stroke for 22 ins. diameter and 
185 lbs. gauge pressure is 1.161 lbs. and for a stroke of 28 ins. 
(2i ft.) it is 2.709 lbs. For a complete revolution of the drivers 
it is 4X2.709 = 10.836 lbs. Since the engine can develop the 
equivalent of 23640 lbs. of steam per hour and will use 10.836 lbs. 
at one revolution, it can run at a speed of 23640 -r- 10.836 = 2182 
revolutions per hour, or 36.36 revolutions per minute, at full 
stroke and maintain full boiler pressure. The drivers are '57 
ins. in diameter and, therefore, have a circumference of (57 -^ 12) 
X3.1416 = 14.923 ft. The maximum engine speed for full 
stroke is 36.36X14.923 = 542.6 ft. per minute. Multiplying by 
60 and dividing by 5280, or dividing by 88, we have 6.167 miles 
per hour as the maximum speed at which full stroke can be main- 
tained, which is the value M for these conditions. 

455' Pounds of steam and per cent, of cut-off for multiples of 
ikf velocity. In Table XXXVIII, also taken from the Pro- 
ceedings of the American Railway Engineering Association and 
now included at Table 4 in the '' Economics " section of the Man- 
ual, are given the pounds of steam per indicated horse-power 
hour for simple and for compound locomotives for various 
velocities, which are multiples of M, the maximum velocity 
at which the locomotive can use steam at full stroke and yet the 
boiler can mamtain steam at full pressure. The table is com- 
puted on the basis of 200 lbs. gauge pressure, but factors are 



§ 455. 



THE POWEK OP A LOCOMOTIVE. 



503 



TABLE XXXVII.- 



-WEIGHT OF STEAM USED IN ONE FOOT OF STROKE 
IN LOCOMOTIVE CYLINDERS. 



(Cylinder diameter is for high-pressure cylinders in compound locomotives) 





Weight of steam per foot of stroke for various gauge 




pressures. 


Diameter 
















of cylinder 


220 lbs. 


210 lbs. 


200 lbs. 


190 lbs. 


180 lbs. 


170 lbs. 


160 lbs. 


(inches) 


per 


per 


per 


per 


per 


per 


per 




sq. in. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 




(lb.) 


(lb.) 


(lb.) 


(lb.) 


(lb.) 


(lb.) 


(lb.) - 


12 


0.405 


0.389 


0.370 


0.354 


0.337 


0.321 


0.304 


13 


0.475 


0.456 


0.435 


0.415 


0.396 


0.376 


0.357 


14 


0.551 


0.529 


0.504 


0.482 


0.459 


0.436 


0.414 


15 


0.633 


0.607 


0.579 


0.553 


0.527 


0.501 


0.476 


15i 


0.675 


0.649 


0.618 


0.590 


0.562 


0.535 


0.508 


16 


0.720 


0.691 


0.658 


0.629 


0.599 


0.570 


0.541 


17 


0.812 


0.780 


0.744 


0.710 


0.676 


0.643 


0.611 


18 


0.911 


0.875 


0.834 


0.796 


0.759 


0.722 


0.685 


18i 


0.962 


0.924 


0.881 


0.841 


0.801 


0.762 


0.724 


19 


1.015 


0.975 


0.928 


0.887 


0.845 


0.804 


0.763 


191 


1.069 


1.027 


0.978 


0.934 


0.890 


0.847 


0.804 


20 


1.125 


1.080 


1.029 


0.983 


0.936 


0.891 


0.836 


20i 


1.181 


1.134 


1.081 


1.032 


0.984 


0.936 


0.888 


21 


1.240 


1.191 


1.134 


1.083 


1.032 


0.982 


0.932 


22 


1.361 


1.307 


1.245 


1.189 


1.133 


1.078 


1 023 


23 


1.487 


1.428 


1.361 


1.300 


1.238 


1.178 


1.118 


24 


1.620 


1.555 


1.482 


1.416 


1.348 


1.283 


1.218 


25 


1.758 


1.688 


1.608 


1.536 


1.462 


1.392 


1.322 


26 


1.901 


1.825 


1.739 


1.661 


1.582 


1.506 


1.430 


27 


2.050 


1.968 


1.875 


1.792 


1.706 


1.624 


1.542 


28 


2.204 


2.117 


2.017 


1.926 


1.835 


1.745 


1.657 



For weight of steam used per revolution of drivers at full cut-off: 
Multiply the tabular quantity by four times the length of stroke in feet 

for simple and four-cylinder compounds. For two-cylinder compounds 

multiply by two times the length of stroke. 



given for other pressures. For example, continuing the above 
numerical problem, the pounds of steam per i.h.p.-hour, for a 
simple locomotive, at M velocity, and at 200 lbs. pressure, taken 
from Table XXXVIII, is 38.30; for 185 lbs. pressure we must 
multiply by the factor 1.0095, which makes the quantity 38.66. 
Dividing this into 23640, the steam produced per hour, we have 

611.5, the i.h.p. at M velocity. Multiplying this by 33000, 
the foot-pounds per minute in one horse-power, and dividing by 

542.6, the velocity in feet per minute, we have 37190, the cylinder 
tractive power in pounds, when burning 4000 lbs. of coal per 
hour and running at 6.167 m.p.h. 



504 



RAILEOAD CONSTRUCTION. 



§456. 



-tABLE XXXVIII. — MAXIMUM CUT-OFF AND POUNDS OF STEAM PER 
I.H.P.-HOUR FOR VARIOUS MULTIPLES OF M. 

(M is maximum velocity in miles per hour at full cut-off, with boiler 
pressure at 200 pounds per square inch) 



Velpcity 


:Cut-off 

jper cent 


Pounds steam per 
I.H.P.-hour 


Velocity 


Cut-off 
per cent 


Pounds steam per 
I.H.P.-hour 


Simple 


Com- 
pound 


Simple 


Com- 
pound 


1.0 M 
1.1" 

1.2 " 

1.3 " 

1.4 " 


Full 
94.4 
89.1 
84.3 
79.7 


38.30 
36.46 
34.89 
33.56 
32.41 


25.80 
24.36 
23.24 
22.35 
21 . 65 


2.9 M 
3.0 " 
3.2 " 
3.4 " 
3.6 " 


38.5 
37.0 
34.2 
31.8 
29.8 


24.37 
24.22 
24.00 
23.85 
23.80 


21.04 
21.21 
21.57 
21.93 
22.27 


1.5 " 

1 .6 ' ' 

1.7 '' 

1.8 " 

1.9 " 


75.4 
71.4 
67.7 
64.3 
61.0 


31.40 
30.49 
29.67 
28.93 

28.25 


21.14 
20.77 
20.52 
20.40 
20.40 


3.8 " 
4.0 " 
4.25 " 
4.50 " 

4.75 " 


28.0 
26.4 
24.7 
23.3 
22.1 


23.80 
23.87 
24.05 
24.24 
24.44 


22.57 
22.85 
23.22 
23.56 
23.85 


2.0 " 

2.1 " 

2.2 " 

2.3 " 

2.4 " 


58.0 
55.2 
52.6 
50.1 

47.8 


27.62 
27.05 
26.52 
26.06 
25.67 


20.40 
20.40 
20.40 
20.40 
20.40 


5.0 " 
5.5 " 
6.0 " 
6.5 " 
7.0 " 


21.1 
19.5 
18.4 
17.6 
17.1 


24.64 
24.98 
25.20 
25.45 
25.60 


24.15 
24.70 


2.5 " 

2.6 " 

2.7 " 

2.8 " 


45.7 
43.7 
41.8 
40.1 


25.32 
25.02 
24.76 
24.54 


20.47 
20.60 
20.73 
20.88 


7.5 " 
8.0 " 
9.0 " 


16.7 
16.4 
16.1 


25.70 
25.80 
25.90 




For steam per i 
centages of values 


.h. p. -hour for other boiler pres 
given in table: 


sure take 


the follo-^ 


ving per- 


160 
170 


lb., 103. ( 
lb., 102.] 




180 lb. 
190 lb. 


, 101.3% 
, 1<)0.^% 


2 
2 


10 lb., 99 
,00 lb., 99 


.5% 
.2% 



456. Draw-bar Pull. To obtain the draw-bar pull we must 
deduct the engine resistance. These have already been dis- 
cussed in § 429 and the numerical value of the resistance of this 
same locomotive has been there computed to be about 1771 lbs. 
Subtracting this from 37190 we have 35419 lbs., the estimated 
draw-bar pull for that speed and coal consumption. 

457. Effect of increasing the rate of coal consumption. To 
note the effect of increasing the rate of coal consumption, the 
problem may be again worked through on the basis that the rate 
of coal consumption is increased, even temporarily, from 4000 
lbs. to 5000 lbs. per hour. The steam developed per pound of 
coal is reduced from 5.91 to 5.23, but the total steam produced 
per hour is increased from 23640 to 26150. The increased ca- 
pacity comes through a loss of efficiency. The increased steam 



§457. 



THE POWEB OF A LOCOMOTIVE. 



505 



production raises the velocity at which full stroke may be main^ 
tained from 6.167 m.p.h to 6.820 m.p.h and the i.h.p. from 
611.5 to 676.4. But the computed cylinder tractive power is 
practically identical, the numerical computation of 37190 being 
only changed to 37189. But these cyUnder tractive powers 
are each computed for the " M " velocities, the maximum ve- 
locities at which full stroke can be maintained, and " M " is 
higher with increased coal consumption. For a real comparison, 
the figures must be reduced to the same velocity, e.g., the work- 
ing velocity of 10 m.p.h. lO-f-6.167 = 1.621, the multiple for the 
original problem. For 5000 lbs. of coal per hour, M velocity is 



TABLE XXXIX' 



-PER CENT CYLINDER TRACTIVE 
VARIOUS MULTIPLES OP M. 



POWER FOR 



(M is maximum 


L velocity 


in miles per hour at which boiler pressure can be 






maintained with full cut-off) 


Veloc- 


Per cent 
(Com- 


Per cent 
(Sim- 


Veloc- 
ity 


Per cent 
(Com- 


Per cent 
(Sim- 


Veloc- 
ity 


Percent 
(Com- 


Per cent 
(Sim- 


ity 


pound) 


ple) 


pound) 


ple) 


pound) 


ple) 


Start 


135.00 


106.00 


3.6 M 


32.40 


44.75 


6.4 M 




23.59 


0.5 M 


103.00 


103.00 


3.7 " 


31.25 


43.56 


6.5 " 




23.18 


1.0 " 


100.00 


100.00 


3.8 " 


30.10 


42.39 


6.6 " 




22.79 


1.1 " 


96.28 


95.57 


3.9 " 


29.14 


41.24 


6.7 " 




22.42 


1.2 " 


92.55 


91.53 


4.0 " 


28.24 


40.10 


6.8 " 




22.06 


1.3 " 


88.83 


87.83 


4.1 " 


27.38 


39.00 


6.9 '• 




21.71 


1.4 " 


85.12 


84.46 


4.2 " 


,26.56 


37.96 


7.0 " 




21.38 


1.5 " 


81.40 


81.37 


4.3 " 


25.77 


36.97 


7.1 " 




21.06 


1.6 " 


17.68 


78.55 


4.4 " 


25.03 


36.03 


7.2 " 




20.75 


1.7 " 


73.96 


75,97 


4.5 " 


24.34 


35.13 


7.3 " 


' 


20.45 


1.8 " 


70.25 


73.60 


4.6 " 


23>.€9 


34.26 


7.4" 




20.16 


1.9 " 


66.54 


71.41 


4.7 " 


23.07 


33.41 


7.5 " 




19.88 


2.0 " 


63.21 


69.37 


4.8 " 


22.48 


32.59 


7.6 '* 




19.61 


2.1 " 


60.20 


67.47 


4.9 " 


21.92 


31.82 


7.7 •• 




19.34 


2.2 " 


57.48 


65.67 


5.0 " 


21.38 


31.11 


7.8 " 




19.08 


2.3 " 


54.97 


63.94 


5.1 " 


20.87 


30.42 


7.9 " 




18.82 


2.4 " 


52.68 


62.22 


5.2 " 


20.37 


29.75 


8.0" 




18.57 


2.5 " 


50.42 


60.55 


5.3 " 


19.89 


29.10 


8.1 " 




18.33 


2.6 " 


48.16 


58.92 


5.4 " 


19.43 


28.48 


8.2 " 




18.09 


2.7 " 


46.08 


57.33 


5.5 " 


18.99 


27.87 


8.3 •• 




17.86 


2.8 " 


44.10 


55.78 


5.6 " 




27.33 


8.4 " 




17.64 


2.9 " 


42.29 


54.26 


5.7 " 




.26.81 


8.5 •• 




17.43 


3.0 " 


40.57 


52.78 


5.8 " 




26.30 


8.6 •• 




17.22 


3.1 " 


38.95 


51.33 


5.9 " 




25.81 


8.7 ♦• 




17.01 


3.2 " 


37.42 


49.91 


6.0 " 




25.34 


8.8 " 




16.82 


3,3 " 


35.98 


48.55 


6.1 " 




24.88 


8.9 *• 




16.63 


3.4 " 


34.66 


47.24 


6.2 " 




24.44 


9.0 •• 




16.45 


3.5 " 


33.53 


45.97 


6.3 " 




24.01 









* Table 5 in " Economics" Section of Manual of American Railway. 
Engineering Association. 



506 RAILROAD CONSTRUCTION. §458. 

6.820 m.p.h., and the multiple is 1.466. From Table XXXIX 
we find that the percentages of cylinder tractive power for simple 
engines for these two multiples of M are 78.01 and 82.42, respec- 
tively. The higher value is 105.7% of the lower, which shows 
that, in this case, adding 25% to the rate of coal consumption 
adds only 5.7 to the cylinder tractive power at 10 m.p.h. 

458. Effect of using a better quality of coal. As another 
instructive variation of the same problem, assume that the coal 
has effective B.t.u. of 13000, instead of only 11700. It will be 
found that steam will be produced more rapidly, the M velocity 
is 6.867 m.p.h. and the horsepower at that velocity is 680.3, 
but the cylinder power is computed to be 37191 lbs., which is 
again almost identical with the previous values, although the M 
velocity is still higher. The multiple for 10 m.p.h. is 1.456 and 
by Table XXXIX the per cent, of cylinder tractive power is 
82.73, which is an increase of 6% over 78.01%, showing that the 
increase in effective B.t.u. from 11700 to 13000 adds 6% to the 
cylinder tractive power at 10 m.p.h. 

459. Check with approximate rule. Applying Eq. 103 to 
the above data on the basis that the "effective steam pressure " 
is 85% of the gauge pressure (185) or 157 lbs., we will have 

_ . , 222X157X28 „ „„ 

Tractive force = = 37327 lbs. 

57 

This agrees with the more precise value (37190) computed above 
to within one-half of one per* cent. This rule is more simple as a 
method of obtaining merely the maximum tractive power at 
slow velocities, but the previous method, although longer, is 
preferable, since it computes the critical velocity M, and also 
the tractive force at higher velocities. 

460. Tractive Force at Higher Velocities. At higher velocities 
than M, the cyhnder power falls off quite rapidly, since the steam 
is cut off at part stroke and is used expansively. The proper 
per cent of cut-off for any given velocity and the number of 
pounds of steam per i.h.p. are shown in Table XXXVIII, in 
which is give the per cent of cylinder tractive power for multi- 
ples of M. The table shows, for example, that, for simple 
engines, the cylinder tractive power is 69.37% of its value for 
full stroke when the velocity is 2M and that when the velocity 
is increased to 5M the tractive power is reduced to 31.11%. 



§460. 



THE POWER OF A LOCOMOTIVE. 



507 



Applying this to the above numerical problem, when M = Q.1Q7 
m.p.h., the cylinder tractive power is reduced to 31.11% of 
37190, or 11570 lbs., but, since the velocity is five times as great, 
the horse-power developed is 31.11%X5 = 1.55 times as great. 
It should be noted that Table XXXIX shows a shght excess of 
tractive power (6% when starting), for the simple engine. This is 
due to the fact that with very low velocities the cylinder pressure 
more nearly equals the full boiler pressure and there is not the 
usual reduction of about 15%. Also, compound locomotives are 
operated with all the cylinders using full-pressure steam, which 
increases their effectiveness at starting about 35%, although at 
some loss in economy of steam due to compounding. But since 
the starting resistances are so much greater than the resistances 
above 5 miles per hour, the extra assistance is very timely. 

Any competent locomotive designer will, of course, make a 
design such that there is a proper relation between cyHnder 
power and tractive adhesion. In the above case, 106% of 
37190 = 39421 lbs., which is 25.7%, of the weight on the drivers, 
and this is just about the ratio of adhesion which may be ex- 
pected. 



Velocity. 


Cylinder tractive, 
power 


Locomo- 
tive resist- 
ance 
pounds. 


Draw-bar 

pull, 
pounds 


Multiples 
of M. 


Miles 
per hour. 


Per cent. 


Pounds. 


0.0 
1.0 
. 1.2 
1.5 
2.0 
3.0 
4.0 
5.0 
6.0 


0.000 

6.167 

7.400 

9.250 

12.334 

18.501 

24.668 

30.835 

37.002 


106.00 
100.00 
91.53 
81.37 
69.37 
52.78 
40.10 
31.11 
25.34 


39421 
37190 
34040 
30261 
25799 
19629 
14913 
11570 
9424 


1762 
1771 
1776 
1783 
1800 
1847 
1913 
1999 
2104 


37659 
35419 
32264 
28478 
23999 
17782 
13000 
9571 
7320 



A graphical illustration of the variation in tractive power and 
velocity may be obtained by computing first and setting down 
in tabular form the multiple values of M (6.167) ; the percentages 
taken from Table XXXIX, for each multiple of M; the products 
of each percentage times the tractive force (37190), for M veloc- 
ity; the locomotive resistance, from Table XXIX, for each 
velocity; and the net draw-bar pull for each velocity. These 
several values for cylinder tractive power and for draw-bar pull 
may be plotted as shown in Fig. 203, 



50S 



HAlLtlOAD CONSTRUCTION. 



§461. 



The student sKould realize that the above values represent 
the maximum draw-bar pull which the locomotive can produce, 
provided the fire-box is fed With 4000 lbs. of coal per hour. These 
draw-bar pulls as given will overcome the resistance of a train of 
some definite weight, at uniform speed, along a straight level 
track, at the several velocities given. A less weight of train 
will be drawn somewhat faster; or, it will travel at the same 
speed by Using less coal or by throttling the steam and, perhaps, 
wasting it at the blow-off. A heavier train could not maintain 
such speed* While the values given are approximately correct, 
a variation in the quality of the coal, or in the condition of the 



40,000$ 



% 



30,000 



o 

Pi 



% 20,000 



S 10,000 



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10 



FlG. 208. — Tractive Power, Mikado Locomotive. 



track, or in the firing, or in the management by the engineman, 
will alter the results materially, and they should not be relied 
on to give an accurate measure of what can and will be accom- 
plished at all times. But the method is useful and dependable 
in comparing two types of engines, or, for comparing the oper- 
ating results of light trains at faster speed or heavier trains at 
slower speed, using the same engine, or, as shown later, of com- 
paring the operating results of using a certain type of engine on 
two grades and thus estimating the value of reducing the higher 
grade. 

461. Effect of Grade on Tractive Power. The effect of grade 
on tractive power is best shown by some numerical computatior 1 
whose results are plotted in Fig. 209. The cylinder tractive 
power was computed for three engines of greatly different total 
weight and power, but which had driving-axle loads nearly 
identical (about 60750 lbs.), and, therefore, by the Baldwin 



§461. 



THE POWER OF A LOCOMOTIVE. 



509 



Locomotive Works rule, given in § 268, could all be operated on 
the same kind of track. Using the rule, 1X50750-^300 = 84.5, 
which means that the rails should weigh at least 85 lbs. per yard. 
Making computations for these locomotives, using 12000 B.t.u. 
coal, similar to those already detailed in §§ 453 et seq., it was 
found that the cylinder tractive powers of the Pacific, Mikado, 
and Mallet locomotives were 29718, 33575, 49095- lbs., respec- 
tively, when the velocity was uniformly 10 m.p.h. and the loco- 
motives each burned 4000 lbs. of coal per hour. The several 
engine resistances at 10 m.p.h. are easily computed from Table 
XXIX and are tabulated below. 



Engine characteristics 
(At velocity F = 10 m.p.h.) 

Cylinder tractive power. . . . 
Engine resistance on level . . 

Draw-bar pull on level 

Draw-bar pull on 3% grade. 



Pacific 

4-6-2 

(lb.) 



29,718 

2,205 

27,513 

15,213 



Mikr-lo 

2-8-2 
(lb.) 



33,575 

2,648 

30,927 

18,207 



Mallet 

2-8-8-2 
(lb.) 



49,095 

4,864 

44,231 

25,631 



The net values, or the draw-bar pulls, are plotted on the left- 
hand vertical line of Fig. 209, and in each case are the left-hand 
ends of the solid lines which show the tractive powers of the 
locomotives. On a 3% grade the grade resistances for the loco- 
motives equal 60 lbs. per ton, and are 12300, 12720 and 18600 
lbs., respectively. This reduces the effective draw-bar pull ap- 
proximately 40% in each case. Since this reduction varies 
uniformly with the grade, we may plot the three values, 15213, 
18207 and 25631, on the 3% vertical line and draw straight 
lines which represent in each case the tractive power of the 
locomotive at 10 m.p.h. and on any grade within that range. 

Assume trains of cars, all averaging 50 tons per car and vary- 
ing from 10 cars weighing 500 tons to 50 cars weighing 2500 tons. 
The resistances at 10 m.p.h on a level grade are given by Eq. 121, 
and may be plotted on the left-hand vertical line of Fig. 209. 
Grade adds resistance proportional to the grade. For example, 
on a 0.7% grade the grade resistance per ton is 14 lbs. and for 
2500 tons is 35000 lbs. Adding this to 11580, the tractive resist- 
ance, we have 46580, which we plot on the 0.7% vertical UnCo 
It is indicated by a small circle. Joining the two points gives 
the resistance line for 2500 tons hauled at 10 m.p.h. The circles 
on the other lines indicate similar computations. The inter- 



510 



RAILROAD CONSTRUCTION. 



§461. 



sections of these resistance lines with the Hnes of tractive power 
indicate the relative power of each locomotive. For example, 
the 1000-ton train can be hauled by the Pacific locomotive at 
10 m.p.h. up a 0.96% grade, but a Mikado can do the same on a 
1.1% grade, while the Mallet can do it on a 1.52% grade. 



50,000 " 1 j'" "'" 


__ 


t 4 7 


' 4 I 


t J 


2 z _ V 


J, f- 


-X -i 2^ 




]C__^i.__ _ Jl . 


45,000 -- -- 7~7 T~~ 


7 ^ Z"^ 


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^s L 3 2 


-T / 


^^ -. t t 


- t - 


±f\ ftftA -^S^ _j_ 


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40,000 -J S.. J t 


7* ~ 




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t 2 


^tS' ^/ ^'^ 


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fir rtnn — .^.fl*'iL rPJ- .^l . ..^."^^ 


I jr 


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^^^^ 


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f-T^^/--^-^ 


^^ ^1: ^'^ 


^ unnnn :=s.^I t jJ^ <S 


J^L_ -s*^^. J 


^ 30,000 "^^l" r 7^/ 


^¥_ ^V^.^ 


g i^"^'^ ^ -* 


^ \ 5^V.'o. 


2. >.^ tZ ^-"i t 


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7 "=3;^ ? ^> 


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•3 25,000 --^1 j"=>" ^^ 


[2i^3^ ^S-^ 


& 17 / "^^^ 'TT 


it^' X 


r tC t7 '^^^^^ 


tt ^sEr* 


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16,000 inj /■ y ---r- 




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t^l^'^.' / 




'Ul-(- ^ / 




. r ^^ r 




aOjOOO,-??^"" "''"" ~ 




l-,''t- ^^ 




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Ki\(\i\t-Zy. 




6,000 i ,^^ 




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("^ 








A 





0.5 1.0 1.5 2.0 2.5 3.0 

Per cent of grade 

Fig. 209. — Curves Showing Effect of Grade on Tractive Power. 



All of these calculations were made on the basis of burning 
4000 lbs. of coal per hour, which, as before stated, is the prac- 
tical limit of what an ordinary fireman can be expected to do for 
an extended run. 

The description of the Mallet locomotive (built by the Bald- 
win Locomotive Works), stated that its tractive power is 91000 
lbs. A computation of its cylinder tractive power at M velocity, 
using 12000 B.t.u. coal, shows it to be 95389 lbs. Subtracting 
the engine resistance (4843 lbs.), we would have 90546 lbs., 
which is a very fair check, especially as the Baldwin Locomotive 
Works method of calculation is different. 



§ 462. THE POWER OF A LOCOMOTIVE. 511 

462. Acceleration-speed curves. The time required for an 
engine of given weight and power to haul a train of known 
weight and resistance over a track with known grades and cur- 
vature is an important and necessary matter for an engineer to 
compute, since the saving in time has such a value as to justify 
constructive or operating changes which will reduce that time. 
Fig. 208 shows that the draw-bar pull is very much greater at 
very low velocities than at the moderate speed of even 15 m.p.h. 
In spite of the increased resistance at these low velocities the 
margin of power left for acceleration is also greater and the 
" speed curve " is really a curve and not a straight line. Its 
general form may be most easily developed by a numerical 
example, especially as each case has its own special curve. 

Illustrative Example. The Mikado locomotive, whose char- 
acteristics have already been investigated in §§ 453 et seq., has 
draw-bar pulls at various velocities as shown in the tabular 
form in § 460, to which frequent reference must be made in this 
demonstration. Assume that this locomotive starts from rest on 
a 0.4% upgrade, hauling a train of 14 cars, each weighing 50 
tons, and a caboose weighing 10 tons. Then the normal level 
tractive resistance, by Eq. 107, § 439, equals 

i2 = (2.2X710) + (122X15) =3392 lbs. 

The grade resistance of the cars will be 20X0.4X710 = 5680 lbs. 
The extra starting resistance will be considered as 6 lbs. per ton, 
or 4260 lbs. These three items total 13332 lbs. The average 
draw-bar pull of the locomotive at velocities between zero and M 
velocity, which is 6.167 m.p.h., is K37659 +35419) =36539 lbs., 
but this must be diminished in this case by 20 XO.4 X 157.5 = 1260 
lbs. for grade and by 157.5X6 = 945 lbs. for starting resistance, 
leaving a net draw-bar pull of 34334 lbs., excluding the force 
required for the acceleration of the locomotive. The net force 
available for acceleration of both the locomotive and the train 
is 34334-13332 = 21002 lbs., or prorated, is 210024- (157.5 + 
710) =24.21 lbs. per ton. Transposing Eq. 106, with Fi = 0, 
72 = 6.167, and P = 24.21 lbs., we have 5 = 70(38.03-0) -^24.21 
= 110 feet, the distance required to attain a velocity of 6.167 
m.p.h. 

While the velocity is increasing from 1.0 M to 1.2 M, the mean 
draw-bar pull is K35419 +32264) -1260 = 32582 lbs., less the 
accelerative resistance of the locomotive. Subtracting the 



612 [eailroad construction. § 462 

tractive and grade resistances of the cars, we have 32682—3392 
— 5680 = 23510 lbs. Note that there is no longer any starting 
resistance. The accelerative force in pounds per ton is 23510 
•^867.5 = 27.10. The distance s required to increase the veloc- 
ity from 6.167 m.p.h. to 7.400 m.p.h., is 70(54.76-38.03)4- 
27.10 = 43 feet. Similarly the distances required to increase 
the velocity from 1.2 M to 1.5 M, from 1.5 M to 2M, etc., are 
computed as in the accompanying tabular form. 

The corresponding distances and velocities have been plotted 
in Fig. 210. The velocity of 10 m.p.h. is acquired in a little over 
300 feet, but it requires 500 feet to acquire a velocity of 12.33 
m.p.h. and about 16000 feet to raise it to 29 m.p.h. The force, 
in pounds per ton, available for acceleration, is maximum at low 
velocities, after the extra starting resistance is overcome. As 
the margin per ton for acceleration becomes less and less, the 
greater is the distance required to increase the velocity 1 mile 
per hour— especially through the last increments — up to the 
velocity at which the net draw-bar pull exactly equals the total 
car resistance arid the velocity becomes uniform, which is later 
computed to be 4.78 M. There is an approximation in using 
average draw-bar pulls between the different velocities at which 
the draw-bar pull has been definitely computed, but the com- 
puted distances are practically correct up to 4 M velocity or 
24.67 m.p.h. But the computation for the distance required to 
increase the velocity from 4 M up to 4.78 M is far less accurate if 
the average draw-bar pull is used. The effective pull at 4 M 
velocity equals 13000 — 1260 = 11740, less the accelerative resist- 
ance of the locomotive. The tractive and grade resistance of 
the cars at this velocity is 3392+5680 = 9072. This leaves 
11740-9072 = 2668 lbs. available for acceleration of both loco- 
motive and cars. The reduction in tractive force between 4 M 
velocity and 5 M velocity (see § 460), is 13000-9571 =3429 lbs. 
By proportionate interpolation we would then say that the 
excess force available for acceleration would be exhausted at 
(2668 -^ 3429) = .78 of the interval, or at a velocity of 4.78 M, 
or 29.48 m.p.h. The mean accelerative force is one-haM of 2668, 
or 1334 lbs., which is 1.53 lbs. per ton of train. The dis- 
tance, by an inversion of Eq. 106, is computed to be 11925 feet. 
Owing to the approximate equality of working force and resist- 
ance and the momentary variations in both, the precise point 
where the acceleration would cease and the velocity woul<i 



462. 



THE POWEB OF A LOCOMOTIVE. 



513 



> 
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03 

03 

s 


Total 
from 
start! 

feet. 


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T-i i-i(N lO lO !>. l> 

I-lTjiCD 

1-1 


(NTf<i-l(N 
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(MO»C(N 
1-1 CO CO 00 


Accel- 
eration, 
or re- 
tarda- 
tion, 

feet 


O CO CO rH rH CD lO 
T-H Tj* Oi lO 05 Ol (N 

r-ieOrH 


(M(Nt>.-< 

COCOl-QO 
(NOOtJHcO 
r-(T-(COi-l 


in 

ID 

O 
u 


Net 

force 

per 

ton, 

lbs, 


rH O O rt< 1> CO CO 
(N --H i-H CO i-i 00 lO 


(N 

coi>coc^ 

Tj< rH 00 1-1 


Tj< l> CO 00 (N >0 r-( 


■*OC0O 

1-1 1-1 


Differ- 
ence ef- 
fective 
for ac- 
celera- 
tion or 
retarda- 
tion, 
lbs. 


(M O CT) t> 05 CT> -"i^ 
O --I CO O im lO CO 
0*0005 lOOCO 
■-1 CO O lO O »0 r-< 
<MlM(NrHr-l 


Oi-li-ICD 
»C(N(NO 
lO OOCOt-I 
(NOOCO 
1-1 


Car re- 
sistance 
tractive 

grade, 
plus 

start* 

lbs. 


(N <M <M (N (N (N (M 
CO I> !>• I> t> l^ I> 

COOOOOOO 
CO Oi Oi Qi 05-O5 O 
1—1 
■X- 


20432 
20432 
20432 
20432 


Actual 
draw- 
bar pull, 
average, 

lbs. 


tJ< (M ,-H 0> i-H !-< CO 
CO 00 1-1 1^ CO CO O 

CO lo .-1 03 CO --1 Tt< 

Tt* (N Oi -^ CT> Tt< O 
CO CO IM (M rH ,-1 1-1 


7882 
11611 
17111 
20326 


Loco- 
motive 
resist- 
ance, 
grade 
plus 
start* 

lbs. 


ICOOOOOO 
O CO CO CD CO CO CO 
(N (M (M (N(N IM (N 
(N iH tH r-l 1-1 1-1 1-1 
* 


OOOO 
00 00 00 00 

CO CO CO CO 


Mean 
draw- 
bar pull, 
level, 

lbs. 


61 (X 1-1 03 ■i-< 1-1 CO 
CO Tt< l> CO o> CT> CO 

lo 00 CO iM 00 CO CO 

CO CO O CO O lO 1-1 
COCOCOIMIMiHtH 


11662 
15391 
20891 
24106 


.2 
'+= 

'3 


Mean, 

feet per 

sec. 


(M IC (N CO i-( CO 1-1 

lO Oi 0^ 00 CO cD l> 


rH COi-IG3 

1>COC0 03 


■<i< O IN lO (N ,-( 03 
1-1 1-1 (M coco 


03i-<<Nt> 
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u 

ft 

CI <p 

p— ( 


COOiOCOOI>00 
1-1 ■<* <M CO »0 CO Tt< 


l>OC0i-l 

coiococq 


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i-ii-H(MO« 


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1> 

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^COiOCO 


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r-li-l<N. 


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<MC^r-<i-( 


<0 m 
fa M 

a 


O ■* CO I> 03 CO 00 
O O 00 »0 O 1-1 !-< 


T)H 00 CO 03 
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ra 

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oa 

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ft 
ft 

03 



.1-1 

o3 



514 



RAILROAD CONSTRUCTION. 



§463. 



actually become uniform would be be very uncertain. For- 
tunately the inaccuracy is of little or no practical importance 
and for the purposes of our calculations we may call this last 
interval 11925 feet, assuming that the grade is as long as 16715 
feet or 3.1 miles. If the 0.4% grade continued indefinitely the 
train would travel at this uniform velocity as long as the loco- 
motive operated on the basis assumed for this problem. Note 
that Fig. 210 would have to be extended to nearly three times its 











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LIME OF UNIFORM VELOCITY (29.48 M.P.H. 


ON 0.4(^ GRADEJ 










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LINE OF UNrFORM VELOCfTY( 12.21 M-P-H."* ON A.Q 4 GRADE 






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J.000 2000 3000 4000 

DISTANCES IN FEET 

Fig. 210. 



"^5000 



6000 



present length before the time curve would reach and become 
tangent to the " line of uniform velocity." 

463. Retardation-speed curves. When, on account of grade 
resistance, the total of tractive and grade resistance is greater 
than the draw-bar pull, there is retardation. 

Illustrative Example. Continuing the numerical problem of 
§ 462, assume that, while moving up the 0.4% grade at a velocity 
of 4.78 M, or 29.48 m.p.h., the train reaches a grade of +1.2%. 
The grade resistance of the cars will be 20X1.2X710 = 17040 
lbs. The tractive resistance will be 3392 lbs., as before, making 
a total of 20432 lbs. Interpolating in the tabular form in § 460 
for the draw-bar pull at 4.78 M velocity, we find 10325; at 4 ikf 
it is 13000 and the mean is 11662; but from this must be sub- 
tracted 20X1.2X157.5 = 3780 for grade resistance of the loco- 
motive, leaving 7882 lbs. for the net draw-bar pull. The retard- 
ing force is 20432 — 7882 = 12550; or in pounds per ton of train, 
is 12,550-^867.5 = 14.46. As before, using an inversion of 



§464. THE POWER OF A LOCOMOTIVE. 515 

Eq. 106, s = (29.482-24.672)70 ^14.46 = 1262 feet, the distance 
at which the velocity would reduce to 4 M. As before, the other 
quantities may be computed and recorded, with less danger of 
confusion and error, by tabulating them, as given in § 462. 

The mean velocity, when retarding from 4.78 M to 4.0 M, 
reduced to feet per second, is as before 39.71 feet per second, and 
dividing this into the distance, 1262 feet, gives 32, the time in 
seconds. The quantities for the reduction in velocity from 
4 M to 3 M and from 3 ilf to 2 ilf are computed similarly. The 
level draw-bar pull for 1.5 M is 28478 (see § 460), and by sub- 
tracting 3780, we get 24698 lbs. the actual net pull on the grade. 
Similarly, the actual pull at 2 M is 20219 lbs. The increase from 

213 

20219 to 20432 is = 4.7% of the interval from 20219 to 

4479 

24698 and 4.7% X. 5 = .02; therefore, the actual draw-bar pull 

just equals the resistance at 2.00 — .02 = 1. 98M, or 12.21 m.p.h. 

The deficiency of draw-bar pull at 2.0 M = 20,432 -20219 = 213 

lbs. At 1.98 M the deficiency is zero and, therefore, the mean 

deficiency is one-half of 213, or 106. Dividing this by 867.5, 

we have 0.122, which is the value of P in Eq. 106. Then 

s = (152.01- 149.08)70 -^0.122 = 1681 ft. 

Velocities in miles per hour can be readily converted into 
velocities in feet per second by multiplying by 1.4667. Averag- 
ing the two velocities at the beginning and the end of each period 
gives the mean velocity; and dividing each of these into the 
distance for that period gives the time in seconds. 

464. Drifting. The tractive resistance of the cars of the 
problem just worked out is 3392 lbs. ; the locomotive resistance 
at 20 m.p.h. is 1862 lbs., or a total of 5254 lbs. Variation in 
velocity will affect this but Uttle. Dividing by 867.5, the total 
weight in tons, we have 6.06 lbs., the resistance per ton, from 
which the equivalent rate of grade is 6.06 -^20 =.303%. This 
means practically that when this train is running down a grade 
which is over .303% it will run by gravity and steam may be 
shut off. If the grade is much greater than .303% the accelera- 
tion on the downgrade may become so great, if the grade is very 
long, that the velocity may become objectionably high. 

Illustrative Example. Assume that the limiting safe velocity 
for freight trains, considering the condition of track and rolling 



516 RAILROAD CONSTRUCTION. § 464. 

stock, is 35 m.p.h.; assume that the train we have been consider- 
reaches a 0.4% downgrade at a velocity of 15 m.p.h. How far 
down the grade will it run with steam shut off, before the speed 
reaches 35 m.p.h. and brakes must be applied? There is no 
question here of variable tractive power since the only motive 
power is gravity. The resistance is nearly independent of 
velocity and we will here assume it to be so and utilize Table 
XLII. At 15 m.p.h. the train has a velocity head of 7.90 feet. 
At 35 m.p.h. the velocity head is 43.01 feet. The train can, 
therefore, drop down the grade a vertical height of 43.01—7.90 
= 35.11 feet before the velocity reaches 35 m.p.h. On a 0.4% 
grade the distance required for such a fall is 35.11-^.004 = 8777 
feet. The problem in § 462 assumed that the 0.4% grade is 
16715 feet or more, and this shows what will happen to the 
trains moving in the opposite direction. 

But it must not be thought that there is no loss of energy 
during drifting. Even though no steam is used in the cylinders, 
some is frequently wasted at the safety valve and more is used 
in operating brakes and in maintaining the brake air-reservoir 
at full pressure. But the greatest loss of heat is that due to 
radiation, especially in winter, in spite of all the jacketing devices 
to retain heat. Although the results of the numerous tests 
which have been made are quite variable, the following approxi- 
mate averages may be used: The loss due to radiation while 
standing may be figured at 120 lbs. of coal per hour per 1000 
square feet of heating surface; while drifting the loss will in- 
crease to 220 lbs. per hour. The amount of coal used for firing 
up will be about 510. This is based on the use of 12000 B.t.u. 
coal. The better the coal, the less will be used. 

Illustrative Example. The Mikado locomotive we have been 
considering has 2565 square feet of heating surface. It will then 
require about 2.565X510 = 1308 lbs. of coal to fire up. While 
drifting down the grade, referred to above, a distance of 8777 feet, 
the average velocity is ^(15+35) =25 m.p.h. =36.67 ft. per sec. 
and the required time is 8777 -i- 36.67 =239 seconds = 3 min. 59 
sec. = .066 hour. The coal used while drifting down this short 
run would be 

.220X2.565X.066 = 371bs. 

At this point brakes would need to be applied and the time 
spent in drifting beyond this point must be computed as an item 



§465. THE POWER OF A LOCOMOTIVE. 517 

!•; 
in the total time spent on the run and also to compute the total 
amount of coal consumed while drifting. Although this item 
of 37 lbs. is relatively very small, its method of computation is 
typical of the computation of the several items to make up the 
total of coal consumed during a trip. 

465. Review of computed power of one locomotive. It was 
assumed that it started on a +0.4% grade with a load of 15 cars 
weighing 710 tons. After moving 16715 feet (assuming that 
the grade was that long), and doing it in 493 seconds, or 8 min- 
utes 13 seconds, the train acquired a velocity of 29.48 m.p.h. 
and the power of the locomotive would then be sufficient, when 
burning 4000 lbs. of coal per hour, to keep it moving up such a 
grade indefinitely at that velocity. In case the grade were not 
as long as 16715 feet, it would be necessary to compute the 
velocity where the rate of grade changed and make that the 
basis for the computation on the succeeding grade. But, 
assuming that the grade were as long as 16715 feet, or more, and 
that the velocity of 29.48 m.p.h. had been acquired, and that the 
train had run at that speed for some distance — although this 
does not modify the problem — ^the train is assumed to reach a 
still steeper grade +1.2%. The velocity then begins to decrease 
and in a total distance of 8252 feet and a total time of 337 sec- 
onds, or 5 minutes 37 seconds, the velocity is reduced to 
12.21 m.p.h., at which velocity the locomotive is able to make 
steam fast enough to overcome the higher resistance on the 
steeper grade. From that point on, assuming that the 1.2% 
grade is longer than 8252 feet, the train would continue for 
the remaining length of that grade at the velocity of 12.21 
m.p.h. 

As before stated, precision in the above results depends on 
many factors (such as B.t.u. of coal used, or the actual consump- 
tion in pounds per hour), which are somewhat variable. Some- 
times the variation of these factors from the values used above is 
known; sometimes it is unknown and then the accuracy of the 
results is correspondingly uncertain. But whether accurately 
known or not, when this method is used, employing the best 
values for the factors which are obtainable, the method shows 
a valuable comparison of two proposed alinements or grades. 
In such a comparison, any error in the factors will affect both 
results nearly, if not quite, equally, and the comparative results 
will still be substantially correct. 



518 RAILKOAD CONSTRUCTION. § 466. 

466. Selection of route. The preceding articles may be 
utilized in comparing two routes. If one of the lines is already 
in operation, the engineer has the great advantage of being able 
to determine by test exactly what results may be obtained on 
that line and what factors should be used in computations. 

It is then only necessary to compute the quantities for the 
proposed new line. When both lines are " on paper " there is 
less certainty as to the accuracy of the results, except that the 
line which is shown to be most advantageous will probably con- 
tinue to be most advantageous even if the uncertain factors used 
in the comparison are somewhat changed. Using the methods 
outlined in §§ 462 to 464, there will be computed the behavior of 
an assumed tj^e of locomotive, hauling one or more types of 
train load, and passing over tracks having definite grades and 
lengths. The effect of curves may be disregarded provided that 
the grades were properly compensated during original con- 
struction, and then the rate of grade for the entire length of 
straight and curved track may be taken as the rate on the 
straight track. If the rate of grade is actually uniform, even 
through the curves, then the lengths of curved track must 
be computed separately and on the basis of a rate of grade 
equal to the actual rate plus an allowance of .035% for each 
degree of curve. The behavior of a train from starting to 
stopping must be computed, making due allowance for each 
change in condition which will affect the hauling power of the 
locomotive. The locomotive is assumed to be working at the 
limit of its steaming capacity, except when drifting with steam 
shut off on a down grade, or when brakes are applied, either to 
prevent objectionably high velocity on a down grade or to make 
a stop. The action of brakes during a service stop (as distin- 
guished from an emergency stop), may be considered as a retard- 
ing force varying from 10% to 20% of the train weight. Un- 
fortunately brake action is so variable, being directly imder the 
control of the locomotive engineer and varying from zero to 
the full braking power, that any computation of energy used in 
operating them or of the effect of the brakes is impracticable 
except on the basis of arbitrary assumptions such as the require- 
ment that the brakes are used in such a way that a train will be 
retarded at a specified rate. The performance of the locomotive 
over the entire division, the total time required, its velocity in 
critical places, etc., can be computed. In §§ 462 and 463 it 



§ 467 THE POWER OF A LOCOMOTIVE. 519 

was shown that the locomotive considered could haul the par- 
ticular train considered up a 0.4% grade at a velocity of 29.48 
m.p.h„ and maintain such speed indefinitely; also that it could 
haul the same train up a 1.2% grade at 12.21 m.p.h. and main- 
tain its velocity indefinitely. This of course,, means that a 
much heavier train could be hauled up the 0.4% grade and that a 
somewhat heavier train could be hauled up the 1.2% grade with- 
out being stalled, although the velocities in each case would be 
reduced. There are an infinite number of combinations, but 
there are usually some considerations which narrow the choice. 
Even after construction is complete these tables may be utihzed 
in a study of the most economical combination of type of loco- 
motive and amount of train load for the track conditions as 
they may exist. 

467. Rating of locomotives. The maximum power of a loco- 
motive on any grade at M velocity is measured by its " rating." 

Let P=the tractive power of the locomotive, measured at 
the rim of the drivers; 
^ = Weight of engine and tender, in pounds; 
TF = Weight of cars behind tender, in pounds; 
r=rate of grade, or the ratio of vertical to horizontal; 
a = a constant, which as determined by tests = 2.2 lbs. 

per ton or .0011 lb. per pound of train; 
6=a constant, which as determined by tests = 122 lbs. 
per ton. a and b are the same constants as are used 
in § 439. 
n= number of cars in train. 
Then P = (E+W) (r+a) +hn. 
Transforming, 

P h 

.E = W+n . ..... (122) 



r-\-a r+a 

The right-hand side of this equation Is called the "rating," A, 
and is the weight of the train behind the tender plus the number 
of cars times a quantity made up of two constants and the rate 
of grade. This quantity is independent of any special engine or 
train values and may be tabulated for various rates of grade, 
as given in Table XL. 

Examples. The Mikado locomotive considered in §§ 453, 
et seq., has a tractive power, measured at the rim of the drivers, 



520 




RAILROAD CONSTRUCTION.' 


■J *6^ 




TABT<F, XL- 


-LOCOMOTIVE EATING DISCOUNTS. 








. VALUES OF 


b-7-(rXa) FOR various grades. 










(In tons per car.) 










t-i . 




t-< • 




;-i . 




f-< . 




f~* 




m^-. 


■^ 


S^'S" 


-ij 


J?';? 




o3^ 


+2' 


03^ 


S 


o e 


!3 


« 6 


a 


o e 


fl 


o e 


(3 


o € 






CD h 














si 


(1 


, g.|. 


1:3 


g.|. 


la 




13 




13 


iJ 


o 


&H 


o 


H 


a 


H 


, o 


. Eh 


o 


&H 


Level 


55 


0.5 


10.0 


1.0 


5.5 


1.5 


3.8 


2.0 


2.88 


0.1 


29 


0.6 


8.5 


1.1 


. 5.0 


1.6 


3.6 


2.1 


2.75 1 


0'.2' 


20 


0.7 


7.5 


1.^ 


: 4.6 


1.7 


3.4 


2.2 


2.6311 


0-.3 


14 


0.8 


6.7 


1.3 


4.3 


1.8 


3.2 


2.3 


2.52" 


0.4 


12 


0.9 


6.0 


1.4 


4.0 


1.9 


3.0 


2.4 


2.42 



at M velocity, or 6.167 m.p.h., of 37190-1432 = 35758 lbs., 
which equals P; 1432 is the locomotive resistarxce between 
cylinder and rim of. drivers, see § 429. The weight of engine 
and tender is 315000 lbs. What is its rating on a 1.2% grade? 
The value of r for a 1.2% grade = .012; a = .0011 lb. per pound. 
Theu 

P 35758 

A= — ~E= ,._ _... - 315000 = 2,414,000 lbs. = 1207 tons. 



r-\-a 



.012 + .0011 



which is the rating for that locomotive for a 1.2% grade. But 
this does not mean 1207 tons of cars. Placing this equal to the 
right-hand side of Eq. 122, we have 

b 

1207=.ir+n' . 

r-\-a 

The value of — r— for a 1.2% grade is given in Table XL as 4.6. 



r-\-a 



Then 



TF==1207-4.6n, 



which shows that the weight of train depends on the number of 
cars. Assume that n = 16. Then T7 = 1133.4 and the average 
weight per car is 70.8 tons. Assume that the cars are all 
" empties," weighing 18 tons each; then TF = 18n, and 

7i = 1207 -^ (18 +4.6) =53.4, 

which must be interpreted as 53 empty cars. 

In the above examples the pulling power P is determined on the 
basis of the locomotive working at the maximum velocity M at 



§467. THE POWER OF A LOCOMOTIVE. 521 

which it can maintain full stroke. See § 455. This represents 
practically the maximum power of the locomotive. The velocity 
M is usually from 4 to 7 miles per hour and is as low as should be 
allowed on maximum grades, since an attempt to utilize a slightly 
higher tractive force at a somewhat lower velocity would prob-- 
ably result in stalhng the train if an unexpected resistance in 
the track slightly increased the normal resistance. 



CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

468. Method of formation of railroad corporations. Many 
business enterprises, especially the smaller ones, are financed 
entirely by the use of money which is put into them directly 
in the form of stock or mere partnership interest. A railroad 
enterprise is frequently floated with a comparatively small 
financial expenditure on the part of the original promoters. 
The promoters become convinced that a railroad between A 
and B, passing through the intermediate towns of C and D, 
with others of less importance, will be a paying investment. 
They organize a company, have surveys made, obtain a charter, 
and then, being still better able (on account of the additional 
information obtained) to exploit the financial advantages of 
their scheme, they issue a prospectus and invite subscriptions 
to bonds. Sometimes a portion of these bonds are guaranteed, 
principal and interest, or perhaps the principal alone, by town- 
ships or by the national government. The cost of this pre- 
liminary work, although large in gross amount if the road is 
extensive, is yet but an insignificant proportion of the total 
amount involved. The proportionate amount that can be 
raised by means of bonds varies with the circumstances. In 
the early history of railroad building, when a road was pro- 
jected into a new country where the traffic possibilities were 
great and there was absolutely no competition, the financial 
success of the enterprise would seem so assured that no diffi- 
culty would be experienced in raising from the sale of bonds 
all the money necessary to construct and equip the road. But 
the promoters (or stockholders) must furnish all money for the 
preliminary expenses, and must make up all deficiencies be- 
tween the proceeds of the sale of the bonds and the capital needed 
for construction. 

**In theory, stocks represent the property of the responsible 
owners of the road, and bonds are an encumbrance on that 

522 



§ 469. PROMOTION OF RAILROAD PROJECTS. 523 

property. According to this theory, a railroad enterprise 
should begin with an issue of stock somewhere near the value 
of the property to be created and no more bonds should be 
issued than are absolutely necessary to complete the enter- 
prise. Now it is not denied that there are instances in which 
this theory is followed out. In New England, for example, 
as well as in some of the Southern States, there are a few roads 
represented wholly by stock or very lightly mortgaged. But 
this theory does not conform to the general history of railway 
construction in the United States, nor is it supported by the 
figures that appear in the summary. The truth is, railroads 
are built on borrowed capital, and the amount of stock that is 
issued represents in the majority of cases the difference between 
the actual cost of the undertaking and the confidence of the 
public expressed by the amount of bonds it is willing to absorb 
in the ultimate success of the venture." * 

"The same general law obtains and has always obtained 
throughout the world, that such properties (as railways) are 
always built on borrowed money up to the limit of what is 
regarded as the positive and certain minimum value. The 
risk only — the dubious margin which is dependent upon sagac- 
ity, skill, and good management — is assumed and held by the 
company proper who control and manage the property." f 

469. The two classes of financial interests — the security and 
profits of each. From the above it may be seen that stocks, 
bonds, car-trust obligations, and even current liabilities repre- 
sent railroad capital. The issue of the bonds "was one means 
of collecting the capital necessary to create the property against 
which the mortgage lies." The variation between these inter- 
ests lies chiefly in the security and profits of each. The current 
liabilities are either discharge'd or, as frequently happens, they 
accumulate until they are funded and thus become a definite 
part of the railroad capital. 

The growth of this tendency is shown in the following tabular 
form (see next page) : 

The bonded interest has greater security than the stock, but 
less profit. The interest on the bonds must be paid before any 
money can be disbursed as dividends. If the bond interest 



* Henry C. Adams, Statistician, U. S. Int. Con. Commission. 
t A. M. Wellington, Economic Theory of Railway Location. 



524 RAILROAD 


CONSTRUCTION. 


§ 469, 


Capitalization of 


June 30, 1898. 


June 30, 1912. 


Dec. 31, 1918. 


Railroads in the United 

States. 


Amount, 
millions. 


Per 
cent. 


Amount, 
millions. 


Per 

cent. 


Amount, 
millions. 


Per 

cent. 


Stocks 


5311 
5510 
1087 


44.6 
46.3 1 
9.1/ 


8622 
11130 


43.7 
66.3 


8678 
11406 


43.2 
56.8 


Funded debt 


Current liabilities, etc . . 



is not paid, a receivership, and perhaps a foreclosure and sale 
of the road, is a probability, and in such case the stockholder's 
interests are frequently wiped out altogether. The bond- 
holder's real profit is frequently very different from his nomi- 
nal profit. He sometimes buys the bonds at a very considerable 
discount, which modifies the rate which the interest received 
bears to the amount really invested. Even the bondholder's 
security may suffer if his mortgage is a second (or fifth) mort- 
gage, and the foreclosure sale fails to net sufficient to satisfy 
all previous claims. 

On the other hand, the stockholder, who may have paid in 
but a small proportion of his subscription, viay, if the venture 
is successful, receive a dividend which equals 50 or 100% of the 
money actually paid in, or, as before stated, his entire holdings 
may be entirely wiped out by a foreclosure sale. When the 
road is a great success and the dividends very large, additional 
issues of stock are generally made, which are distributed to the 
stockholders in proportion to their holdings, either gratuitously 
or at rates which give the stockholders a large advantage over 
outsiders. This is the process known as ''watering." While 
it may sometimes be considered as a legitimate "salting down" 
of profits, it is frequently a cover for dishonest manipulation of 
the money market. 

For the twelve years between 1887 and 1899 about two thirds 
of all the railroad stock in the United States paid no dividends, 
while of those that paid dividends the average rate varied 
from 4.96 to 5.74%. The year from June 30, 1898, to June 30, 
1899, was the most prosperous year of the group, and yet nearly 
60% of all railroad stock paid no dividend, and the average 
rate paid by those which paid at all was 4.96%. The total 
amount distributed in dividends was greater than ever before, 
but the average rate is the least of the above group because many 
roads, which had passed their dividends for many previous 



§470. PROMOTION OF RAILROAD PROJECTS. 525 

years, distinguished themselves by declaring a dividend, even 
though small. During that same period but 13.35% of the 
stock paid over 6% interest. The total dividends paid amounted 
to but 2.01% of all the capital stock, while investments ordi- 
narily are expected to yield from 4 to 6% (or more) according 
to the risk. Of course the effect of ''watering" stock is to 
decrease the nominal rate of dividends, but there is no dodging 
the fact that, watered or not, even in that year of "good times," 
about 60% of all the stock paid no dividends. Unfortunately 
there are no accurate statistics showing how much of the stock 
of railroads represents actual paid-in capital and how much 
is ''water." The great complication of railroad finances and 
the dishonest manipulation to which the finances of some rail- 
roads have been subjected would render such a computation 
practically worthless and hopelessly unreliable now. 

During the year ending June 30, 1898 (which may in general 
be considered as a sample), 15.82% of the funded debt paid no 
interest. About one third of the funded debt paid between 
4 and 5% interest, which is about the average which is paid. 

The income from railroads (both interest on bonds and divi- 
dends on stock) may be shown graphically by diagrams, such 
as are given in the annual reports of the Interstate Commerce 
Commission. They show that while railroad investments are 
occasionally very profitable, the average return is less than 
that of ordinary investments to the investors. The indirect 
value of railroads in building up a section of country is almost 
incalculable and is worth many times the cost of the roads. 
It is a discouraging fact that very few railroads (old enough to 
have a history) have escaped the experience of a receivership, 
with the usual financial loss to the then stockholders. But 
there is probably not a railroad in existence which, however 
much a financial failure in itself, has not profited the community 
more than its cost. 

470. The small margin between profit and loss to projectors. 
When a railroad is built entirely from the funds furnished by 
its promoters (or from the sale of stock) it will generally be a 
paying investment, although the rate of payment may be very 
small. The percentage of receipts that is demanded for actual 
operating expenses is usually about 67%. The remainder will 
usually pay a reasonable interest on the total capital involved. 
But the operating expenses are frequently 90 and even 100% of 



526 RAILROAD CONSTRUCTION. - §471. 

the gross receipts. In such cases even the bondholders do not ■ 
get their due and the stockholders have absolutely nothing. 
Therefore the stockholder's interest is very speculative. A 
comparatively small change in the business done (as is illus- 
trated numerically in § 472) will not only wipe out altogether the 
dividend — taken from the last small percentage of the total 
receipts and which may equal 50% or more of the capital stock 
actually paid in — but it may even endanger the bondholders' 
security and cause them to foreclose their mortgage. In such 
a case the stockholders' interest is usually entirely lost. It 
does not alter the essential character of the above-stated rela- 
tions that the stockholders sometimes protect themselves 
somewhat by buying bonds. By so doing they simply decrease 
their risk and also decrease the possible profit that might result 
from the investment of a given total amount of capital. 

471. Extent to which a railroad is a monopoly. It is a popu- 
lar fallacy that a railroad, when not subject to the direct com- 
petition of another road, has an absolute monopoly — that it 
controls "all the traffic there is" and that its income will be 
practically independent of the facilities afforded to the public. 
The growth of railroad traffic, like the use of the so-called 
necessities or luxuries of life, depends entirely on the supply 
and the cost (in money or effort) to obtain it. A large part of 
railroad traffic belongs to the unnecessary class — such as travel- 
ing for pleasure. Such traffic is very largely affected by mere 
matters of convenience, such as well-built stations, convenient 
terminals, smooth track, etc. The freight traffic is very largely 
dependent on the possibility of delivering manufactured articles 
or produce at the markets so that the total cost of production 
and transportation shall not exceed the total cost in that 
same market of similar articles obtained elsewhere. The crea- 
tion of facilities so that a factory or mine may successfully 
compete with other factories or mines will develop such traffic. 
The receipts from such a traffic may render it possible to still 
further develop facilities which will in return encourage further 
business. On the other hand, even the partial withdrawal of 
such facilities may render it impossible for the factory or mine 
to compete successfully with rivals; the traffic furnished by 
them is completely cut off and the railroad (and indirectly the 
whole community) suffers correspondingly. The "strictly 
necessary" traffic is thus so small that few railroads could pay 



§472. 



PROMOTION OF RAILROAD PROJECTS. 



527 



their operating expenses from it. The dividends of a road 
come from the last comparatively small percentage of its revenue, 
and such revenue comes from the ''unnecessary" traffic which 
must be coaxed and which is so easily affected by apparently 
insignificant ''conveniences." 

472. Profit resulting from an increase in business done; loss 
resulting from a decrease. In a subsequent chapter it will 
be shown that a large portion of the operating expenses are 
independent of small fluctuations in the business done and that 
the operating expenses are roughly two thirds of the gross 
revenue. Assume that by changes in the alinement the business 
obtained has been increased (or diminished) 10%. Assume for 
simplicity that the operating expenses on the revised track 
are the same as on the route originally planned; also that the 
cost of the track is the same and hence the fixed charges are 
assumed to be constant for all the cases considered. Assume 
the fixed charges to be 28%. The additional business, when 
carried in cars otherwise but partly filled will hardly increase 
the operating expenses by a measurable amount. When 
extra cars or extra trains are required, the cost will increase 
up to about 60% of the average cost per train mile. We may 
say that 10% increase may in general be carried at a rate of 
40% of the average cost of the traffic. A reduction of 10% 
in traffic may be assumed to reduce expenses a similar amount. 
The effect of the change in business will therefore be as follows: 



Operating exp. = 67 
Fixed charges = 28 



Total income. 



95 
.100 



Available for divi- 
dends 5 



Business increased 10%. 

67(1 + 10% X 40%)= 69.68 
28.00 

97.68 
Income 110.00 

Available for divi- 
dends 12.32 



Business decreased 10%. 

67( 1 - 10% X 40% ) = 64 . 32 
: .... 28 . 00 

92 32 
Income 90 ! 00 

Deficit 2.32 



In the one case the increase in business, which may often 
be obtained by judicious changes in the alinement or even by 
better management without changing the alinement, more than 
doubles the amount available for dividends. In the other case 
the profits are gone, and there is an absolute deficit. The 
above is a numerical illustration of the argument, previously 



528 RAILROAD CONSTRUCTION. §473. 

r 

stated, of the small margin between profit and loss to the original 
projectors. 

473. Estimation of probable volume of traffic and of probable 
growth. Since traffic and traffic facilities are mutually inter- 
dependent and since a large part of the normal traffic is merely 
potential until the road is built, it follows that the traffic of a 
road will not attain its normal volume until a considerable 
time after it is opened for operation. But the estimation even 
of this normal volume is a very uncertain problem. The esti- 
mate may be approached in three ways: 

1st. The actual gross revenue derived by all the railroads 
in that section of the country (as determined by State or U. S. 
Gov. reports) may be divided by the total population of the 
section and thus the average annual expenditure per head of 
population may be determined. A determination of this value 
for each one of a series of years will give an idea of the normal 
rate of growth of the traffic. Multiplying this annual eontri- 
bution by the population which may be considered as tributary 
gives a valuation of the possible traffic. Such an estimate is 
unreliable (a) because the average annual contribution may not 
fit that particular locality, (6) because it is very difficult to 
correctly estimate the number of the true tributary population 
especially when other railroads encroach more or less into the 
territory. Since a rough value of this sort may be readily 
determined, it has its value as a check, if for nothing else. 

2d. The actual revenue obtained by some road whose 
circumstances are as nearly as possible identical with the road 
to be considered may be computed. The weak point consists 
in the assumption that the character of the two roads is identical 
or in incorrectly estimating the allowance to be made for ob- 
served differences. The method of coiu-se has its value as a 
check. 

3d. A laborious calculation may be made from an actual 
study of the route — determining the possible output of all 
factories, mines, etc., the amount of farm produce and of lumber 
that might be shipped, with an estimate of probable passenger 
traffic based on that of like towns similarly situated. This 
method is the best when it is properly done, but there is always 
the danger of leaving out sources of income — both existent 
and that to be developed by traffic facilities, or, on the other 
hand^ of overestimating the value of expected traffic. In the 



§473. 



PROMOTION OF BML.ROAD PROJECTS. 



529 



following tabular form are shown the population, gross re- 
ceipts, receipt^ per head of population, mileage, earnings per 
mile of line operated, and mileage per 10,000 of population for 
the whole United States. It should be noted that the values 
are only averages, that individual variations are large, and that 
only a very rough dependence may be placed on them as applied 
to any particular case. 



Year. 


Population 

(eatimated) . 


Gross 
receipts. 


Receipts 
per head 
of popu- 
lation. 


Mileagef 


Earnings 

per mile 

of line 

operated. 


Mileage 
per 
10,000 
popula- 
tion. t 


1888... 
1889... 
1890... 
1891. . . 


60,100,000 

61,450,000 

*62,801,571 

64,150,000 


$910,621,220 

964,816,129 

1051,877,632 

1096,761,395 


115.15 
15 . 81 
16.75 
17.10 


136,884 
153,385 
156,404 
161,275 


$6653 
6290 
6725 
6801 


24.94 
25.67 
26.05 
26.28 


1892... 
1893... 
1894... 
1S95... 
1896... 


65,500,000 
68,850,000 
68,200,000 
69,550,000 
70,900,000 


1171,407,343 
1220,751,874 
1073,361,797 
1075,371,462 
1150,169,376 


17.89 
18.26 
15.74 
15.46 
16.22 


162,397 
169,780 
175,691 
177,746 
181,983 


7213 
7190 
6109 
6050 
6320 


26.19 
26.40 
26.20 
25.97 

25.78 


1897...: 

1898... 
1899... 
1900... 
1901. . . 


72,350,000 
73,600,000 
74,950,000 
*76,295,220 
77,863,000 


1122,089,773 
1247,325,621 
1313,610,118 

1487,044,814 
1588,526,037 


15.53 
16.95 
17.53 
19.49 
20.47 


183,284 
184,648 
187,535 
192,556 
195,562 


6122 
6755 
7005 
7722 
8123 


25.53 
25.32 
25.25 
25.44 
25.52 


1902... 
1903 . . . 
J904... 
1905... 
1906... 


79,431,000 
80,998,000 
82,566,000 
84,134,000 
85,701,000 


1726,380,267 
1900,846,907 
1975,174,091 
2082,482,406 
2325,765,167 


21.88 
23.70 
24.23 
25.15 
27.65 


200,155 
205,314 
212,243 
216,974 
222,340 


8625 
9258 
9306 
9508 
10460 


25 . 76 
26.03 
26.34 
26.44 
26.78 


1907... 
1908... 
1909...: 
1910... 
1911... 
1912... 


87,279,000 
88,837,000 
90,405,000 
*9 1,9 72, 266 
93,572,266 
95,172,266 


2.589,105,578 
2393,805,989 
2418,677,-538 
2750,667,435 
2789,761,669 
2842,695,382 


29.63 
26.95 
26.71 
29.91 
29.81 
29.87 


227,455 
231,540 
234,800 
238,609 
244,476 
247,981 


11383 

10338 

10301 

11528 

11411 

11463 


26.38 
26.30 
26.20 
26.14 
26.10 
25.93 



♦Actual. t Excludes a snaall percentage not reporting "gross receipts.'^ 

% Axitual mileage. 



The probable growth in traffic, after the traffic has oncfe 
.attained its normal volume, is a small but almost certain quantity. 
In the above tabular form this is indicated by the gradual 
growth in "receipts per head of population" from 1897 to 
3-907. Then the sudden drop due to the panic of 1907 is clearly 
indicated, and also the gradual growth in the last few years, 
^ven in England, where the population has been nearly station- 
ary for many years, the growth though small is unmistakable. 
On the other hand the growth in some of the Western States 



530 EAILROAD CONSTRUCTION. §474 

■ i'. K^ — - 
has been very large. For example, the gross earnings per head 

of population in the State of Iowa increased from $1.42 in 1862 

to $10.00 in 1870, and to $19.46 in 1884. 

There will seldom be any justification in building to accommo-r| 
date a larger business than what is "in sight." Even if it';i 
could be anticipated with certainty that a large increase in i 
business would come in ten years, there are many reasons why 
it would be unwise to build on a scale larger than that required 
for the business to be immediately handled. Even though it 
may cost more in the future to provide the added accommo- 
dations (e.g. larger terminals, engine-houses, etc.), the extra 
expense will be nearly if not quite offset by the interest saved 
by avoiding the larger outlay for a period of years which may 
often prove much longer than was expected. A still more im- 
portant reason is the avoidance of uselessly sinking money at 
a time when every cent may be needed to insure the success 
of the enterprise as a whole. 

474. Probable number of trains per day. Increase with 
growth of traffic. The number of passenger trains per day 
cannot be determined by dividing the total number of passengers 
estimated to be carried per day by the capacity of the cars 
that can be hauled by one engine. There are many small 
railroads, running three or four passenger trains per day each 
way, which do not carry as many passengers all told as are 
carried on one heavy train of a trunk line. But because the 
bulk of the passenger traffic, especially on such light-traffic 
roads, is " unnecessary" traffic (see § 471) and must be encouraged 
and coaxed, the trains must be run much more frequently 
than mere capacity requires. The minimum number of passen- 
ger trains per day on even the lightest-traffic road should be 
two. These need not necessarily be passenger trains exclusively. 
They may be mixed trains. 

The number required for freight service may be kept more 
nearly according to the actual tonnage to be moved. At least 
one local freight will be required, and this is apt to be considerably 
within the capacity of the engine. Some very light-traffic 
roads have little else than local freight "to handle, and on such 
there is less chance of economical management. Roads with 
heavy traffic can load up each engine quite accurately according 
to its hauling capacity and the resulting economy is great. Fluc- 
tuations in traffic are readily allowed for by adding on or drop- 



§ 475, PROMOTION OF RAILROAD PROJECTS. 531 

ping off one or more trains. Passenger trains must be run on 
regular schedule, full or empty. Freight trains are run by 
train-despatch er's orders. A few freight trains per day may be 
run on a nominal schedule, but all others will be run as extras. 
The criterion for an increase in the number of passenger trains 
is impossible to define by set rules. Since it should always 
come before it is absolutely demanded by the train capacity 
being overtaxed, it may be said in general terms that a train 
should be added when it is believed that the consequent in- 
crease in facilities will cause an increase in traffic the value of 
which will equal or exceed the added expense of the extra train. 

475. Effect on traffic of an increase in facilities. The term 
facilities here includes everything which facilitates the transport 
of articles from the door of the producer to the door of the 
consumer. As pointed out before, in many cases of freight 
transport, the reduction of facilities below a certain point will 
mean the entire loss of such traffic owing to local inability to 
successfully compete with more favored localities. Sometimes 
owing to a lack of facilities a railroad company feels compelled 
to make some concession which is a virtual reduction on what 
would normally be the freight rate. In competitive freight 
business such a method of procedure is a virtual necessity in 
order to retain even a respectable share of the business. Even 
though the railroad has no direct competitor, it must if possible 
enable its customers to meet their competitors on even terms. 
In passenger business the effect of facilities is perhaps even 
more marked. The pleasure travel will be largely cut down 
if not destroyed. 

476. Loss caused by inconvenient terminals and by stations 
far removed from business centers. This is but a special case 
of the subject discussed just in the preceding paragraph. The 
competition once existing between the West Shore and the 
New York Central was hopeless for the West Shore from the 
start. The possession of a terminal at the Grand Central 
Station gave the New York Central an advantage over the West 
Shore, with its inconvenient terminal at Weehawken, which 
could not be compensated by any obtainable advantage by 
the West Shore. This is especially true of the passenger busi- 
ness. The through freight business passing through or termi- 
nating at New York is handled so generally by means of floats 
that the disadyaatage ia this respect is not so great. The 



832 RAILROAD co!^struct1on7 | ?46. 

enottnous expendittife (roughly $10,000,000) made by thfe 
Pennsylvania ti. ll., on the Broad Street Station (and its ap- 
proaches) in Philsldelphia, a large part of which was niade in 
crossing the Schuylkill River and running to City Hall Square, 
father than retain theit terminal in West Philadelphia, is an 
iilustfation of the policy of a great road on such a question. | 
The fa,ct that the original plan and expenditure has been very 
largely increased since the first construction proves that the 
management has not only approved the original large outlay, 
but saw the wisdom of making a very large increase in the ex- 
penditure. 

The construction of great terminals is comparatively infrequent 
and seldom concerns the majority of engineers. But an engineer 
has frequently to consider the question of the location of a 
way station with reference to the business center of the town, 
I'he following points may (or may not) have to be considered, 
and the real question consists in striking a proper balance 
between conflicting considerations. 

(1) During the early history of a railroad enterprise it is 
especially needful to avoid or at least postpone all expenditures 
which are not demonstrably justifiable. 

(2) The ideal place for a railroad station is a location im- 
mediately contiguous to the business center of the town. The 
location of the station even one fourth of a mile from this may 
result in a loss of business. Increase this distance to one mile 
and the loss is very serious. Increase it to five miles and the 
loss approaches 100%. 

(3) The cost of the ideal location and the necessary right 
of way may be a very large sum of money for the new enterprise. 
On the other hand the increase in property values and in the 
general prosperity of the town, caused by the railroad itself, 
will so enhance the value of a more convenient location that its 
cost at some future time will generally be extravagant if not 
absolutely prohibitory. The original location is therefore under I 
ordinary conditions a finality. 

(4) To some extent the railroad will cause a movement of i 
the business center toward it, especially in the establishment | 
of new business, factories, etc., but the disadvantages caused i 
to business already established is permanent. 

(5) In any attempt to compute the loss resulting from a 
location at a given distance from the business center it must be 



§ 477. PROMOTION OF RAILROAD PROJECTS. 533. 

recognized that each problem is distinct in itself and that any 
change or growth in the business of the town changes the amount 
of this loss. 

The argument for locating the station at some distance from 
the center of the town may be based on {a) the^cost of right 
of way, thus involving the question of a large initial outlay, 
(&) the cost of very expensive construction {e.g. bridges), 
again involving a large initial outlay, (c) the avoidance of ex- 
cessive grade into and out of the town. It sometimes happens 
that a railroad is following a line which would naturallj^ cause 
it to pass at a considerable elevation above (rarely below) 
the town. In this case there is to be considered not only the 
possible greater initial cost, but the even more important increase 
in operating cost due to the introduction of a very heavy grade. 
The loss of business due to inconvenient location cati only be 
guessed at. Wellington says that at a distance of one mile 
the loss would average 25%, with upper and lower limits of 
10 and 40%, depending on the keenness of the competition 
and other modifying circumstances. For each additional mile 
reduce 25% of the preceding value. While such estimates are 
grossly approximate, yet with the aid of sound judgment they 
are better than nothing and may be used to check gross errors. 

477* General principles which should govern the expenditure 
of money for railroad purposes. It will be shown later that 
the elimination of grade, curvature, and distance have a positive 
money value ; that the reduction of ruling grade is of far greater 
value ; that the creation of facilities for the handling of a large 
traffic is of the highest importance and yet the added cost of 
these improvements is sometimes a large percentage of the 
cost of some roud over which it would be physically possible 
to run trains between the termini. 

The subsequent chapters will be largely devoted to a discussion 
of the value of these details, but the general principles governing 
the expenditure of money for such purposes may be stated as 
follows: 

1. No money should be spent (beyond the unavoidable 
minimum) unless it may be shown that the addition is in itself 
a profitable investment. The additional sum may not wreck 
the enterprise and it may add something to the value of the 
road, but unless it adds more than the improvement costs it is 
not justifiable, - '• ' ^ " ' "• 



534 KAILROAD CONSTRUCTION. § 478. 

2. If it may be positively demonstrated that an improvement 
will be more valuable to the road than its cost, it should certainly 
be made even if the required capital is obtained with difhculty. 
This is all the more necessary if the neglect to do so will per- 
manently hamper the road wdth an operating disadvantage 
which will only grow worse as the traffic increases. 

3, This last principle has two exceptions: (a) the cost of 
the improvement may wreck the whole enterprise and cause 
a total loss to the original investors. For, unless the original 
promoters can build the road and operate it until its stock 
has a market value and the road is beyond immediate danger 
of a receivership, they are apt to lose the most if not all of 
their investment ; (6) an improvement which is very costly 
although unquestionably wise may often be postponed by means 
of a cheap temporary construction. Cases in point are found 
at many of the changes of alinement of the Pennsylvania R. R., 
the N. Y., N. H. & H. R. R., and many others. While some of 
the cases indicate faulty original construction, at many of the 
places the original construction was wise, considering the then 
scanty traffic, and now the improvement is wise considering 
the great traffic. 

478. Study of railroad economics — its nature and limitations. 
The multiplicity of the elements involved in most problems 
in railroad construction preclude the possibility of a solution 
which is demonstrably perfect. Barring out the comparatively 
few cases in this country where it is difficult to obtain any 
practicable location, it may be said that a comparatively low 
order of talent wiU suffice to locate anywhere a railroad over 
which it is physically possible to run trains. It may be very 
badly located for obtaining business, the ruling grades may 
be excessive, the alinement may be very bad, and the road 
may be a hopeless financial failure, and yet trains can be run. 
Among the infinite number of possible locations of the road, 
the engineer must determine the route which will give the best 
railroad property for the least expenditure of money — the 
road whose earning capacity is so great that after paying the 
operating expenses and interest on the bonds the surplus avail- 
able for dividends or improvements is a maximum. 

An unfortunate part of the problem is that even the blunders 
are not always readily apparent nor their magnitude. A defec- 
tive dam or bridge will give way and every one realizes the 



§479. PROMOTION OF RAILROAD PROJECTS. 535 

failure, but a badly located railroad affects chiefly the finances 
oi the enterprise by a series of leaks which are only perceptible 
and demonstrable by an expert, and even he can only say that 
certain changes would probably have a certain financial value. 
479. Outline of the engineer's duties. The engineer must 
realize at the outset the nature and value of the conflicting 
interests which are involved in variable amount in each possi- 
ble route. 

(a) The maximum of business must be obtained, and yet it may 
happen that some of the business may only be obtained by an 
extravagant expenditure in building the line or by building a 
line very expensive to operate. 

(b) The ruling grades should be kept low, and yet this may 
require a sacrifice in business obtained and also may cost more 
than it is worth. 

(c) The alinement should be made as favorable as possible; 
favorable alinement reduces the future operating expenses, 
but it may require a very large immediate outlay. 

(d) The total cost must be kept within the amount at which 
the earnings will make it a profitable investment. 

(e) The road must be completed and operated until the 
"normal" traffic is obtained and the road is self-supporting 
without exhausting the capital obtainable by the projectors ; 
for no matter how valuable the property may ultimately be- 
come, the projectors will lose nearly, if not quite, all they have 
invested if they lose control of the enterprise before it becomes 
a paying investment. 

Each new route suggested makes a new combination of the 
above conflicting elements. The engineer must select a route 
by first eliminating all lines which are manifestly impracticable 
and then gradually narrowing the choice to the best routes 
whose advantages are so nearly equal that a closer detailed 
comparison is necessary. 

The ruling grade and the details of alinement have a large 
influence on the operating expenses. A large part of this course 
of instruction therefore consists of a study of operating expenses 
under average normal conditions, and then a study of the effect 
on operating expenses of given changes in the alinement. 



CHAPTER XX. 

OPERATING EXPENSES.* 

480. Distribution of gross -revenue. When a railroad com- 
prises but one single property, owned and operated by itself, 
the distribution of the gross revenue is a comparatively simple 
matter. The operating expenses then absorb about two thirds 
of the gross revenue; the fixed charges (chiefly the interest on 
the bonds) require about 25 or 30% more, leaving perhaps 3 
to 8% (more or less) available for dividends. The report on 
the Fitchburg R. R. for 1898 shows the following: 

Operating expenses $5,083,571 69. 1% 

Fixed charges 1,567,640 21 . 3% 

Available for dividends, surplus, or per- 
manent improvements 708,259 9.6% 

Total revenue $7,359,470 100.0% 

But the financial statements of a large majority of the railroad 
corporations are by no means so simple. The great consolida- 
tions and reorganizations of recent years . have been effected 
by an exceedingly complicated system of leases and sub-leases, 
purchases, "mergers," etc., whose forms are various. Railroads 
in their corporate capacity frequently own stocks and bonds 
of other corporations (railroad properties and otherwise) and 
receive, as part of their income, the dividends (or bond interest) 
from the investments. 

' The Interstate Commerce Commission annually makes a 
report of the income and profit-and-loss account of all the rail- 
roads of the United States, considered as one system. For 
example, the statement for the year 1912 includes the following 
items. Operating revenues from rail operations 12,842,695,382; 
operating expenses due to rail operations $1,972,415,776, which 
is 69.4%. Interest on funded debt used up 13.9% of the rev- 

* The operating expenses of railroads have been utterly abnormal during 
and since the Great War. The figures of this chapter are not now (1921) 
applicable to present conditions, but corresponding figures, revised to date, 
would not be typical. The chapter therefore stands untouched until new 
figures, representing normal conditions, are available. 

536 



481. 



OPERATING EXPENSES. 



587 



eniles, and taxes 4.2%. There were other miscellaneous incomes 
and expenditures which caused a net loss of another 2.0% 
of revenue, leaving 10.5% or $299,361,208 which were issued 
as dividends. These dividends are about 3.4% of the outstand- 
ing stock. The percentage to the amount of money abtually 
paid for the stock is unknown and unknowable. 

481. Operating expenses per train-mile. The uniformity in 
the average operating expenses per train mile for light-traffic 
and heavy-traffic roads and for long and short roads is very 
remarkable. This is illustrated by a comparison of figures for 
ten heavy traffic roads and ten small roads selected at random, 
except that each had a mileage of less than 100 miles, 

OPERATING EXPENSES PER TRAIN-MILE ON LARGE AND SMAlIj ' 
ROADS (1904 AND 1910). 



Mileage. 



1904. 



1910. 



Operating 

expenses per 

train-mile. 



1904. 



1910. 



Ratio expenses 

to earnings 

per cent. 



1904. 



1910. 



Whole United States . 



220,112 



240,439 



1.314 



1.489 



67.79 



66.29 



Canadian Pacific 

C, B. & Q 

Chicago & Northwestern. 

Southern Railway 

C:, R. I. &P 

Northern Pacific 

A., T. & S. F 

Great Northern 

Illinois Central 

Atlantic Coast Line 



8,332 
8,326 
7,412 
7,197 
6,761 
5,619 
5,031 
4,489 
4,374 
4,229 



10,271 
9,040 
7,629 
7,050 
7,396 
6,189 
7,460 
7,147 
4,551 
4,491 



1 . 320 
1.313 
1.136 
1.048 
1.199 
1.392 
1.305 
1.464 
1.107 
0.984 



1.504 
1.710 
1.306 
1.234 
1.344 
1.824 
1.626 
1 . 808 
1.409 
1.213 



68.72 
64.35 
66.61 
70.30 
72.90 
52.26 
60.05 
49.72 
70 . 02 
58.95 



65.41 
71.71 
70.31 
67.43 
73.07 
61.71 
64.33 
60.53 
74.84 
62.44 



Average of ten , 



1 . 227 



1.498 



63 . 39 



67.18 



Montpelier & Wells River. . . 

Somerset Railway Co.* 

Huntingdon & Broadtop 
Mountain . . 

Lehigh & New England 

Ligonier Valley 

Newburgh, Dutchess & Con- 
necticut t 

Susquehanna & New York . . 

Detroit & Charlevoix 

Harriman & Northeastern * 

Galveston, Houston & Hen- 
derson 

Average of ten (or nine) . . 



44 

42 

66 
96 
11 

59 
55 
51 
20 

50 



50 
94 

70 

170 

16 



80 
51 
20 

50 



1.169 
0.802 

. 950 
0.793 
1.427 

0.922 
1.368 
1.424 
2.162 

1.556 



1.430 
1.314 

2.052 
2.045 
1.480 



1.028 
1.010 
1.733 

1.759 



80.73 
59.37 

52.10 
69.80 
69.33 

85.09 
78.47 
67.52 
79.26 

47.27 



75.08 
76. e5 

96.40 
62.84 
49.15 



77.81 
99.63 
63. 7(/ 

70.37 



1.257 



1.539 



68.89 



74. 6 1 



*' Subsidiary road' since 1904. 

t Merged since 1904; separate figures not available. 



538 



KAILROAD CONSTRtJCTION. 



§482. 



The fluctuations of the average cost per train-mile for several 
years past may be noted from the following tabular form: 

AVERAGE COST PER TRAIN-MILE (fOR WHOLE U. S.) IN CENTS. 



Year. 


Cents. ■ 

1 


Year. 


Cents. 


Year. 


Cents. 


Year. 


Cents. 


1890 
1891 
1892 
1893 
1894 
1895 


96 . 006 
95.707 
96 . 580 
97.272 
93.478 
91.829 


1896 
1897 
1898 
1899 
1900 
1901 


93 . 838 
92.918 
95.635 
98.390 
107.288 
112.292 


1902 
1903 
1904 
1905 
1906 
1907 


117.960 
126.604 
131.375 
132.140 
137.060 
146.993 


1908 
1909 
1910 
1911 
1912 


147.340 
143.370 
148.865 
154.338 
159.077 



The enforced economies after the panic of 1893 are well 
shown. The reduction generally took the form of a lowering 
of the standards of maintenance of way and of maintenance of 
equipment. The marked advance since 1895 is partly due to 
the necessity for restoring the roads to proper conditions, replen- 
ishing worn-out equipment and providing additional equip- 
ment to handle the greatly increased volume of business. The 
recent advance is chiefly due to the increase in wages and the 
generally increased cost of supplies. 

It may be noted from the I. C. C. reports that the cases where 
the operating expenses per train-mile and the ratio of expenses 
to earnings vary very greatly from the average are almost 
invariably those of the very small roads or of "junction roads" 
where the operating conditions are abnormal. For example, one 
little road, with a total length of 13 miles and total annual opera- 
ting expenses of $5342, spent but 22 Jc. per train-mile, which pre- 
cisely exhausted its earnings. This precise equality of earnings 
and expenses suggests jugglery in the bookkeeping. As another 
abnormal case, a road 44 miles long spent $3.81 per train-mile, 
which was nearly fourteen times its earnings. In another case a 
road 13 miles long earned $7.76 per train-mile and spent $6.03 
(78%) on operating expenses, but the fixed charges were abnor- 
mal and the earnings were less than half the sum of the operating 
expenses and fixed charges. The normal case, even for the 
small road, is that the cost'per train-mile and the ratio of operat- 
ing expenses to earnings will agree fairly well with the average, 
and when there is a marked difference it is generally due to 
some abnormal conditions of expenses or of earning capacity. 

482. Reasons for uniformity in expenses per train-mile. 
The chief reason is that, although on the heavy-traffic road 
everything is kept up on a finer scale, better roadbed, heavier 



§ 483. OPERATING EXPENSES. 539 

I 

rails, better rolling stock, more employees, better buildings, 

stations, and terminals, etc., yet the number of trains is so much 

greater that the divisor is just enough larger to make the average 

cost about constant. This is but a general statement of a fact 

which will be discussed in detail under the different items of 

expense. 

483. Detailed classification of expenses with ratios to the 
total expense. The Interstate Commerce Commission now 
publishes each year a classification with detailed summation 
for the cost of each item. These summations are made up 
from reports furnished by railroads which have (in the report's 
recently made) represented over 99% of the total traffic han- 
dled. In the annexed tabular form (Table XLI) are shown the 
percentages which each item bears to the total. The railroads 
have been divided into two classes, "large" and "small," as 
indicated below. Large roads report on 116 items which are 
combined and condensed with 44 items for small roads. 

"Large roads" are those with mileage greater than 250 miles, 
or those with operating revenues greater than $1,000,000. 
Roads subsidiary to "large roads" are also included in this 
class. 

"Small roads" are those with mileage less than 250 miles 
and also with operating revenues less than $1,000,000. 

484. Amounts and percentages of the various items. The 
I. C. C. report for the year ending June 30, 1909, was the first 
to include the distribution of expenses according to the present 
classification. The items as given are reliable and may be utihzed, 
as far as any such computations are to be depended on, in 
estimating future expenses. The chief purpose of this dis- 
cussion is to point out those elements of the cost of operating 
trains which may be affected by such changes of location as an 
engineer is able to make. There are some items of expense with 
which the engineer has not the slightest concern, nor will they 
be altered by any change in alinement or constructive detail 
which he may make. In the following discussion such items 
will be passed over with a brief discussion of the sub-items 
included. 

MAINTENANCE OP WAY AND STRUCTURES. 

485. Items 2 to 5. Track material. The relative cost of 
ballast, ties, rails and other track material, as shown by com- 



540 



RAILBOAD CONSTRUCTION. 



§485. 



TABLE XLI. — 'ANALYSIS OF OPERATING EXPENSES OF ALL ''lARGE"* 
RAILROADS IN THE UNITED STATES FOR YEAR ENDING JUNE 30, 
1912, SHOWING PERCENTAGE OP EACH ITEM TO TOTAL AND COST 
IN CENTS PER TRAIN-MILE, 



Item 
No. 



Account. 



Total 

Amount 

(thousands) 



Per cent 

of total 

Expenses 



Cents per 
Train- 
Mile. 



1 
2 
3 

,4 
5 
6 
7 
8 
9 
10-12 

13-15 

16,17 

18 

19 
20,21 

22,23 



Maintenance of Way and Structures. 

Superintendence 

Ballast 

Ties 

Rails 

Other track material 

Roadway and track 

Removal of snow, sand, and ice . . 

Tunnels 

Bridges, trestles, and culverts. ... 

Crossings, all; fences; snow struc 
tures 

Signals, telegraph, electrical power 
transmission 

Buildings, grounds, docks, wharves 

Roadway tools and supplies 

Injuries to persons 

Stationery, printing and other ex- 
penses 

Joint tracks, etc. (net balance) .... 

Maintenance of Equipment. 

Superintendence. 

Repairs, renewals and depreciation: 
Locomotives, steam and electric . 

Cars, passenger 

Cars, freight 

. Equipment, electrical, car. 

Equipment, floating 

Equipment, work 

Equipment, shop (machinery and 

tools) 

Equipment, power plant 

Injuries to persons 

Stationery, printing and other ex- 
penses 

Joint equipment, at terminals (net 
balance) 

Traffic Expenses. 
Agencies; advertising; fast freight 
lines; etc 



$18,789 

7,157 

55,463 

16,438 

17,346 

129,397 

6,920 

1,141 

27,712 

8,066 

13,681 

35,389 

4,480 

1,989 

1,038 
3,463 



0.990 

0.377 

2.921 

.866 

.914 

6.815 

.364 

.060 

1.460 

,425 

.720 

1.864 

.236 

.105 

.054 
.182 



$348,471 



18.353 



1.58 

.60 

4.65 

1.38 

1.45 

10.84 

.58 

.10 

2.32 

.68 

1.14 

2.96 

.38 

.17 

.09 

.29 



29.20 



24 

25-30 
31-33 
34-36 
37-39 
40-42 
43-45 
46 

47 
48 

49,50 

51, 62 



63-60 



$13,175 

175,589 

38,968 

183,968 

318 

1,333 

6,128 

10,418 

268 
1,818 

4,036 

676 



.694 

9.263 

2.052 

9.690 

.017 

.071 

.322 

.548 
.014 
.096 

.213 

.036 



1.10 

14.74 
3.26 

15.41 
.03 
.11 
.51 

.87 
.02 
.15 

.34 

.06 



$436,995 



23.016 



36.61 



$59,047 



3.110 



4.95 



* The " large " roads here reported represent 88% of the total mileage.' 



paring either the gross amounts or the percentages in Table XLI, 
is suggestive and instructive. The fact that ties cost con- 
siderably more than all other track material combined shows 



§ 485. 



OPERATING EXPENSES. 



541 



TABLE XLi. (ConHnued). — analysis of operating expenses 

OF ALL " LARGE " RAILROADS IN THE UNITED STATES FOR 
YEAR ENDING JUNE 30, 1912, SHOWING PERCENTAGE OF 
EACH ITEM TO TOTAL AND COST IN CENTS PER TRAIN-MILE. 



Item 
No. 



J 61, 62 

■■ 63 
64-66 

67-70 

71-76 



77,78 
104, 105 

79, 80 
81 

82 

83 

84,85 

86, 87 

88 

89 

90-92 

93 

94-98 

99-103 



Account. 



Transportation Expenses. 

Superintendence and train dis- 
patching 

Station employees . 

Weighing; car service associa- 
tion; coal and ore docks 

Yards (wages, expenses, sup- 
plies) 

Yard locomotives (enginemen, 
fuel, water, lubricants, sup- 
plies) 

Operating joint tracks, ter- 
minals, yards, and facilities 
(net balance) 

Motormen and road enginemen. 

Road locomotives, engine-hous^ 
expenses 

Road locomotives, fuel 

Road locomotives, water 

Road locomotives, lubricants 
and other supplies 

Operating power plants, pur- 
chased power 

Road trainmen 

Train supplies and expenses . . . 

Interlockers, signals, flagmen, 
draw-bridges 

Clearing wrecks 

Telegraph, floating equipment, 
stationery, miscellaneous. . . . 

Loss and damage to property, 
personal injuries 



Total 

Amount 

(thousands) 



$40,743 
133,877 

15,949 

76,069 

74,370 



10,430 
120,966 

33,951 

194,142 

12,482 

7,430 

1,797 

128,339 

34,462 

17,831 
5,167 

20,009 

56,838 



;.4,852 



Per cent 

of total 

expenses. 



2.146 
7.051 

.839 

4.007 

3.917 



.550 
6.371 

1.788 

10.225 

.657 

.392 

.095 
6.759 
1.815 

.939 
.272 

1.054 

2.994 



51.871 



Cents per 
train- 
mile. 



3.41 

11.22 

1.33 
6.37 

6.23 



.88 
10.14 

2.84 

16.27 

1.04 

.62 

.15 
10.75 

2.89 

1.49 
.43 

1.68 

4.76 



82.51 



106-116 



General Expenses. 
Salaries of general oflB.cers, 
clerks, etc.; law, insurance, 
pensions, miscellaneous 



69,297 



3.650 



5.81 



Total operating expenses. . . 



$1,898,662 



100 . 000 



159.08 



the importance of any possible saving in tie renewals. It 13 
also significant that the relative importance of ties has increased 
in the last few years, and that the relative increase has not been 
due to a reduction in the cost of other track material. Appar- 
ently the lengthening of the average life of ties, due to pre- 
servative processes, the use of tie-plates, and greater care to 
avoid the premature withdrawal from the track of ties which 



542 



KAILROAD CONSTRUCTION. 



§486. 



are still serviceable, has not kept pace with the increase in thej 
average cost per tie. The cost of rails has advanced because I 
of (a) the very general adoption of heavier rails; (6) the almost 
universal substitution of more expensive open-hearth steel forj 
Bessemer, on account of greater reliability and durability, andj 
(c) the increase in cost of all steel products. 

486. Item 6. Roadway and track. This item is three-eighths 
of the total cost of maintenance of way and structures. It 
consists chiefly of the wages of trackmen. There has been an 
almost steady increase in the daily wages of section foremen 
and other trackmen since 1900, as shown below: 





1900 


1901 


1902 


1903 


1904 


190(5 


1906 


Section foremen 

Other trackmen 

No. of trackmen per 
100 miles 


1.68 
1.22 

118 


1.71 
1.23 

122 


1.72 
1.25 

140 


1.78 
1.31 

147 


1.78 
1.33 

136 


1.79 
1.32 

143 


1.80 
1.36 

155 









1907 


1908 


1909 


1910 


1911 


1912 


Section foremen 

Other trackmen 

No. of trackmen per 
100 miles 


1.90 . 
1.46 

162 


1.95 
1.45 

130 


1.96 
1.38 

136 


1.99 
1.47 

157 


2.07 
1.50 

147 


2.09 
1.50 

143 







The average number of section foremen per 100 miles of line 
has remained almost constant at 18. Although there have been 
fluctuations in the number of " other trackmen " required per 
100 miles of line, there has been in general a very substantial 
increase. These two causes combined (increased number and 
increased wages) have had a great influence in producing the 
regular and steady increase in the average cost of a train-mile, 
as shown in § 481. 

487. Items 8 to 15. Maintenance of track structures. As a 
matter of economics, the locating engineer has little or no concern 
with the cost of maintaining track structures. If he is com- 
paring two proposed routes it would be seldom that they would 
be so different that he would be justified in attempting to compute 
a train-mile difference in cost of operation, based on differences 
in these items. Of course, one proposed line might call for one 
or more tunnels which the alternate line might not have, and 
the annual cost of maintaining the tunnels would increase the 
cost of operation. Such a case would justify special considera- 



§488. OPERATING EXPENSES. 543 

tion. So far as the maintenance of small bridges and culverts 
are concerned it would usually be sufficiently accurate to consider 
that a proposed change of line, involving perhaps several miles 
of road, would require substantially the same number of bridges 
and culverts, and therefore that the cost of maintaining them 
would be the same by either line. The error involved in such 
an assumption would usuall}^ be insignificant, unless there was 
a very large and material difference in the two lines in this 
respect. Under such conditions special computations should be 
made. The items total less than 3% for small roads and still 
less for large roads. 

MAINTENANCE OF EQUIPMENT. 

488. Items 25 to 27. Repairs, renewals and depreciation of 
steam and electric locomotives. The item is of interest to the 
locating engineer because he must appreciate the effect on 
locomotive repairs and renewals of an addition to distance. 
A large part of the repairs of locomotives are due to the wear 
of wheels, which is largely caused by curvature. Therefore the 
value of any reduction of curvature is a matter of importance, 
and this will be considered in Chapter XXII. A considerable 
portion of the deterioration of a locomotive is due to grade, and 
the economic advantages of reductions of grade will be con- 
sidered in Chapter XXIII. 

This item includes the expenses of work whose effect is sup- 
posed to last for an indefinite period. It dof.s not include the 
expense of cleaning out boilers, packing cylinders, etc., which 
occurs regularly and which is charged to items 72 or 81. It 
does include all current repairs, general overhauHng, and even 
the replacement of old and worn-out locomotives by new ones 
to the extent of keeping up the original standard and number. 
Of course additions beyond this should be considered as so 
much increase in the original capital investment. As a loco- 
motive becomes older the annual repair charge becomes a larger 
percentage on the first cost, and it may become as much as one- 
fourth and even one-third of the first cost. When a locomotive 
is in this condition it is usually consigned to the scrap-pile';, the 
annual cost for maintenance becomes too large an item for its 
annual mileage. The effect on expenses of increasing the weight of 
engines is too complicated a problem to be solved accurately, but 



544 RAILEOAn CONS^RtrGTlon. § 489. 

certain elements of it may be readily computed. While the cost 
of repairs is greater for the heavier engines, the increase is only 
about one-half as fast as the increase in weight — some of the 
subitems not being increased at all. 



TRANSPORTATION. 

489. Items 71 to 76. Yard-engine expenses. By com- 
paring these items with the corresponding items (80 to 85) for 
road engines, it may be seen that the total expenses assignable to' 
yard engines are about 20% of those of road engines-; the relative 
fuel charge for 1912 was 15.6%. The number of switching 
locomotives in the United States in 1912 was 9529 or 15.3% of 
the total number, 62,262. The relative charge for wages of engine- 
men was 26.2%. This higher proportionate charge is proba;bly 
due to the fact that the wages for yard enginemen must neces- 
sarily be on a per diem basis, but the wages of road enginemen 
are generally on a mileage basis, as explained later. On the 
other hand the mileage of a yard engine is usually comparatively 
low, and the coal consumed will be correspondingly, although 
not proportionately, low. It must also be remembered that 
these figures are exclusive of the work and equipment of switching 
and terminal companies. 

490. Item 80. Road enginemen. This item requires 6% of 
the total operating expenses. The enginemen are usually paid 
on a mileage basis, or by the trip, except on very small railroads. 
On very short roads, where a train crew may make two, three, 
or even four complete round trips per day, they may readily be 
paid by the day, so many round trips being considered as a 
day's work, but on roads of great length, where all trains, and 
especially freight-trains, are run day and night, weekday and 
Sunday, all trainmen are necessarily paid by the trip. The pay 
for a trip is figured on a mileage basis except that a trip is usually 
considered to have a minimum length of 100 miles or 10 hours of 
time. Eight hours was fixed as standard by the " Adamson " 
law, in 1916. All extra time is called " overtime " and is paid 
for at an extra rate. The basis of train wages is too complicated 
for any brief discussion. Even the basis is Constantly changing, 
the only uniform feature being a steady increase. 

The increase in the average wages paid to enginemen and 
firemen since 1900 is plainly shown by the following figures: 



f#l. 



OPS]EA*lNG} fiX^UNSES. 



545 



INCREASE IN DAILY WAGES; FROM 1900 TO 1912. 





1900 


1901 


1902 


1903 


1904 


1905 


1906 


Enginemen 


$ 
3.75 
2.14 


■ $ 

3.78 

2.16 


3.84 
2.30 


- $ 
4.01 
2.28 


$ 
4.10 
2.35 


4.12 
2.38' 


$ 
4.12 


li"irenien. 


2 42 









1907 


' 1908 


1909 


1910 


1911 


i&r2 


Enginemen 


4.30 
2.54 


$ 
4.45 
2.64 


4.44 
2.67 


$ 
4.55 
2.74 


$ ■ 

' 4.79 

2.94 


5.6)0 


Firemen 


3.02 



49'i. It€tti 82. Fuel for road locomotives. This item in* 
6ludteg every siibitetii of the entire cost of the fuel uiitil it i« 
placed in the engine-tender. The cost therefore includes not only 
Ihe first cost at the point of delivery to tho road, but also the 
expense of hauling it over the road from the point of delivery 
to the various coaling-stations and the cost of operating the 
coal-pockets from which it is loaded on to the tenders. Even 
though the cost may be fairly regular for any one road, the cost 
for different roads is exceedingly variable. There has been an 
almost steady increase in the percentage of the cost of this item 
per train-mile since 1897. Items 73 and 82 amounted to nearly 
12% of the total operating expenses in 1912, and required an 
actu^rl expenditure of nearly $225,000,000. It is the largest 
item in the whole cost of railroad operation. Although some 
roads, which traverse coal-regions and perhaps actually own the 
coal-mines,- are aible to obtain their coal for a cost which may be 
eharged up as $1 per ton or less, there are many roads whi<^h 
are far removed from coal-fields which have to pay $3 or S4 
per ton, on account of the excessive distance over which the coal 
must be hauled. Unfortunately the figures published by th'e 
Interstate Commerce Commission do not show the variations 
in the percentage of this item in different localities. A sur- 
prisingly large percentage of the fuel consumed is not utilized in 
drawing a train along the road. A portion of this percentage is 
used in firing-up. A portion is wasted When the engine is stand- 
ing still, which is- a considerable proportion of the whole time. 
The policy of banking fires instead of drawing them reduces the 
injury resulting from great fluctuations in temperature, but in a 
general way we may say that there is but little, if any, saving in 
fuel by banking the fires, and therefore we may consider tha)fe 



546 RAILROAD CONSTRUCTION. 



§491. j 






almost a i&re-box full of coal is wasted whether the fires arei 
banked or drawn. As given in § 464, the fuel used by a loco 
motive in firing-up may be estimated as 510 lbs. per 1000 square 
feet of heating surface, based on using 12000 B.t.u. coal. But 
even the amount of coal required to produce the required steam- 
pressure in the boiler from cold water does not represent the 
total loss. The train-dispatcher, in his anxiety that engines 
shall be ready when needed, will sometimes order out the loco- 
motives which remain somewhere in the yard, perhaps exposed 
to cold weather, and blow off steam for several hours before they 
make an actual start. This loss has been estimated as 120 lbs. 
per hour per 1000 square feet of heating surface, but it would 
evidently be far greater on a windy winter day than on a calm 
summer day. A freight-train, especially on a single-track road, 
will usually spend several hours during the day on sidings, and 
when a single-track road is being run to the limit of its capacity, 
or when the management is not good, the time will be stiU greater. 
It is estimated that the amount lost through a 2|-inch safety- 
valve in one minute would represent the consumption of 15 
pounds of coal, which would be sufficient to haul .100 tons on a 
mil6 of track with easy grades. Again we see that the amount 
thus lost is exceedingly variable and almost non-computable, 
although as a rough estimate the amount has been placed 
at from 3 to 6% of the total. Another very large subitem 
of loss of useful energy is that occasioned by stopping and 
starting. A train running 30 miles per hour has enough kinetic 
energy to move it on a straight level track for more than two 
miles. Therefore, every time a train running at 30 miles per 
hour is stopped, enough energy is consumed by the brakes to 
run it about two miles. There is a double loss^ not only due 
to the fact of the loss of energy, but also because the power of 
the locomotive has been consumed in operating the brakes. 
When the train is again started, this kinetic enej-gy must be 
restored to the train in addition to the ordinary resistances which 
are even greater, on account of the greater resistance at very 
low velocities. Of course, the proportion of fuel thus con- 
sumed depends on the frequency of the stops. It was demon- 
strated by some tests on the Manhattan Elevated Road in New 
York City, where the stops average one in every three-eighths 
of a mile, that this cause alone would account for the consump- 
tion of nearly three-fourths of the fuel. On ordinary railroads 



§492. 



OPEKATING EXPENSES. 



547 



the proportion, of course, will not be nearly so great, but there 
is reason to believe that 10 to 20% is not excessive as an 
average figure. 

492. Item 88. Road trainmen. This item includes the wages 
of conductors and " other trainmen." As in the case of all 
other employees, the average daily wages have advanced since 
1900 as shown below: 



.AVERAGE DAILY WAGES OF CONDUCTORS AND OTHER TRAINMEN, 

1900 TO 1912. 



Conductors. . . . 
Other trainmen . 



1900 



3.17 
1.96 



1901 



$ 
3.17 
2.00 



1902 



3.21 
2.04 



1903 



3.38 
2.17 



1904 



3.50 
2.27 



1905 



3.50 
2.31 



1906 



3.51 
2.35 





1907 


1908 


1909 


1910 


1911 


1912 


Conductors 

Other trainmen 


$ 
3.69 
2.54 


$ 
3.81 
2.60 


$ 
3.81 
2.59 


$ 
3.91 
2.69 


4.16 

2.88 


$ 
4.29 
2.96 



These figures are of vital importance from an economic stand- 
point, since they show a constant tendency to increase and thereby 
raise the average cost of a train-mile. And as there is no present 
indication of any limit to this increase, all economic calculations 
which attempt to predict future expenses, even for a few years 
in advance, must allow for these and other increased expenses. 

493. Item 89, Train supplies and expenses. These items, 
which average about 1.8%, include the large list of consumable 
supplies such as lubricating oil, illuminating-oil or gas, ice, fuel 
for heating, cleaning materials, etc., which are used on the cars 
and not on the locomotives. The consumption of some of these 
articles is chiefly a matter of time. In other cases it is a function 
of mileage. The effect of changes which an engineer may make 
on this item will be considered when estimating the effect of the 
changes. 

494. Items 93, 99 to 103. Clearing wrecks, loss, damage and 
injuries to persons and property. These expenses are fortuitous 
and bear no absolute relation either to the number of miles of 
road or the number of train-miles. While they depend largely 
on the standards of discipline on the road, even the best of roads 
have to pay gome small proportion of their earnings to these 



548 RAILROAD CONSTRUCTION. § 495^ 

items. While we might expect that a road with heavy trafR 
would have a larger proportion of train accidents than a road 
light traffic, it is usually true that on the heavy-traffic roads th 
precautions taken are such that they are usually freer from acci 
dents than the light-traffic roads. During recent years therdl 
has been a very perceptible increase in the percentages of these 
items, particularly in the compensations paid for ''injuries to 
persons." The increase in this item coincides with the increase 
already noted in the number of passengers killed during recent 
years. The possible relation between curvature and accidents 
is discussed in § 507, but otherwise the locating engineer has 
no concern with these items. 

495. Items 104, 105. Operating joint tracks and facilities, 
Dr. and Cr. A large part of these debit and credit charges 
are those for car per diem and mileage charges. This is a charge 
paid by one road to another for the use of cars, which are chiefly 
freight-cars. To save the rehandling of freight at junctions, 
the poKcy of running freight-cars from one road to another is 
very extensively adopted. Since the foreign road receives its 
mileage proportion of the freight charge, it justly pays to the road 
owning the car at a rate which is supposed to represent the 
value of the use of the freight-car for the number of miles 
traveled. The foreign road then loads up the freight-car with 
freight consigned to some point on the home road and sends it 
back, paying mileage for the distance traveled on the foreign 
road, a proportional freight charge having been received for that 
service. All of these movements of freight-cars are reported 
to a car association, which, by a clearing-house arrangement, 
settles the debit and credit accounts of the various roads with 
each other. Such is the simple theory. In practice the cars are 
not sent back to the home road at once, but wander off according 
to the local demand. As long as a strict account is kept of the 
movements of every car, and as long as the home road is paid 
the charge which really covers the value of lost service, no harm 
is done to the home road, except that sometimes, when business 
has suddenly increased, the home, road cannot get enough cars 
to handle its own business. The value of the car is then abnor- 
mally above its ordinary value, and the home road suffers for 
lack of the rolling stock which belongs to it. Formerly such 
charges were paid strictly according to the mileage. This 
developed the intolerable condition that loaded cars would be 



§495. OPERATING EXPENSES. 549 

run onto a siding and left there for several days, simply because 
it was not convenient to the consignee to unload the car imme- 
diately. On the mileage basis the car would be earning nothing, 
and, since the road on which the car then was had no particular 
interest in the car, the car was allowed to stand to suit the con- 
venience of the consignee. To correct this evil a system of per 
diem charges has been developed, so that a railroad has to pay a 
per diem charge for every foraign car on its lines. To reduce this 
charge as much as possible the railroads compel consignees, 
under penalty of heavy demurrage charges, to unload cars 
promptly. The running of freight-cars on foreign lines is now 
settled almost exclusively on the per diem basis, but the running 
of passenger-cars over other Unes, as is done on account of the 
advantages of through-car service, as well as the running of 
Pullmans and other special cars, is still paid for on the mileage 
basis. To the extent to which this charge is settled on the mile- 
age basis, any change in distance which the engineer may be able 
to effect in the length of the road will have its influence on this 
item, but when the freight-car business, which comprises by far 
1 the larger part of the running of cars over f oreign'lines, is settled 
on the per diem basis no changes in alinement which the engineer 
may make will affect the item appreciably.- 

Switching Charges. Where two or more railroads intersect 
there will be a considerable amount of shifting of cars, chiefly 
freight-cars, from one road to the other. This shifting at any 
one junction may be done entirely by the engines of one road 
or perhaps by those of both roads. A portion of the expense 
of this work is charged up against the other road by the road 
which does the work. The total amount of this work is care- 
fully accounted for by a clearing-house arrangement, and tjie 
balance is charged up against the road which has done the least 
work. The item is very small, is fairly uniform year by year, 
and is seldom, if ever, affected by changes of alinement. 

Other Items. All of the remaining items, as stated in Table 
XLI, are of no concern to the locating engineer. They are either 
general expenses, such as the salaries of general officers, insurance 
or law expenses, or are special items, such as advertising or the 
operation of marine equipment which will not be changed by 
any variations in distance, curvature, or grades which a locating 
engineer may make. There is therefore no need for their further 
discussion here. 



CHAPTER XXL 

DISTANCE. 

; 496. Relation of distance to rates and expenses. Rates i 
are usually based on distance traveled, on the apparent 
hypotheses that each additional mile of distance adds its pro- 
portional amount not only to the service rendered but also to 
the expense of rendering it. Neither hypothesis is true. The 
value of the service of transporting a passenger or a ton of , 
freight from A to B is a more or less uncertain gross amount 
depending on the necessities of the case and independent of 
the exact distance. Except for that very small part of passen- 
ger traffic which is undertaken for the mere pleasure of traveling, 
the general object to be attained in either passenger or freight 
traffic is the transportation from A to B, however it is attained. 
A mile greater distance does not improve the service rendered ; 
in fact, it consumes valuable time of the passengers and perhaps 
deteriorates the freight. From the standpoint of service ren- 
dered, the railroad which adopts a more costly construction and 
thereby saves a mile or more in the route between two places 
is thereby fairly entitled to additional compensation rather 
than have it cut down as it would be by a strict mileage rate. 
The actual value of the service rendered may therefore vary 
from an insignificant amount which is less than any reasonable 
charge (which therefore discourages such traffic) and its value 
in cases of necessity — a value which can hardly be measured in 
money. If the passenger charge between New York and Phila- 
delphia were raised to $5, $10, or even $20, there would still be 
some passengers who would pay it and go, because to them 
it would be worth $5, $10, or $20, or even more. Therefore, 
when they pay $2.25 they are not paying what the service is 
worth to them. The service rendered cannot therefore be 
made a measure of the charge, nor is the service rendered pro- 
portional to the miles of distance. 
The idea that the eost of transportation is proportional to 

550 



} 497. DISTANCE. 551 

the distance is mucli more prevalent and is in some respects 
more justifiable, but it is still far from true. This is especially 
true of passenger service. The extra cost of transporting a single 
passenger is but little more than the cost of printing his ticket. 
Once aboard the train, it makes but little difference to the rail- 
road whether he travels one mile or a hundred. Of course there 
are certain very large expenses due to the passenger traffic 
which must be paid for by a tariff which is rightfully demanded, 
but such expenses have but little relation to the cost of an 
additional mile or so of distance inserted between stations. 
The same is true to a slightly less degree of the freight traffic. 
As shown later, the items of expense in the total cost of a train- 
mile, which are directly affected by a small increase in distance, 
are but a small proportion of the total cost. 

497. The conditions other than distance that affect the cost; 
reasons why rates are usually based on distance. Curvature 
and minor grades have a considerable influence on the cost of 
transportation, as will be shown in detail in succeeding chap- 
ters, but they are never considered in making rates. Ruling 
grades have a very large influence on the cost, but they are like- 

'wise disregarded in making rates. An accurate measure of 
the effect of these elements is difficult and complicated and 
I would not be appreciated by the general public. Mere dis- 
tance is easily calculated; the public is satisfied with such 
a method of calculation; and the railroads therefore adopt a 
tariff which pays expenses and profits even though the charges 
are not in accordance with the expenses or the service rendered 

EFFECT OF DISTANCE ON RECEIPTS. 

498. Classification of traffic. There are various methods 
of classifying traffic, according to the use it is intended to make 
of the classification. The method here adopted will have ref- 
erence to its competitive or non-competitive character and also 
to the method of division of the receipts on through traffic. 
Traffic may be classified first as " through " and " local "— 
through traffic being that traveling over two (or more) lines, 
no matter how short or non-competitive it may be; " local " 
traffic is that confined entirely to one road. A fivefold classifica- 
tion is however necessary — which is : 

1^.^ Ai Non-competitive local — on one road with no choice of route* 



56l RA1LR0A15 COlsrSTEUCTlON. §499. 

B. Non-competitive through — on two (or more) roads, but 
with no choice. 

C. Competitive local — a choice of two (or more) routes, but 
the entire haul may be made on the home road. 

D. Competitive through — ditect competition between two 
err more routes each passing over two or more lines. 

E. Semi-competitive through — a non-competitive haul on the 
home road aiid a competitive haul on foreign roads. 

There ate other possible combinations, but they all reduce td 
one of the above f ortns so far as their essential effect is concerned. 
499. Method of divisiofn of through rates between the 
foads run over. Through rates are divided between the 
roads run ovet iti proportion to the mileage. There laaf 
be terminal charges and possibly other more or less arbitrary 
deductions to be taken from the total amount received, but 
when the final division is made the remainder is divided accord- 
ing to the mileage. On account of this method of division and 
also because non-coinpetitive rates are always fixed according 
to the distance, there results the unusual feature that, unlike 
curvature and grade, thete is a compensating advantage ill 
increased distance, which applies to all the above kinds oi 
traffic except one (competitive local), and that the compensation 
is Sometimes sufficient to make the added distance an actual 
source of pi-ofi't. It has been estimated that the cost of h^hn^ 
a train an additioftal mile is only 33 to 49% of the average cos#. 
Therefore in all non-competitive business (local or through) 
where the rate is according to the distance, there is an actu^ 
profit in all such added distance. In competitive local busi- 
ness, in which the rate is fixed by competition and has practically 
ti& i'eMtion to distance, aiiy additional distance is dea<i loss. In 
eoiiipetitive through business the profit or loss depends on the 
distances involved. This may best be demonstrated by exam- 
ples. 

§60. Effect of a change in the length of the home road oh 
its jfeceipt^ from trough competitive traffic. Suppose the 
home road is 100 miles long and the foreign road- is 150 miles 

long. Then the home road will receive = 40% of the 

through rate. 

.Suppose th6 home road is lengthened 5 miles; then it will 



§ 501. DISTANCE. 55^ 

105 
receive--— —— = 41.176% of the through rate. The traffic 
105 + 150 

being competitive, the rate will be a fixed quantity regardless 
of this change of distance. By the first plan the rate received 
is 0.4% per mile; adding 5 miles, the rate for the original 100 
miles may be considered the same as before; and that the addi- 
tional 5 miles receive 1.176%, or 0.235% per mile. This is 59% 
of the original rate per mile, and since this is more than the 
cost per mile for the additional distance, the added distance is 
evidently in this case a source of distinct profit. On the other 
hand, if the line is shortened 5 miles, it may be similarly shown 
\ that not only are the receipts lessened, but that the saving in 
! operating expenses by the shorter distance is less than the 
j reduction in receipts. 

A second example will be considered to illustrate another 
phase. Suppose the home road is 200 miles long and the foreign 
road is 50 miles long. In this case the home road will receive 

=80% of the through rate. Suppose the home road is 



200 + 50 



205 

lengthened 5 miles; then it will receive _ ' ■ ■ = 80.392% 

of the through rate. By the first plan the rate received is 
. 400% per mile ; adding 5 miles, there is a surplus of . 302, 
or 0.0784 per mile, which is but 19.6% of the original rate. 
At this rate the extra distance evidently is not profitable, al- 
though it is not a dead loss— ^there is some compensation. 

501. The most advantageous conditions for roads forming 
part of a through competitive route. From the above it may 
be seen that when a road is but a short link in a long com- 
petitive through route, an addition to its length will increase 
its receipts and increase them more than the addition to the 
operating expenses, 

As the proportionate length of the home road increases the 
less will this advantage become, until at some proportion aii 
increase in distance will just pay for itself. As the proportionate 
length grows greater the advantage becomes a disadvantage 
until, when the competitive haul is entirely on the home road, 
p,ny increase in distance becomes a net loss without any cQni- 
pensation. It is therefore advantageous for a road to be ^ 
short link in a long competitive route; an increase i;i thj\,t lin]^ 



554 RAILROAD CONSTRUCTION. § 502. 

will be financially advantageous; if the total length is less than 
that of the competing line, the advantage is still greater, for 
then the rate received per mile will be greater. 

502. Effect of the variations in the length of. haul and the 
classes of the business actually done. The above distances 
refer to particular lengths of haul and are not necessarily the 
total lengths of the road. Each station on the road has 
traffic relations with an indefinite number of traffic points 
all over the country. The traffic between each station on' 
the road and any other station in the country between which 
traffic may pass therefore furnishes a new combination, the 
effect of which will be an element in the total effect of a 
change of distance. In consequence of this, any exact solution 
of such a problem becomes impracticable, but a sufficiently 
accurate solution for all practical purposes is frequently ob- 
tainable. For it frequently happens that the great bulk of a 
road's business is non-competitive, or, on the other hand, it 
may be competitive-through, and that the proportion of one 
or more definite kinds of traffic is so large as to overshadow 
the other miscellaneous traffic. In such cases an approximate 
but sufficiently accurate solution is possible. 

503. General conclusions regarding a change in distance. 
(a) In a// non-competitive business (local and through) the 
added distance is actually profitable. Sometimes practically 
all of the business of the road is non-competitive; a considerable 
proportion of it is always non-competitive. 

(b) When the competitive local business is very large and the 
competitive through business has a very large average home 
haul compared with the foreign haul, the added distance is 
a source of loss. Such situations are unusual and are generall}' 
confined to trunk lines. 

(c) The above may be still further condensed to the general 
conclusion that there is always some compensation for the added 
cost of operating an added length of line and that it frequently 
is a source of actual profit. 

(d) There is, however, a limitation which should not be lost 
sight of. The above argument may be carried to the logical 
conclusion that, if added distance is profitable, the engineer 
ishould purposely lengthen the line. But added distance means 
added operating expenses. A sufficient tariff to meet these is a 
tax on the community — a tax which more or less discourages 



§ 504. DISTANCE. 655 

traffic. It is contrary to public policy to burden a community 
with an avoidable expense. But, on the other hand, a railroad 
is not a charitable organization, but a money-making enter- 
prise, and cannot be expected to unduly load up its first cost 
in order that subsequent operating expenses may be unduly 
cheapened and the tariff unduly lowered. A common reason 
I for increased distance is the saving of the first cost of a very 
expensive although shorter line. 

(e) Finally, although there is a considerable and uncom- 
pensated loss resulting from curvature and grade which will 
I justify a considerable expenditure to avoid them, there is by 
; no means as much justification to incur additional expenditure 
to avoid distance. Of course needless lengthening should be 
i avoided. A moderate expenditure to shorten the line may be 
! justifiable, but large ex:penditures to decrease distance are 
never justifiable except when the great bulk of the traffic is 
exceedingly heavy and is competitive. 

504. Justification of decreasing distance to save time. It 
should be recalled that the changes which an engineer may 
make which are physically or financially possible will ordi- 
narily have but little effect on the time required for a trip. 
The time which can thus be saved will have practically no value 
for the freight business — at least any value which would justify 
changing the route. When there is a large directly competitive 
passenger traffic between two cities (e.g. New York to Phila- 
delphia) a difference of even 10 minutes in the time required 
for a run might have considerable financial importance, but 
such cases are comparatively rare. It may therefore be con- 
cluded that the value of the time saved by shortening distance 
will not ordinarily be a justification for increased expense to 
accomplish it. 

505. Effect of change of distance on the business done. 
The above discussion is based on the assumption that the busi- 
ness done is unaffected by any proposed change in distance. 
If a proposed reduction in distance involves a loss of business 
obtained, it is almost certainly unwise. But if by increasing 
the distance the original cost of the road is decreased (because 
the construction is of less expensive character) , and if the receipts 
are greater, and are increased still more by an increase in busi- 
ness done, then the change is probably wise. While it is almost 
impossible in a subject of such complexity to give a general 



656 railroao coNsnaucTioN. § 505. 

rule; the following is generally safe : Adopt a route of STich -length 
that the annual trafhc per mile of line is a maximum. 'I'his 
statement may be improved by allowing the element of original 
tost to enter and say, adopt a route of such length that the annual I 
traffic per mile of line divided by the average cost per mile is 
a maximum. Even in the above the operating cost per mile, 
as affected by the curvature and grades on the various routes, 
does not enter, but any attempt to formulate a general rule 
which would allow for variable operating expenses would evi- 
dfently be too complicated for practical applicationi 



CHAPTER XXII. 
GURVAfURii. 

506* General objectidns to eurvatitfe. lii the popular miiid 
curvature is one of the most objectionable features of railrtiad 
alinement. The Cause of this is plain. The objectionablfe 
qualities are oil the s\irface, and are apparent to the non-tech' 
nical mindi They may be itemized as follows: 

1. Curvature increases operating expenses by increasitig (k) 
the required tractive force, (b) the wear and tear of roadbed 
and track, (c) the wear and tear of equipment, atid (d) the 
required number of track-walkers and wa,tchmen. 

2; It may affect the operation of trains (k) by limiting thfe 
leingth of trains, and (b) by preventing the use of the heaviest 
tj'pes of etigines. 

3; Ife may affect travel (a) b}'' the difficulty of making tittle, 
(b) on account of roiigh riding, and (c) on aecdutit of the appre- 
hension of danger. 

4. There is actually an increased daiigdr of collision, derail- 
ment, or other form of accident. 

Some of these objections are quite definite and their tfue 
value may be computed. Others are more general atid vague 
and are usually exaggerated. These objections Will be di^ 
cussed in inverse order. 

507. Financial value of the danger of accideiit dtle tO ctitvl- 
ture^ At the outset it should be realized that in general the 
problem is not otie of ciirvature vs. no curvature, but simply 
sharp curvature vs. easier curvature (the central angle remain^ 
iiig the same), or a greater or less percentage of elifninatiori 
of the degrees of central angle. A straight road between ter^ 
mini is in general a financial (if not a physical) impossibility. 
The practical question is then, how much is the finaiicial value 
of such diminution of danger that may result from such elimi- 
nations of curvature as an engineer is able to make? 

557 



558 ' RAILROAD CONSTRUCTION. §508.' 

In the year 1898 there were 2228 railroad accidents reported 
by the Bailroad Gazette, whose hsts of all ac(?idents worth re- 
porting are very complete. Of these a very large proportion 
clearly had no relation whatever to curvature. But suppose 
we assume that 50% (or 1114 accidents) were directly caused 
by curvature. Since there are approximately 200,000 curves 
on the railroads of the country, there was on the average an \ 
accident for every 179 curves during the year. Therefore we ' 
may say, according to the theory of probabilities, that the 
chances are even that an accident may happen on any particular 
curve in 179 years. This assumes all curves to be equally danger- 
ous, which is not true, but we may temporarily consider it to be 
true. If, at the time of the construction of the road, $1.00 were 
placed at compound interest at 5% for 179 years, it would pro- 
duce in that time $620.89 for each dollar saved, wherewith to pay 
all damages, while the amount necessary to eliminate that cur- 
vature, even if it were possible, would probably be several thou- 
sand dollars. The number of passengers carried one mile for 
one killed in 1898-99 was 61,051,580. If a passenger were to 
ride continuously at the rate of sixty miles per hour, day and 
night, year after year, he would need to ride for more than 116 
years before he had covered such a mileage, and even then the 
probabilities of his death being due to curvature or to such a 
reduction of curvature as an engineer might accomplish are 
very small. Of course particular curves are often, for special 
reasons, a source of danger and justify the employment of 
special watchmen. They would also justify very large expen- 
ditures for their elimination if possible. But as a general 
proposition it is evidently impossible to assign a definite money 
value to the danger of a serious accident happening on a par- 
ticular curve which has no special elements of danger. 

Another element of safety on curved track is that trait of 
human nature to exercise greater care where the danger is more 
apparent. Many accidents are on record which have been 
caused by a carelessness of locomotive engineers on a straight 
track when the extra watchfulness usually observed on a curved 
track would have avoided them. 

508. Effect of curvature on travel, (a) Difficulty in making 
time. The general use of transition curves has largely elimi- 
nated the necessity for reducing speed on curves, and even m hen 
the speed is reduced it is done so easily and quickly by means 



§ 509. CURVATURE. 559 

I 

of air-brakes that but little time is lost. If two parallel lines 
were competing sharply for passenger traffic, the handicap of 
sharp curvature on one road and easy curvature on the other 
might have a considerable financial value, but ordinarily the 
mere reduction of time due to sharp curvature will not have any 
computable financial value. 

(b) On account of rough riding. Again, this is much reduced 
by the use of transition curves. Some roads suffer from a gen- 
eral reputation for crookedness, but in such cases the excessive 
curvature is practically unavoidable. This cause probably 
does have some effect in ^influencing competitive passenger 
traffic. 

(c) On account of the apprehension of danger. This doubtless 
has its influence in deterring travel. The amount of its influence 
is hardly computable. When the track is in good condition 
and transition curves are used so that the riding is smooth, 
even the apprehension of danger will largely disappear. 

Travel is doubtless more or less affected by curvature, but 
it is impossible to say how much. Nevertheless the engineer 
should not ordinarily give this item any financial weight what- 
soever. Freight traffic (two thirds of the total) is unaffected 
by it. It chiefly affects that limited class of sharply competi- 
tive passenger traffic — a traffic of which most roads have not a 
trace. 

509. Effect on operation of trains, (a) Limiting the length 
of trains. When curvature actually limits the length of trains, 
as is sometimes true, the objection is valid and serious. But 
this can generally be avoided. If a curve occurs on a ruling 
grade without a reduction of the grade sufficient to compensate 
for the curvature, then the resistance on that curve will be a 
maximum and that curve will limit the trains to even a less 
weight than that which may be hauled on the ruling grade. 
In such cases the unquestionablj^ correct policy is to ''com- 
pensate for curvature, " as explained later (see §§510, 511), and 
not allow such an objection to exist. It is possible for curvature 
to limit the length of trains even without the effect of grade. 
On the Hudson River R, R. the total net fall from Albany to 
New York is so small that it has practically no influence in 
determining grade. On the other hand, a considerable portion 
of the route follows a steep rocky river bank which is so crooked 
that much curvature is unavoidable and very sharp curvature 



5^ RAILROAD CONSTRUCTION. ' §509, 

cap. ouly be avoided by very large expenditure. As a consequence 
charp curvature has been used and the resistance on the curves 
is far greater than that of any fluctuations of grade which it 
was necessary tc use. Or, at least, a comparatively small 
expenditure would sijifiGe to cut down any grade so that its 
resistance would be less than that of some curve which could 
ijiot be avoided except at an enormous cost. And as a result j 
since the length of trains is really limited by curvature, minor 
grades of 0.3 to 0.5% have been freely introduced which 
might be renpved at comparatively small expense The above 
case is verj^ unusual. Low grades are usually associated with 
generally level country where curvature is easily avoided — 
as jn the Camden and Atlantic R. R. Even in the extreme 
case of the Hudson River road the maximum curvature is 
opjy equivalent to a comparatively low ruling grade. 

0)) Preventing the use of the heaviest types of engines. The 
validity of this objection depends somewhat on the degree of 
curvature and the detailed construction of the engine. While 
some types of engines might have difficulty on curves of ex- 
tremely short radius, yet the objection is ordinarily invalid, 
f his will hesp be appreciated when it is recalled that the " Con-;' 
RoUdation" type was originally designed for use on the sharp 
curvature of the mountain divisions of the Lehigh Valley R. R., 
and that the type has been found so satisfactory that it has 
been extensively employed elsewhere. It should also be re- 
membered that during the Civjl War an immense traffic daily 
passed over a hastily constructed trestle near Petersburg, Va,., 
the track having a radius of 50 feet. As a result of a test made 
at Renovo on the Philadelphia and Erie R. R. by Mr, Isaac 
Dripps, Gen. Mast. Mech., in 1875,* it was claimed that a 
Consolidation engine encountered less resistance per ton than 
one of the "American" type. Whether the test was strictly 
reliable or not, it certainly demonstrated that there was no 
trouble in using these heavy engines on very sharp curvature, 
and we may therefore consider that, except in the most 'Extreme 
cases, this objection has no force whatsoever. 

r- . ■ •■ - ,,i..U.. ■ »w .^X -~- i ^t-~ J-I.t.1— J_ . 1 ' T '■■ H M . iiar 

* Seventh An. Rep. Am. Mast. Mech. Assa. 



§510. CUKVATURE. 561 



COMPENSATION FOR CURVATURE. 

510. Reasons for compensation. The effect of curvature on 
a grade is to increase the resistance by an amount which is equiv- 
alent to a material addition to that grade. On minor grades 
the addition is of little importance, but when the grade is nearly 
or quite the ruling grade of the road, then the additional resist- 
ance induced by a curve will make that curve a place of maxi- 
mum resistance and the real maximum will be a ''virtual grade" 
somewhat higher than the nominal maximum. If, in Fig. 211, 



Fig. 211. 

AN represents an actual uniform grade consisting of tangents 
and curves, the "virtual grade" on curves at BC and DE may 
be represented by BC and DE. If BC and DE are very long, 
_or if a stop becomes necessary on the curve, then the full dis- 
advantage of the curve becomes developed. If the whole grade 
may be operated without stoppage, then, as elaborated further 
in the next chapter, the whole grade may be operated as if equal 
to the average grade, AF, which is better than BC, although 
much worse than AN. The process of ''compensation" con- 
sists in reducing the grade on every curve by such an amount 
that the actual resistance on each curve, due to both curvature 
and grade, shall precisely equal the resistance on the tangent. 
, The practical effect of such reduction is that the "virtual" grade 
is kept constant, while the nominal grade fluctuates. 

One effect of this is that (see Fig. 212) instead of accomplish- 
ing the vertical rise from A to G^ (i.e., HG) in the horizontal 
distance AH, it requires the horizontal distance AK. Such an 
addition to the horizontal distance can usually be obtained by 
proper development, and it should always be done on a ruling 



562 



EAILROAD CONSTRUCTION. 



§511. 



grade. Of course it is possible that it will cost more to accom- 
plish this than it is worth, but the engineer should be sure of 
this before allowing this virtual increase of the grade. 




Fig. 212. 



European engineers early realized the significance of unre- 
duced curvature and the folly of laying out a uniform ruling 
grade regardless of the curvature encountered. Curve compen- 
sation is now quite generally allowed for in this country, but 
thousands of miles have been laid out without any compensa- 
tion. A very common limitation of curvature and grade has 
been the alliterative figures 6° curvature and 60 feet per mile 
of grade, either singly or in combination. Assuming that the 
resistance on a 6° curve is equivalent to a 0.3% grade (15.84 feet 
per mile), then a 6° curve occurring on a 60-foot grade would 
develop more resistance than a 75-foot grade on a tangent. 
The "mountain cut-off" of the Lehigh Valley Railroad near 
Wilkesbarre is a fine example of a heavy grade compensated 
for curvature, and yet so laid out that the virtual grade is uni- 
form from bottom to top, a distance of several miles. 

511. The proper rate of compensation. This evidently is the 
rate of grade of which the resistance just equals the resistance 
due to the curve. But such resistance is variable. It is greater 
as the velocity is lower ; it is generally about 2 lbs. per ton 
(equivalent to a 0.1% grade) per degree of curve when starting 
a train. On this account, the compensation for a curve which 
occurs at a known stopping-place for the heaviest trains should 
be 0.1% per degree of curve. The resistance is not even strictly 
proportional to the degree of curvati;r3, although it is usually 
considered to be so. In fact most formulae for curve resistance 
are based on such a relation. But if the experimentally doter- 
mined resistances for low curvatures are applied to the exces, iv^e 
curvature of the JSTew York Elevated road, for example, the 



§ 512. CtTEVATURE. 563 

rules become ridiculous. On this account the compensation 
per degree of curve may be made less on a sharp curve than on 
an easy curve. The compensation actually required for very 
fast trains is less than for slow trains, say 0.02 or 0.03% per 
degree of curve; but since the comparatively slow and heavy 
freight trains are the trains which are chiefly limited by ruong 
grade, the compensation must be made with respect to those 
trains. From 0.04 to 0.05% per degree is the rate of compen- 
sation most usually employed for average conditions. Curves 
which occur below a known stopping-place for alt trains need 
not be compensated, for the extra resistance of the curve will 
be simply utilized in place of brakes to stop the train. If a curve 
occurs just above a stopping-place, it is very serious and should 
be amply compensated. Of course the down-grade traffic need 
not be considered. 

It sometimes happens that the ordinary rate of compensa- 
tion will consume so much of the vertical height (especially if. 
the curvature is excessive) that a steeper through grade must 
be adopted than was first computed, and then the trains might 
stall on the tangents rather than on the curves. In such cases 
a slight reduction in the rate of compensation might be justi- 
fiable. 

The following rules have been approved by the Amer. Rwy. 
Eng. Assoc. 

1. Compensate .03% per degree (a) when the length of curve 
is less than half the length of the longest train; {■ ) when a curve 
occurs within the first 20 feet of rise of a grade; (c) when cur- 
vature is in no sense limiting. 

2. Compensate .035% per degree (a) when curves are be- 
tween one-half and three-quarters as long as the longest train; 
(6) when the curve occurs between 20 feet and 40 feet of rise 
from the bottom of the grade. 

3. Compensate .04% per degree (a) where the curve is habit- 
ually operated at low speed; (6) where the length of the curve 
is longer than three-quarters of the length of the longest train; 
(c) where elevation is excessive for freight trains; (d) at all places 
where curvature is likely to be limiting. 

. 4. Compensate .05% per degree wherever the loss of elevation 
can be spared. 

^ 512. The limitations of maximum curvature. What is the 
maximum degree of curvature which should be allowed on any 



564 RAILROAD CONSTRUCTION. § 512, 

road? It has been shown that sharp curvature does not prevent 
the use of the heaviest types of engines, and although a sharp 
curve unquestionably increases operating expenses, the increase 
is but one of degree with hardly any definite limit. The general 
character of the country and the gross capital available (or 
the probable earnings) are generally the true criterions. 

A portion of the road from Denver to Leadville, Col., is an 
example of the necessity of considering sharp curvature. The 
traffic that might be expected on the line was so meagre and 
yet the general character of the country was so forbidding 
that a road built according to the usual standards would have 
cost very much more than the traffic could possibly pay for. 
The line as adopted cost about $20,000 per mile, and yet in a 
stretch of 11.2 miles there are about 127 curves. One is a 25° 
20' curve, twenty-four are 24° curves, twenty-five are 20° curves, 
and seventy-two are sharper than 10°. If 10° had been made 
the limit (a rather high limit according to usual ideas), it is 
probable that the line would have been found impracticable 
(except with prohibitive grades) unless four or five times as 
much per mile had been spent on it, and this would have ruined 
the project financially. 

For many years the main-line traffic of the Baltimore and 
Ohjo R. R. has passed over a 300-foot curve (19° 10') and a 
400-foot curve (14° 22') at Harper's Ferry. A few years ago 
some reduction was made in this by means of a tunnel, but 
the fact that such a road thought it wise to construct and operate 
such curves (and such illustrations on the heaviest-traffic roads 
are quite common) shows how foolish it is for an engineer to 
sacrifice money or (which is much more common) sacrifice 
gradients in order to reduce the rate oi curvature on a road 
which at its best is but a second- or third-class road. 

Of course such belittling of the effects of curvature may 
be (and sometimes is) carried to an extreme and cause an engi- 
neer to fail to give to curvature its due consideration. Degrees 
of central angle should always be reduced by all the ingenuity 
of the engineer, and should ®nly be limited by the general rela- 
tion between the financial and topographical conditions of the 
problem. Easy curvature is in general better than sharp curva- 
ture and should be adopted when it may be done at a small, 
financial sacrifice, especially since it reduces distance generally 
and may even cut down the initial cost of that section of the, 



> ! 



§ 512. CURVATURE. 565 

road. But large financial expenditures are rarely, if ever, jus- 
tifiable where the net result is a mere increase in radius v/ithout 
a reduction in central angle. An analysis of the changes which 
have been so extensively made during late years on the Penn. 
R. R. and the N. Y., N. H. & H. R. R. will show invariably a 
reduction of distance, or of central angle, or both, and perhaps 
incidentally an increase in radius of curvature. There are but 
few, if any, cases where the sole object to be attained by the 
improvement is a mere increase in radius. 

The requirements of standard M. C. B. car-couplers have 
virtually placed a limitation on the radius on account of the 
corners of adjacent cars striking each other on very sharp 
curves. This limitation has been crystallized into a rule on 
the P. R. R. that no curve, even that of a siding, can have a 
less. radius than 175 feet, which is nearly the radius of a 33° 
curve. Of course only the most peremptory requirements of 
yard work would justify the employment of such a radius. 



CHAPTER XXIII. 

GRADE. 

513. Two distinct effects of grade. The effects of grade on 
train expenses are of two distinct kinds; one possible effect is 
very costly and should be limited even at considerable expen- 
diture; the other is of comparatively little importance, its cost 
being slight. As long as the length of the train is not limited, 
the occurrence of a grade on a road simply means that the engine 
is required to develop so many foot-pounds of work in raising 
the train so many feet of vertical height. For example, if a 
freight train weighing 600 tons (1,200,000 lbs.) climbs a hill 
50 feet high, the engine performs an additional work of creating 
60,000,000 foot-pounds of potential energy. If this height is 
surmounted in 2 miles and in 6 minutes of actual time (20 
miles per hour), the extra work is 10,000,000 foot-pOunds per 
minute, or about 303 horse-power. But the disadvantages of 
such a rise are always largely compensated. Except for the fact 
that one terminus of a road is generally higher than the other, 
every up grade is followed, more or less directly, by a down grade 
which is operated partly by the potential energy acquired during 
the previous climb. But when we consider the trains running 
in both directions even the difference of elevation of the termini 
is largely neutralized. If we could eliminate frictional resist- 
ances and particularly the use of brakes, the net effect of minor 
grades on the operation of minor grades in both directions would 
be zero. Whatever was lost on any up grade would be regained 
on a succeeding down grade, or at any rate on the return trip. 
On the very lowest grades (the limits of which are defined later) 
we may consider this to be literally true, viz., that nothing is 
lost by their presence ; whatever is temporarily lost in climbing 
them is either immediately regained on a subsequent light down 
grade or is regained on the return trip. If a stop is required 
at the bottom of a sag, there is a net and uncompensated loss 

of energy. 

566 



§ 514. GRADE. 567 

On the other hand, if the length of trains is limited by the 
grade, it will require more trains to handle a given traffic. The 
receipts from the traffic are a definite sum. The cost of hand- 
ling it will be nearly in proportion to the number of trains. 
Assume that by lowering the rate of ruling grade it becomes pos- 
sible to handle such an increased number of cars with one engine 
that four engines can haul as many cars on the reduced grade as 
five engines could haul on the higher grade and at a cost but 
slightly more than four-fifths as much. The effect of this on 
dividends may readily be imagined. 

514. Application to the movement of trains of the laws of 
accelerated motion. When a train starts from rest and acquires 
its normal velocity, it overcomes not only the usual tangent 
resistances (and perhaps curve and grade resistances), but it 
also performs work in storing into the train a vast fund of kinetic 
energy. This work is not lost, for every foot-pound of such 
energy may later be utilized in overcoming resistances, pro- 
vided it is not wasted by the action of train-brakes. If for a 
moment we consider that a train runs without any friction, 
then, when running at a velocity of v feet per second, it possesses 
a kinetic energy which would raise it to a height h feet, when 

h = jr—, in which g is the acceleration of gravity =32.16. Assum- 
ing that the engine is exerting just enough energy to overcome 
the frictional resistances, the train would climb a grade until the 
train was raised h feet above the point where its velocity was v. 
When it had climbed a height h' (less than h) it would have a 
velocity Vi—\/2g{h — h'). As a numerical illustration, assume 

v =30 miles per hour =44 feet per second. Then h = -^= 30.1 feet, 

and assuming that the engine was exerting just enough force 
to overcome the rolling resistances on a level, the kinetic 
energy in the train would carry it for two miles up a grade of 
15 feet per mile, or half a mile up a grade of 60 feet per mile. 
When the train had climbed 20 feet, there would still be 10.1 
feet left and its velocity would be Vi=\/2gil0.1) =25A9 feet 
per second = 17.4 miles per hour. These figures, however, must 
be sKghtly modified on account of the weight and the revolving 
action of the wheels, which form a considerable percentage 
of the total weight of the train. When train velocity is bein§ 



568 RAILROAD CONSTRUCTION. § 515. 

acquired, part of the work done is spent in imparting the energy 
of rotation to the driving-wheels and various truck-wheels of 
the train. Since these wheels run on the rails and must turn 
as the train moves, their rotative kinetic energy is just as effect- 
ive — as far as it goes — in becoming transformed back into 
useful work. The proportion of this energy to the total kinetic 
energy has already been demonstrated (see Chapter XVI, 
§ 435). The value of this correction is variable, but an average 
value of 5% has been adopted fof use in the accompanying 
tabular form (Table XLII), in which is given the corrected 
"velocity head" corresponding to various velocities in miles 
per hour. The table is computed from the following formula: 

^2 in ft. per sec. 2. 151 F^ in m. per h. 

Velocity head = = =0.0334472 

^ 64.32 64.32 

adding 5%for the rotative kinetic energy of the wheels, 0.001677^ 
The corrected velocity head therefore equals 0.0351 IF^ 

Part of the figures of Table XLII were obtained by inter- 
polation and the final hundredth may be in error by one unit, 
but it may readily be shown that the final hundredth is of no 
practical importance. It is also true that the chief use made 
of this table is with velocities much less than 45 miles per hour. 
Corresponding figures may be obtained for higher velocities, if 
desired, by multiplying the figure for half the velocity by four. 

515. Construction of a virtual profile. The following simple 
demonstration will be made on the basis that the ordinary 
tractive resistances and also the tractive force of the locomo- 
tive are independent of velocity. For a considerable range of 
velocity which includes the most common freight-train velocities 
the first assumption is practically true; the second assumption 
is so nearly true under certain possible operative conditions that it 
may serve as a preliminary to the more accurate solution. It may 
best be illustrated by considering a simple numerical example. 

Assume that Fig. 213 shows the profile of a section of road and 
that the grade of AE is 0.40%, which is 21.12 feet per mile. 
Assume also that a freight engine is climbing up the grade at a 
uniform velocity of 20 miles per hour. But since the train is 
moving at 20 miles per hour it has a kinetic energy corresponding 
to a velocity of 14.05 feet (see Table XLII) . At A it encounters a 
down-grade of 0.20 per cent, which is 1500 feet long. Although 



§415. 



GRADE. 



569 



AB has a down-grade of only 0.20%, its grade with respect to 

the up-grade of AE (0.40%) is 

0.60%. Therefore B is 9.00 feet 

below B'. Since the work done 

by the engine would have carried 

the train up to the point B' with 

a velocity of 20 miles per hour, 

the virtual drop of 9 feet will 

increase the velocity head from 

14.05 feet to 23.05 feet, which 
corresponds to the velocity of 

25.6 miles per hour, and this 
win actually be the velocity of 
the train at the point B. At B 
the grade changes to a 1.0% up- 
grade for a distance of 2300 feet. 

The approach of the grade BC 
to the grade B'C is at the rate 
of 1.0-0.4 = 0.6% and therefore, 
the point C will be reached in 
1500 feet. In the remaining 800 
feet the line will climb to D, 
which is 4.8 feet above D\ Al- 
though at B the train is moving 
at the rate of 25.6 miles per 
hour and the engine is working 
at such a rate that it will carry 
the train up a 0.4% grade, yet 
when climbing up a 1.0% grade 
it consumes its kinetic energy in 
overcoming the additional grade. 
When it reaches C, it has lost 
the additional kinetic energy 
which it gained from A to B, and 
as it continues it loses even more. 
When it reaches D, it has lost 4.8 
feet more and its velocity head 
is reduced to 14.05-4.8 = 9.25 ft., 
which corresponds to a velocity 
of 16.2 miles per hour. At D 
the grade changes to +0.1%. 




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6 



570 



HAILKOAD CONSTEUCTION. 



§515. 



TABLE XLII — VELOCITY HEAD (REPRESENTING THE KINETIC 
energy) of trains moving at various VELOCITIES. 



Vel. 


















■ 




mi. 
hr. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


5 


0.88 


0.91 


0.95 


0.99 


1.02 


1.06 


1.10 


1.14 


1.18 


1.22 


6 


1.26 


1.31 


1.35 


1.40 


1.44 


1.48 


1.53 


1.58 


1.62 


1.67 


7 


1.72 


1.77 


1.82 


1.87 


1.92 


1.97 


2.03 


2.08 


2.14 


2.19 


8 


2.25 


2.30 


2.36 


2.42 


2.48 


2.54 


2.60 


2.66 


2.72 


2.78 


9 


2.85 


2.91 


2.97 


3.04 


3.10 


3.17 


3.24 


3.30 


3.37 


3.44 


10 


3.51 


3.58 


3.65 


3.72 


3.79 


3.87 


3.95 


4.02 


4.10 


4.17 


11 


4.25 


4.33 


4.41 


4.49 


4.57 


4.65 


4.73 


4.81 


4.89 


4.97 


12 


5.06 


5.15 


5.23 


5.32 


5.41 


5.50 


5.58 


5.67 


5.75 


5.84 


13 


5.93 


6.02 


6.12 


6.21 


6.31 


6.40 


6.50 


6.59 


6.69 


6.78 


14 


6.88 


6.98 


7.08 


7.19 


7.29 


7.39 


7.49 


7.60 


7.70 


7.80 


15 


7.90 


8.00 


8.11 


•8.22 


8.33 


8.44 


8.55 


8.66 


8.77 


8.88 


16 


8.99 


9.10 


9.21 


9.32 


9.43 


9.55 


9.67 


9.79 


9.91 


10.03 


17 


10.15 


10.27 


10.39 


10.51 


10.63 


10.75 


10.87 


10.99 


11.12 


11.25 


18 


11.38 


11.50 


11.63 


11.76 


11.89 


12.02 


12.15 


12.28 


12.41 


12.55 


19 


12.68 


12.81 


12.95 


13.08 


13.22 


13.35 


13.49 


13.63 


13.77 


13.91 


20 


14.05 


14.19 


14.33 


14.47 


14.61 


14.75 


14.89 


15.04 


15.19 


15.34 


21 


15.49 


15.64 


15.79 


15.94 


16.09 


16.24 


16.39 


16.54 


16.69 


16.84 


22 


17.00 


17.15 


17.30 


17.46 


17.62 


17.78 


17.94 


18.10 


18.26 


18.42 


23 


18.58 


18.74 


18.90 


19.06 


19.22 


19.38 


19.55 


19.72 


19 . 89 


20.06 


24 


20.23 


20.40 


20.57 


20.74 


20.91 


21.08 


21.25 


21.42 


21.59 


21.77 


25 


21.95 


22.12 


22.30 


22.48 


22.66 


22.84 


23.02 


23.20 


23 . 38 


23.56 


26 


23.74 


23.92 


24.10 


24.28 


24.46 


24.65 


24.84 


25.03 


25.22 


25.41 


27 


25.60 


25.79 


25.98 


26.17 


26.36 


26.55 


26.74 


26.93 


27.13 


27.33 


28 


27.53 


27.73 


27.93 


28.13 


28.33 


28.53 


28.73 


28.93 


29.13 


29.33 


29 


29.53 


29.73 


29.93 


30.13 


30.34 


30.55 


30.76 


30.97 


31.18 


31.39 


30 


31.60 


31.81 


32.02 


32.23 


32.44 


32.65 


32.86 


33 . 08 


33.30 


33.52 


31 


33.74 


33.96 


34.18 


34.40 


34.62 


34.84 


35.06 


35.28 


35.50 


35.72 


32 


35.95 


36.17 


36.39 


36.62 


36.85 


37.08 


37.31 


37.54 


37.77 


38.00 


33 


38.23 


38.46 


38.69 


38.92 


39.15 


39.38 


39.62 


39.86 


40.10 


40.34 


34 


40.58 


40.82 


41.06 


41.30 


41.54 


41.78 


42.02 


42.26 


42.51 


42.76 


35 


43.01 


43.26 


43.51 


43.76 


44.01 


44.26 


44.51 


44.76 


45.01 


45.26 


36 


45.51 


45.76 


46.01 


46.26 


46.52 


46.78 


47.04 


47.30 


47.56 


47.82 


37 


48.08 


48.34 


48.60 


48.86 


49.12 


49.38 


49.64 


49.91 


50.18 


50.45 


38 


50.72 


50.99 


51 . 26 


51.53 


51.80 


52.07 


52 . 34 


52.61 


52.88 


53.15 


39 


53.42 


53.69 


53.96 


54.23 


54.51 


54.79 


55.07 


55.35 


55.63 


55.91 


40 


56.19 


56.47 


56.75 


57.03 


57.31 


57.59 


57.87 


58.16 


58.45 


58.74 


41 


59.03 


59.32 


59.61 


59.90 


60.19 


60.48 


60.77 


61.06 


61.35 


61.64 


42 


61.94 


62.23 


62.52 


62.82 


63.12 


63.42 


63.72 


64.02 


64.32 


64.62 


43 


64.92 


65.22 


65.52 


65.82 


66.12 


66.43 


66.74 


67.05 


67.36 


67.67 


44 


67.98 


68.29 


68.60 


68.91 


69.22 


69.53 


69.84 70.15 


70.46 


70.78 



Here we have the rather surprising condition that, although 
the grade is actually rising, it is virtually a down-griade under the 
given conditions, for the engine is working harder than is re- 
quired to run up merely a 0.1% grade and hence will gain in 
velocity. At E, a distance of 1600 feet from D, it reaches what 



§ 515. GEADE. 571 

would have been a uniform 0.4% grade from A to E and the 
grade continues at that rate. Although the train has actually 
cHmbed 1.6 feet from D to E, it has virtually fallen the 4.8 feet 
between D and D', and the velocity head has increased from its 
value of 9.25 feet at D to 14.05 feet, and its velocity is again 20 
miles per hour. The upper hne represents the " virtual profile," 
which may always be drawn by measuring off to the proper scale 
at every point an ordinate which is the velocity head at that 
point. Since the engine is working uniformly, the virtual profile- 
is in this case a straight line. 

As another case, assume that a train is climbing the grade AE 
and exerting a pull just sufficient to maintain a constant velocity 
up that grade. Then A'B' (par- ^ 

allel to AB) is the virtual profile, -f D^^ -^ 

A A' representing the velocity __A i?^ /I I 

head. A stop being required at 
C, steam is shut off and brakes 
are appHed at B, and the velocity r 9 

head BB' reduces to zero at C. 

The train starts from C, and at D attains a velocity correspond- 
ing to the ordinate DD\ At D the throttle may be slightly 
closed so that the velocity will be uniform and the virtual grade 
is D'E', parallel to DE. 

From the above it may be seen that a virtual profile has the 
following properties : 

(a) When the velocity is uniform, the virtual profile is parallel 
with the actual. 

(6) When the velocity is increasing the profiles are separating; 
when decreasing the profiles are approaching. 

(c) When the velocity is zero the profiles coincide. 
{d) The virtual grade at any place is a measure of the work 
required of the engine beyond that required to overcome merely 
the tractive resistances. If it is horizontal it shows that the 
engine is doing nothing besides overcoming the tractive re- 
sistances. If it is upward and is uniform, as in Fig. 213, it 
shows that it is working uniformly and is storing in the train 
"potential " energy which may be utilized on the return trip 
if it is not utilized to overcome tractive resistance in moving 
down a succeeding down-grade. If it is downward, as from B' 
to C, Fig. 214, it shows that the train is giving up kinetic energy, 
probably consuming most of it in brakes, but utilizing some of it 



572 EAILROAD CONSTRUCTION. § 516. 

to fiirnish the tractive power tb tun from 5 to C and also to 
overcome the grade from B to C. 

516. Variation in draw-bar pull. The above demonstration 
has beeii made on the basis that the draw-bar pull is constant 
throughout. It is shown in Chapter XVIII that, when the 
efigirie is working at its full capacity the draw-bar pull decreases 
as the velocity increases, which is chiefly due to the fact, that if 
we attempt to use full stroke at 2 M or 3 M velocity the steam 
will be so rapidly exhausted from the boiler that the pressure will 
fall. Therefore the valves are set to cut off so as toi use the 
dteam expansively but as this reduces the average pressure in 
the cylinder, then (see Eq. 103), the tractive power must be less. 
The reduction of tractive power for several multiples of M is 
shown in Table XXXIX. For example, in the numerical prob- 
lem given above, and assuming the use of the Mikado engine 
whose characteristics have already been computed, the velocity 
at A = 20 -r- 6. 167 = 3.25 M and the tractive power at this velocity 
is 49.23% of its power at M velocity. From the tabular form in 
§ 460 the draw-bar pull at 3.25 M -velocity may be found by 
interpolation to be 16587 lbs. Similarly at B the velocity is 
expected to be 25.6 m.p.h. =4.15 M, and then the tractive power 
is 38:48% and the draw-bar pull only 12484 lbs., about 75% 
of the pull at A. But since the draw-bar pull is so much reduced 
the velocity evidently would not be increased the theoretical 
amount due to the virtual drop BB'. On. the other hand, when 
tliB train reaches D, where the velocity is supposed to be 16.2 
m.p.h. =2.62 M, the draw-bar pull would be 20144, which is over 
121% of the normal pull at 3.25 M velocity. The average pull 
between B and D is 16314 or within 2 % of the normal 16587. 
The average between A and E, assuming that the theoretical 
velocities at J5 and D were actually realized, would be about 2% 
below the assumed pull at A. The 3000-foot sag ABC will be 
passed in 90 seconds and no very great reduction in boiler power 
could take place in that time , especially if the fireman used extra 
care to maintain the pressure. Investigators have declared 
that tests of trains, with a dynamometer car between the tender 
and cars, have shown a practically uniform draw-bar pull, with 
an unchanged throttle and with velocities varying substantially 
on the principles indicated above. If the sag ABC is excessively 
long or deep the reduction of tractive force with increased 
velocity would be so great that the error of the method would be 



§ 517 GRADE. 573 

too great for practical use. But experience has proven that 
for ordinary cases the method can be used with substantial 
accuracy. .ol.> ,: .-. ^.:. .v;,;^.,;;; 

517. Use, valtte, stftd poggible misuse. The essential feature 
respecting grades is the demand on the locomotive. From the 
foregoing it may readily be seen that the ruling grade of a road 
is not necessarily the steepest nominal grade. When a grade 
may be operated by momentum, i.e., when every train has an 
opportunity to take " a run at the hill," it may become a very 
harmless grade and not limit the length of trains, while another 
grade, actually much less, which occurs at a stopping-place 
for the heaviest trains, will require such extra exertion to get 
trains started that it may be the worst place on the road. 
Therefore the true way to consider the value of the grade at 
any critical place on the road is to construct a virtual profile 
for that section of the road. The required length of such a 
profile is variable, but in general may be said to be limited by 
points on each side of the critical section at which the velocity 
is definite, as at a stopping-place (velocity zero), or a long heavy 
grade where it is the minimum permissible, say M miles per 
hour. 

Since the velocities of different trains vary, each train will 
have its own virtual profile at any particular place. Fast 
passenger trains are less affected than slow freight trains. The 
requirement of high average speed necessitates the use of power- 
ful engines, and grades which would stall a heavy freight will 
only cause a momentary and harmless reduction of speed of 
the fast passenger train. 

A possible misuse of virtual profiles lies in the chance that a 
station or railroad grade crossing may be subsequently located 
on a heavy grade that was designed to be operated by momeii->- 
tum. But this should not be used as an argument against the 
employment of a virtual profile. The virtual profile shows the 
actual state of the case and only points out the necessity, if an 
unexpected requirement for a full stoppage of trains at a critical 
point has developed, of changing the location (if a station), or 
of changing the grade by regrading or by using an overhead 
crossing. 

518. Undulatory grades. Advantages. Money can generally 
be saved by adopting an actual profile which is not strictly 
uniform — the matter of compensation for curvature being here 



574 RAILKOAD CONSTRUCTION. § 518. 

ignored. Its effect on the operation of trains is harmless pro- 
vided the sag or hump is not too great. In Fig. 215 the undu- 
latory grade may actually be operated as a uniform grade AG. 
The sag at C must be considered as a sag, even though BC is actu- 
ally an up grade. But the engine is supposed to be working 
hard enough to carry a train at uniform velocity up a grade AG. 
Therefore it gains in velocity from B to C, and from C to D loses 
an equal amount. It may even be proven that the time re- 

F G^ ^ 







Fig. 215. 



quired to pass the sag will be slightly less than the time required 
to run the uniform grade. 

Disadvantages. The hump at F is dangerous in that, if the 
velocity at E is not equal to that corresponding to the extra 
velocity-head ordinate at F, the train will be stalled before 
leaching F. In practice there should be considerable margin. 
Any train should have a velocity of at least M (see § 455) 
in passing any summit. An extra heavy head wind, slippery 
rails, etc., may use up any smaller margin and stall the train. 
If the grade AG jssl ruling grade, then no bump should be allowed 
under any circumstances. For the heaviest trains are supposed 
to be so made up that the engine will just haul them up the 
ruling grades — of course with some margin for safety. Any 
increase of this grade, however short, would probably stall the 
train. 

Safe limits. Since over 99.4% of all freight cars are now 
equipped with train brakes and automatic couplers, there is 
not now the limitation which formerly existed about operating 
freight trains at high speeds, but it may frequently happen 
that it would be undesirable to run a freight train through a 
deep sag at such a velocity as would result from a free run and 
it would therefore become necessary to use brakes, which will 
add a distinct element of cost. 

The term " safe Hmitg" as' used here, refers to the liinits within 



§ 519. GRADE. 575 

which a freight train may be safely operated without the appli- 
cation of brakes or varying the work of the engine. Of course 
much greater undulations are frequently necessary and are 
safely operated, but it should be remembered that they add a 
distinct element to the cost of operating trains and that they 
must not be considered as harmless or that they should be 
introduced unless really necessary, 



RULING GRADES. 

519. Definition. Ruling grades are those which limit the 
weight of the train of cars which may be hauled by one engine. 
The subject of " pusher grades " will be considered later. For 
the present it will suffice to say that on all well-designed roads 
the large majority of the grades on any one division are kept 
below some limit which is considered the ruhng grade. If a 
heavier grade is absolutely necessary no special expense will 
be made to keep it below a rate where the resistance is twice 
(or possibly three times) the resistance on the ruling grade, and 
then the trains can be hauled unbroken up these few special 
grades with the help of one (or two) pusher engines. So far 
as limitation of train length is concerned, these pusher grades 
are no worse than the regular ruling grades and, except for the 
expense of operating the pusher engines (which is a separate 
matter), they are not appreciably more expensive than any 
ruling grade. As before stated, the engineer cannot alter very 
greatly the ruling grade of the road when the general route 
has been decided on. He may remove sags or humps, or 
he may lower the natural grade of the route by development 
in order to bring the grade within the adopted Hmit of ruling 
grade. 

520. Choice of ruling grade. It is of course impracticable for 
an engine to drop ofif or pick up cars according to the grades 
which may be encountered along the Une. A train load is made 
up at one terminus of a division and must run to the other 
terminus. Excluding from consideration any short but steep 
grades which may always be operated by momentum, and also 
all pusher grades, the maximvun grade on that division is the 
ruhng grade. 

It will evidently be economy to reduce the few grades which 
naturally would be a httle higher than the great majority of 



576 RAILROAD CONSTRUCTION. § 521. 

others until such a large amount of grade is at some uniform 
limit that a reduction at all these places would cost more than 
it is worth. The precise determination of this limit is prac- 
tically impossible, but an approximate value may be at once 
determined from a general survey of the route. The distance 
apart of consecutive control points (see § 18) into their differ- 
ence of elevation is a first trial figure for the rate of the grade. If 
a grade even approximately uniform is impossible owing to the 
elevation of intervening ground, the worst place may be selected 
and the natural grade of that part of the route determined. 
If this grade is much steeper than the general run of the natural 
grades, it may be pplicy to reduce it by development or to boldly 
plan to operate that place as a pusher grade. The choice of 
possible grades thus has large limitations, and it justifies very 
close study to determine the best combination of grades and 
pusher grades. When the choice has narrowed down to, two 
limits, the lower of which may be obtained by the expenditure 
of a definite extra suni, tl^^^J^]lQi^^ia,Yyh^Tee^ as 

will be developed. . v v" .-,/,,-,,'•• - 

521. Maximum train load on any grade. The Mikado loco- 
motive, whose characteristics were analyzed in Chapter XVIII^ 
has a net pulling power at the rim of the drivers, at M velocity, 
of 35758 lbs. which is 23.3% of 153,300,, the weight pn the drivei^s. 
This percentage is slightly over -^. Increasing the percentage 
6% on account of increased power at starting we have 24.7% 
or nearly^. 0;!! the other haiid, wet, slippery rails may render 
the adhesion as low as ^ and thus limit the actual dra^vfing ppwer. 
Although the real power of a locon^otive depends on the velocity 
at which it seems desirably to ruU;, the maximum tra,ctive power 
at " M ". velocity can always be, approximately estimated as 
^ of the weight on the drivers. In Table XLIII are given the 
weights of several types of locomotives together with their 
tractive powers at three ratios of adhesion. These values are, 
useful when the more ela,borate method detailed in Chapter 
;5^ VIII is not considered necessary. 

The maximum train load oiijaiiy grade depends on the character 
and number of the cars, as well as on their gross ^^eightj. The 
approximate resistance of cars is given by Eq. 12L as .5 = 2.2 i 
+122 n. Applying this to a steel box-car weighing 24 tons net 
and loaded with 100,000 lbs,, thp resistance would be-285Jbs. or 
3.85 lbs. per ton. Empty, the res^ptance would l?,e, 7. 28 lbs. per 



§521. 



GRADH. 



577 



TABLE XLllI.— TRACTIVE POWER OF VARlOtfS TYPES OF STAND- 
ARD-GAUGE LOCOMOTIVE AT VARIOUS RATES OP ADHESION. 



Type of locomotive. 


Total weight 

of engihe 
and tender. 


Weight 

of 

engine 

only. 


Weight 
on the 
drivers. 


Tractive power when 
ratio of adhesion 
is 




Lbs. 


Tons. 


1 
i 


3% 


1 

5 


Atlantic, 4-4-2 

Atlantic, 4-4-2, four 
cylinder compound 

Pacific, 4-6-2 

Pacific. 4-6-2 

Ten- wheel, 4-6-0. . . 

Prairie, 2-6-2 

Consolidation, 2-8-0 
Consolidation, 2-8-0 

Mikado, 2-8-2 

Mikado, 2-8-2 


340,000 

368,800 
343,600 
403.780 
321,000 
366,500 
214,000 
366,700 
405,500 
315,000 


170.0 

184.4 
171.8 
201.9 
160.5 
183.2 
107.0 
183.3 
202.7 
157.5 


199,400 

206,000 
218,000 
226,700 
201,000 
212,500 
120,000 
221,500 
259,000 
196,100 


105,540 

115,000 
142,000 
151,900 
154,000 
154,000 
106,000 
197,500 
196,000 
153,200 


26,385 

28,750 
35,500 
37,975 
38,500 
38,500 
26,500 
49,375 
49,000 
38,300 


23,740 

25,875 
31,950 
34,180 
34,650 
34,650 
23,850 
44,440 
44,100 
34,470 


21,100 

23,000 
28,400 
30,380 
30,800 
30,800 
21,200 
39,500 
39,200 
30,640 



ton. Applying the formula to a wooden box-car weighing 
15 tons net and carrying 60,000 lbs., the resistances for the 
car full and empty would be 4.9 and 10.3 lbs. per ton, respect- 
ively. Three and 10 pounds per ton are the ordinary extremes. 
Although resistances of less than 3 lbs. per ton have been 
measured for whole trains of heavy-loaded coal cars, there are 
usually enough light-weight cars and empties in a train to 
increase the average per ton resistance to perhaps 6 lbs. per ton. 
The Mikado locomotive, referred to above, had a draw-bar 
pull on a level at ikf velocity (6.167 m.p.h.) of 35,419 lbs. How 
much of a load could it draw up a 1.2% grade at M velocity? 
Assume that the cars have a weight and character such that 
the average resistance would be 6 lbs. per ton. The grade 
resistance of the locomotive is 315,000X .012 = 3780, which 
subtracted from 35,419 leaves 31,639, the pull available for the 
cars. Then, calling T the tons weight of cars 

31,639 = 6r+(20X1.2Xr)= 30 r, and ^ = 1054. 

It should be noted that this computed tonnage is on the basis 
of an assumed tractive resistance of 6 lbs. per tan. In § 467 
the tractive power of this same locomotive, on the same grade, 
is computed, by the regular rating formula, to be 16 fully loaded 
cars, weighing 70.8 tons per car, a total load of 1133 tons, or 
53 empties, weighing 18 tons per car, a total load of 954 tons. 
The above value of 1" is approximately the mean of these two 
extremes. For general computations, when the character of 



578 RAILKOAD CONSTRUCTION. § 522. 

the train load is unknowable, some such average value, as used 
above, is probably as accurate as it is possible to utilize it. 

522. Proportion of the traffic affected by the ruling grade. 
Some very light traffic roads are not so fortunate as to have 
a traffic which will be largely affected by the rate of the ruling 
grade. When passenger traffic is light, and when, for the sake 
of encouraging traffic, more frequent trains are run than are 1 
required from the standpoint of engine capacity, it may happen 
that no passenger trains are really limited by any grade on the 
road — i.e., an extra passenger car could be added if needed. 
The maximum grade then has no worse effect (for passenger 
trains) than to cause a harmless reduction of speed at a few points . 
The local freight business is frequently affected in practically 
the same way. All coal, mineral, or timber roads are affected 
by the rate of ruling grade as far as such traffic is concerned. 
Likewise the through business in general merchandise, especially 
of the heavy traffic roads, will generally be affected by the rate 
of ruhng grade. Therefore in computing the effect of ruling 
grade, the total number of trains on the road should not ordi- 
narily be considered, but only the trains to which cars are added, 
until the limit of the hauling power of the engine on the ruhng 
grades is reached. 

■" PUSHER GRADES. 

523. General principles underlying the use of pusher engines. 

On nearly all roads there are some grades which are greatly 
in excess of the general average rate of grade, and these heavy 
grades cannot usually be materially reduced without an expend- 
iture which is excessive and beyond the financial capacity 
of the road. If no pusher engines are used, the length of all 
heavy trains is limited by these grades. The financial value 
of the reduction of such ruling grades has already been shown. 
But in the operation of pusher grades there is incurred the 
additional cost of pusher-engine service, for a pusher engine 
must run twice over the grade for each train which is assisted. 
It is possible for this additional expense to equal or even exceed 
the advantage to be gained. In any case it means the adoption 
of the lesser of two evils, or the adoption of the more economical 
method. The work of overcoming the normal resistances of so 
many loaded cars over so many miles of track and of lifting so 
many tons up the gross differences of elevation of predetermmed 
points of the line is approximately the same whatever the exact 



§524. GRADE. 579 

route, and if the grades are so made that fewer engines working 
more constantly can accomplish the work as well as more engines 
which are not hard worked for a considerable proportion of the 
time, the economy is very apparent and unquestionable. Wel- 
lington expresses it concisely : " It is a truth of the first importance 
that the objection to high gradients is not the work which the 
engines have to do on them, but it is the work which they do 
not do when they thunder over the track with a light train be- 
hind them, from end to end of a division, in order that the 
needed power may be at hand at a few scattered points where 
alone it is needed." 

524. Balance of grades for pusher service. Assume that both 
pusher and through engines are the Mikado engine with dimen- 
sions already given (§ 453), and that they will be operated at 
their most effective velocity, M = 6.167 m.p.h., and that the 
effective draw-bar pull of each is 37190-1771=35419 lbs., 
less the locomotive grade resistance, which on a 1.9% grade 
is 20X1.9X157.5 = 5985 lbs. The net draw-bar pull on this 
grade for each engine is, therefore, 29434 lbs. Assume that 
the train considered is made up of coal cars weighing 40000 lbs. 
net and carrying 100,000 lbs. feach; also a caboose weighing 12 
tons. Utihzing Eq. 121, the tractive resistance of a loaded 
coal car will be 2.2X70+122 = 276, and the grade resistance 
20X1.9X70 = 2660, making a total of 2936. The total for the 
caboose is 148+456 = 604. The two engines have a net draw- 
bar pull of 2X29434 = 58868 lbs. Subtracting 604 for the 
caboose, there is left 58264 for coal cars. 58264 -=-2936 = 19.84, 
the number of cars. Although the number of cars must, of 
course, be a whole number, the computation of the relative 
through and pusher grades requires that we use the fractional 
number. The tractive resistance of the 19.84 cars and caboose 
is 2.2 [(19.84X70) + 12] + (122X20.84) =5624. The force avail- 
able for grade is 35419-5624 = 29795. The tonnage on the 
single engine grade is 157.5 (engine) plus 19.84X70 = 1388.8 
(coal cars), plus 12 (caboose), or 1558.3 tons. 29795-^1558.3 
= 19.12 lbs. per ton, which is the grade resistance for a 0.956% 
grade. This means that the through grade can be made 0.956% 
and the corresponding pusher grade may be 1.9%. If the same 
problem is worked out on the basis of some other type of engine, 
which, perhaps, weighs considerably less, very nearly the same 
through grade tQ correspond with the pusher grade will be 



580 EAILROAD CONSTRUCTION. § 525. 

obtained. The above combination of unit cbt weights must be 
worked as 19 coal cars and a caboose and have ^ considerable 
margin of unused power. A different combination of car 
weights would use up the power with less or no margin, but in 
any case the computation of the corresponding lower grade, oi* 
the computation of an allowable pusher grade on the basis of 
a given through grade, should be made by using a fractional; 
number of cars. ? 

Since the pusher engine service is interihittent, and siritje it ik 
working at full power for much less than half the time, it is 
practicable for the fireman to feed coal faster than > the standard 
of 4000 lbs. of coal per hour while going up the pusher grade. 
The above computation was made on the basis of power pro- 
duction at the 4000-lb. rate. In § 457, it is shown that increase 
ing the rate of coal consumption increases the value of M, and 
conversely when the locomotive is run at a velocity less than M 
the tractive power is increased, although the increase is disj- 
proportionately small. Increasing the tractive power of the 
pusher engine will increase the number of cars, although probably 
not as much as one car. Then the increase in car number will 
increase the computed resistance and decrease the amount avail- 
kble for grade. This decreased amount is divided by an in- 
creased number of tons and the amount of available for grade 
per ton is less and the computed through grade is less. Con- 
sidering the very slight and disproportionate difference made by 
increasing the rate of coal consumption beyond the 4000-lb. 
standard, it is, perhaps, wisest to make the ratio of the grades 
on the ba^is of engines of equal power. 

525. Two-pusher grades. It may happen, although rarely, 
that three systems of ruling grades may be necessary on one 
division, which may be so balanced that one Unbroken train is 
handled with equal facility on through grades with one engine, 
on one-pUsher grades with two engines and on two-pusher 
grades with three engines. The relation of these three grades 
may be computed on the same principles as are Used above. 

526. Operation of pusher engines. The maximum efficiency 
in operating pusher engines is obtained when the pusher engine 
is kept constantly at work, and this is facilitated when the pusher 
grade is as long as possible, i.e., when the heavy grades and the 
great bulk of the difference of elevation to be surmounted is 
at one place. For example^ a pusher grade of three miles fol- 



§ 526. GRADJ3. 681 

i 

lowed by a comparatively level stretch of three miles and then 
by another pusher grade of two miles cannot all be operated as 
cheaply as a continuous pusher grade of fiv^ miles. EitheT 
the two grades must be operated as a continuous grade of eight 
miles (sixteen pusher miles per trip) or else as two short pusher 
grades, in which case there would be a very great loss of time 
and a difficulty in so arranging the schedules that a train need 
not wait for a pusher Or the pushers need not waste too much 
time in idleness waiting for trains. If the level stretch were 
imperative, the two grades would probably be operated as one, 
but an effort should be made to bring the grades together. It 
is not necessary to bring the trains to a stop to uncouple the 
pusher engine, but a stop is generally made for coupling on, and 
the actual cost in loss of energy and in wear and tear of stopping 
and starting a heavy train is as great as the cost of running 
an engine light for several miles. 

There are two ways in which it is possible to economize in 
\he use of pusher engines, (a) When the traffic of a road is 
go very light that a pusher engine will not be kept reasonably 
busy on the pusher grade it may be worth while to place a 
siding long enough for the longest trains both at top and bottom 
of the pusher grade and then take up the train in sections. 
Perhaps the worst objection to this method is the time lost 
while the engine runs the extra mileage, but with such verj' 
light traffic roads a little time more or less is of small consequence. 
On light traffic roads this method of surmounting a heavy grade 
will be occasionally adopted even if pushers are never used. 
If the traffic is fluctuating, the method has the advantage 
of only requiring such operation when it is needed and avoiding 
the purchase and operation of a pusher engine which has but 
little to do and which might be idle for a considerable proportion 
of the year, (h) The second possible method of economizing 
is only practicable when a pusher grade begins or ends at or 
near a station yard where switching-engines are required. In 
such cases there is a possible economy in utilizing the switching- 
engines as pushers, especially when the work in each class is 
srtiall, and thus obtain a greater useful mileage. But such cases 
are special and generally imply small traffic, 

A telegraph-station at top and bottom of a. pusher grade is 
generally indispensable to effective and safe operation. 

527. Length of a pusher grade. The virtual length of the 



582 RAILROAD CONSTRUCTION. § 528. 

,*■■ ■■* 

pusher grade, as indicated by the mileage of the pusher engine, 
is always somewhat in excess of the true length of the grade 
as shown on thfe profile, and sometimes the excess length is 
very great. If a station is located on a lower grade within a 
mile or so of the top or bottom of a pusher grade, it will ordina- 
rily be advisable to couple or uncouple at or near the station, 
since the telegraph-station, switching, and signaling may be 
more economically operated at a regular station. If the extra 
engine is coupled on ahead of the through engine (as is some- 
times required by law for passenger trains) the uncoupling at 
the top of the grade may be accomplished by running the assist- 
ant engine ahead at greater speed after it is uncoupled, and, 
after running it on a siding, clearing the track for the train. 
But this requires considerable extra track at the top of the grade. 
Therefore^ when estimating the length of the pusher grade, 
the most desirable position for the terminal sidings must be 
studied and the length determined accordingly rather than 
by measuring the mere length of the grade on the profile. Of 
course these odd distances are always excess; the coupling or 
uncoupling should not be done while on the grade. 

528. The cost of pusher-engine service. When we analyze 
the elements of cost, we will find that many of them are dependent 
only on time, while others are dependent upon mileage. Still 
others are dependent on both. Very much will depend on the 
constancy of the service, and this in turn depends on the train 
schedule and on a variety of local conditions which must be 
considered for each particular case. The effect of a pusher- 
engine on maintenance of way may be considered on the basis 
that an engine is responsible for one-half of the deterioration of 
maintenance of way and structures, and, therefore, one-half of 
the percentage of the first 19 items in Table XLI or 9.06% of 
the average cost of a train-mile wiU be considered as chargeable 
for each mile of pusher engine service. Although the cost of 
repairs and renewals of engines is evidently a function of the 
mileage, and would therefore be somewhat less for a pusher- 
engine which did little work than for an engine which was 
worked to the limit of its capacity, yet it is only safe to make 
the same allowance as for other engines. Other items of main- 
tenance of equipment are evidently to be ignored. The item of 
wages of enginemen will evidently depend upon the system 
employed ou the particular road. Whatever the precise system 



§528. 



GRADE. 



583 



i'ABLE XLIV. — COST FOR EACH MILE OF PUSHER-ENGINE SERVICE. 



Item 
number. 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 

engine 

mile, 

per cent. 


1-19 
25-27 


Track material, labor, bridges. . . . 
Steam locomotives 


18.12% 
9.24 

8.12 

11.27 

1.21 


50 
100 

100 
100 
100 


9.06 
9.24 


80,81 


Road enginemen and engine-house 
expenses 


8.12 


82-85 
90,91,94 


Fuel and other engine supplies .... 
Signaling, flagmen, and telegraph. . 


11.27 
1.21 






• — •••• 




38.90 



the general result is to pay the enginemen as much in wages 
as the average payment for regular service, and therefore the 
full allowance for Item 80 will be made. Similarly we must 
allow the full cost of the items for engine supplies. While the 
engine is doing its heavy work in cHmbing up the grade, the 
consumption of fuel and water is certainly greater than the 
average; but, on the other hand, on the return trip, when the 
engine is running hght, it probably runs for a considerable por- 
tion of the distance actually without steam, and therefore the 
consumption of fuel and water will nearly, if not quite, average 
the consumption for an engine running up and down grade 
along the whole line. That portion of fuel consumption which 
is due to radiation, bio wing-off steam, and the many other 
causes previously enumerated, will be the same regardless of 
the work done. We therefore allow 100% for all of these items 
of engine supplies. In general we must add 100% for Items 90, 
91, and 94, the cost of switchmen and telegraphic service. While 
there might be cases where there would be no actual addition 
to the pay-rolls or the operating expenses on account of these 
items, we are not justified in general in neglecting to add the 
full quota for such service. Collecting these items we will have 
38.90% of the average cost of a train-mile for the cost of each 
mile run by the pusher engine. On the basis that the average 
cost of a train mile is $1.60, the cost of one mile of pusher engine 
service would be .3890XS1. 60 = 62.24 cents. Assume that the 
pusher engine grade is five miles long but that the engine actually 
runs 11 miles on a round trip and that it makes 5 round trips or 
55 miles per day. Then the daily cost would be .6224X55 = 
$34.23 per day. Probably $25 to $30 per day should be charged 



584 RAILKOAD CONSTKUCTION^ § 629^ 

up even if the mileage did not amount to as much, since many of 
the items in the cost of service are largely independent of mileage. 
On the other hand the pusher engine service renders unnecessary 
the extra trains which would have been required to handle the 
traffic with one engine over the steeper grades. The cost of 
these must be computed for each particular case. 

BALANCE OF GRADES FOR UNEQUAL TRAFFIC. 

529. Nature of the subject. It sometimes happens, as when 
a road runs into a mountainous country for the purpose of 
hauling therefrom the natural products of lumber or minerals, 
that the heavy grades are all in one direction — that the whole 
line consists of a more or less unbroken climb having perhaps 
a few comparatively level stretches, but no down grade (except 
possibly a slight sag) in the direction of the general up grade. 
With such lines this present topic has no concern. But the 
majority of railroads have termini at nearly the same level 
(500 feet in 500 miles has no practical effect on grade) and 
consist of up and down grades in nearly equal amounts and 
rates. The general rate of ruling grade is determined by the 
character of the country and the character and financial backing 
of the road to be built. It is always possible to reduce the grade 
at some point by "development" or in general by the expen- 
diture of more money. It has been tacitly assumed in the 
previous discussions that when the ruling grade has been de- 
termined all grades in either direction are cut down to that 
limit. If the traffic in both directions were the same this would 
be the proper policy and sometimes is so. But it has developed, 
especially on the great east and west trunk lines, that the weight 
of the eastbound freight traffic is enormously greater than that 
of the westbound — that westbound trains consist very largely of 
"empties" and that an engine which could haul twenty loaded 
cars up a given grade in eastbound traffic could haul the same 
cars empty up a much higher grade when running west. As 
an illus-tration of the large disproportion which may. exist, the 
eastbound ton-mileage on the P. R. R. between the years" 1851 
and 1885 was 3,7 times the westbound ton-mileage. Between 
the years 1876 and 1880 the ratio rose to more than 4.5 to 1. 
On such a basis it is as important and necessary to obtain, say, 
a 0.6% ruling grade against the eastbound traffic as \o have, 



§530. GRADE. 585 

say, a 1.0% grade against the westbound traffic. This is the 
basis of the following discussion. It now remains to estimates 
the probable ratio of the traffic in the two directions and from 
that to determine the proper "balance" of the opposite ruling 
grades. 

530. Computation of the theoretical balance. Assume first, 
for simplicity, that the exact business in either direction is 
accurately known. A little thought will show the truth of the 
following statements. , 

1. The locomotive and passenger-car traffic in both directions 
is equal. 

2. Except as a road may carry emigrants, the passenger 
traffic in both directions is equal. Of course there are innumer- 
able individual instances in which the return trip is made by 
another route, but it is seldom if ever that there is any marked 
tendency to uniformity in this. Considering that a car load 
of, say, 50 passengers at 150 pounds apiece weigh but 7500 
pounds, which is -^o of the 75000 pounds which the car may 
weigh, even a considerable variation in the number of passengers 
will not appreciably affect the hauling of cars on grades. On 
parlor-cars and sleepers the ratio of live load to dead load (say 
20 passengers, 3000 pounds, and the car, 125000 pounds) is 
even more insignificant. The effect of passenger traffic on 
balance of grades may therefore be disregarded. 

3. Empty cars have a greater resistance per ton than loaded 
cars. Therefore in computing the hauling capacity of a loco- 
motive hauling so many tons of " empties," a larger figure must 
be used for the ordinary tractive resistances — say four pounds 
per ton greater. 

4. Owing to greater or less imperfections of management a 
small percentage of cars will run empty or but partly full in 
the direction of greatest traffic. 

5. Freight having great bulk and weight (such as grain^ 
lumber, coal, etc.) is run from the rural districts toward the 
cities and manufacturing districts. 

6. The return traffic — manufactured products — although worth 
as much or more, do not weigh as much. 

As a simple numerical illustration assume that the weight 
of the cars is ^ and the live load f of the total load when 
the cars are "full'' — although not loaded to their absolute 
limit of capacity. Assume that the relative weight of live load 



586 RAILROAD CONSTRUCTION. § 530. 

to be hauled in the other direction is but |; assume that the 
grade against the heaviest traffic is 0.9%. Since the tractive 
resistance per ton is considerably greater in the case of unloaded 
cars than it is in the case of loaded cars, allowance must be 
made for this in calculating the train resistance. Assuming 
the use of the Mikado locomotive described in § 453, its rating 
on a 0.9% grade, see § 467, equals 

A = QQQ^QQii - 315,000 = 3,230,000 = 1615 tons, the " rating." 

Call We the total weight, live and dead, of the cars in an east- 
hound train, and PTw the corresponding weight for a west-bound 
train. Fe and Fj^ are the weights of live freight; ly the dead 
weight of a car, which for simplicity is considered in this case 
to be uniformly a 100,000-lb. capacity car, weighing 20 tons 
or 40,000 lbs. The problem assumes that Fw = i^E- Th^n 
W^ff = lFE+nw. 

We = 1615 - 6.0 n (for a 0.9% grad^-see Table XL, § 467). 

By trial, it is found that for n = 24, T7e = 1615- 144 = 1471, 
which means a total weight of 61.3 tons per car, or a net load 
of 41.3 tons or 82,600 lbs. live load per car. This fulfils the con- 
dition that the live load is | of the total load as nearly as possible 
for an even number of car loads. | of 41.3 tons, or 13.8 tons, 
plus 20 tons, gives an average load of 33.8 tons per car for west- 
bound trains, and for a train of 24 cars = 811.2 tons per train, 
or 1,622,400 lbs. Substituting in Eq. 122, § 467, 

^^^j - 315,000 = 1,622,400 + 24^-^^5j5jj. 

Solving, 7" = . 0169, or a 1.69% grade, which, under the above 
assumptions and conditions, is the grade on which the given 
type of locomotive could handle one-third of the live load which 
could be hauled up a 0.9% grade, in the same number of cars, 
by that same locomotive. It is interesting to note that the solu- 
tion of this problem, given in a previous edition, using a more 
approximate method, and based on the use of a much lighter 
consolidation locomotive, weighing only 107 tons, gave 1.60% 
as the grade corresponding to 0.9% against east bound traffic. 
This substantial agreement, in spite of the difference in operating 
conditions, shows the substantial accuracy of the method for 



§ 53l. GRADE. 587 

the solution of a problem for which the varying conditions of 
traffic in the two directions render useless any very precise 
solution. 

Of course the actual traffic in the two directions, and their 
ratio, will vary from time to time, and the actual operation of 
. trains will vary accordingly, and therefore the relation of ruling 
grades in the two directions, for maximum efficiency of operation, 
will fluctuate accordingly, while the ruling grades, once estab- 
lished, are practically finalities. Therefore any close precision 
in the computation of these relative grades is useless. Never- 
theless the above calculation shows unmistakably that under 
the given conditions, a very considerable variation in the rate 
of grade in opposite directions is not only justifiable, but a neglect 
to allow for it would be a great economic error. 

531. Comptitation of relative traffic. Some of the principal 
elements have already been referred to, but in addition the 
following facts should be considered. 

(a) The greatest disparity in traffic occurs through the hand- 
ling of large amounts of coal, lumber, iron ore, grain, etc. On 
roads which handle but little of these articles or on which for 
local reasons coal is hauled one way and large shipments of 
grain the other way the disparity will be less and will perhaps 
be insignificant. • .. ' - 

(b) A marked change in the development of the country may, 
and often does, cause a marked difference in the disparity of 
traffic. The heaviest traffic (in mere weight) is always toward 
manufacturing regions and away from agricultural regions. But 
when a region, from being purely agricultural or mineral, be- 
comes largely manufacturing, or when a manufacturing region 
develops an industry which will cause a gro^vth of heavy freight 
traffic from it, a marked change in the relative freight movement 
will be the result. 

(c) Very great fluctuations in the relative traffic may be 
expected for prolonged intervals. 

(d) An estimate of the relative traffic may be formed by the 
same general method used in computing the total traffic of the 
road (see § 473, Chapter XIX) or by noting the relative traffic 
on existing roads which may be assumed to have practically 
the same traffic as the proposed road will obtain. 



CHAPTER XXIV. 
THE IMPROVEMENT OF OLD LINES. 

532. Classification of improvements. The improvements here 
considered are only those of aHgnment— horizontal and vertical. 
Strictly there is no definite limit, either in kind or magnitude, 
to the improvements which may be made. But since a railroad 
cannot ordinarily obtain money, even for improvements^ to 
an amount greater than some small proportion of the pre- 
viously invested capital, it becomes doubly necessary to expend 
such money to the greatest possible advantage. It has been 
previously shown that securing additional business and increas- 
ing the train load are the two most important factors in increas- 
ing dividends. After these, and of far less importance, come 
reductions of curvature, reductions of distance (frequently of 
doubtful policy, see Chap. XXI, § 503), and elimination of sags 
and humps. These various improvements will be briefly dis- 
cussed. 

(a) Securing additional business. It is not often possible 
by any small modification of alignment to materially increase 
the business of a road. The cases which do occur a,re usually 
those in which a gross error of judgment was committed during 
the original construction. For instance, in the early history 
of railroad construction many roads were largely aided by the 
towns through which the road passed, part of the money neces- 
sary for construction being raised by the sale of bonds, which 
were assumed or guaranteed and subsequently paid by the 
towns. Such aid was often demanded and exacted by the 
promoters. Instances are not unknown where a failure to 
come to an agreement has caused the promoters to deliberately 
pass by the town at a distance of some miles, to the mutual 
disadvantage of the road and the town. If the town subsequent- 
ly grew in spite of this disadvantage, the annual loss of business 
might readily amount to more than the original sum in dispute. 

588 



§ 533. IMPROVEMENT OF OLD LINES. 589 

Such an instance would be a legitimate opportunity for study 
of the advisability of re-location. 

As another instance (the original location being justifia- 
ble) a railroad might have been located along the bank of a 
considerable river too wide to be crossed except at consider- 
able expense. When originally constructed the enterprise would 
not justify the two extra bridges needed to reach the town. 
A growth in prosperity and in the business obtainable 
might subsequently make such extra expense a profitable invest^- 
ment. 

(b) Increasing the train load. On account of its importance 
this will be separately considered in § 535 et seq. 

(c) Reduction in curvature and distance and the elimination 
of sags and humps. Such improvements are constantly being 
made by all progressive roads. The need for such changes 
occurs in some cases because the original location was very 
faulty, the revised location being no more expensive than the 
original, and in other cases because the original location was the 
best that was then financially possible and because the present 
expanded business will justify a change. 

(d) Changing the location of stations or of passing sidings. 
The station may sometimes be re-located so as to bring it nearer 
to the business center and thus increase the business done. 
But the principal reasons for re-locating stations or passing 
sidings is that starting trains may have an easier grade on which 
to overcome the additional resistances of starting. Such changes 
will be discussed in detail in § 537. 

533. Advantages of re-locations. There are certain undoubted 
advantages possessed by the engineer who is endeavoring to 
improve an old line. 

(a) The gross traffic to be handled is definitely known, 

(b) The actual cost per train-mile for that road (which may 
differ very greatly from the average) is also known, and therefore 
the value of the proposed improvement can be more accurately 
determined. 

(c) The actual performance of such locomotives as are used 
on the road may be studied at leisure and more reliable data 
may be obtained for the computations. 

534. Disadvantages of re-locations. The disadvantages are 
generally more apparent and frequently appear practically 
insuperable — more so than they prove to be on closer inspection 



590 RAILROAD CONSTRUCTION. § 534. 

(a) It frequentl}'- means the abandonment of a greater or less 
length of old line and the construction of new line. At first 
thought it might seem as if a change of line such as would permit 
an increase of train-load of 50 or perhaps 100% could never 
be obtained, or at least that it could not be done except at an 
impracticable expense. On the contrary a change of 10% 
of the old line is frequently all that is necessary to reduce the 
grades so that the train-loads hauled by one engine maj'- be 
nearly if not quite doubled. And when it is considered that 
the cost of a road to sub-grade is generally not more than one- 
third of the total cost of construction and equipment per mile, 
it becomes plain that an expenditure of but a small percentage 
of the original outlay, expended where it will do the most good, 
will often suffice to increase enormously the earning capacity. 

(6) One of the most difficult matters is to convince the finan- 
cial backers of the road that the proposed improvement will 
be justifiable. The cause is simple. The disadvantages of the 
original construction lie in the large increase of certain items 
of expense which are necessary to handle a given traffic. And 
yet the fact that the expenditures are larger than they need 
be are only apparent to the expert, and the fact that a saving 
may be made is considered to be largely a matter of opinion 
until it is demonstrated by actual trial. On the other hand 
the cost of the proposed changes is definite, and the very fact 
that the road has been uneconomically worked and is in a poor 
financial condition makes it difficult to obtain money for im- 
provements. 

(c) The legal right to abandon a section of operated line 
and thus reduce the value of some adjoining property has 
sometimes been successfully attacked. A common instance 
would be that of a factory which was located adjoining the right 
of way for convenience of transportation facilities. The abandon- 
ment of that section of the right of way would probably be fatal 
to the successful operation of the factory. The objection may 
be largely eliminated by the maintenance of the old right of 
way as a long siding (although the business of the factory might 
not be worth it) , but it is not always so easy of solution, and 
this phase of the question must always be considered. 



§ 535. IMPKOVEMENT OF OLD LINES. 591 



xtEDUCTION OF VIRTUAL GRADE. 

535. Obtaining data for computations. As developed in the 
last chapter (§§ 515-517) the real object to be attained is the 
reduction of the virtual grade. The method of comparing grades 
under various assumed conditions was there discussed. When 
the road is still "on paper" some such method is all that is 
possible; but when the road is in actual operation the virtual 
grade of the road at various critical points, with the rolling 
stock actually in use, may be determined by a simple test and 
the eiTect of a proposed change may be reliably computed. 
Bearing in mind the general principle that the virtual grade 
line is the locus of points determined by adding to the actual 
grade profile ordinates equal to the velocity head of the train, 
it only becomes necessary to measure the velocity at various 
points. Since the velocity is not usually uniform, its precise 
determination at any instant is almost impossible, but it will 
generally be found to be sufficiently precise to assume the velocity 
to be uniform for a short distance, and then observe the time 
required to pass that short space. Suppose that an ordinary 
watch is used and the time taken to the nearest second. At 
30 lailes per hour, the velocity is 44 feet per second. To obtain 
the time to within 1%, the time would need to be 100 seconds 
and the space 4400 feet. But with variable velocity there 
would be too great error in assuming the velocity as uniform, 
for 4400 feet or for the time of 100 seconds. Using a stop- 
watch registering fifths of a second, a 1% accuracy would 
require but 20 seconds and a space of 880 feet, at 30 miles per 
hour. Wellington suggests that the space be made 293 feet 
4 inches, or ^^ of a mile; then the speed in miles per hour 
equals 200 -^ s, in which s is the time in seconds required to 
traverse the 293' 4". For instance, suppose the time required 
to pass the interval is 12.5 seconds, j^ "^^^ i^ 12.5 seconds = 
one mile in 21^5 seconds, or 16 miles per hour. But likewise 
200^12.5=16, the required velocity. The following features 
should be noted when obtaining data for the computations: 

(a) All critical grades on the road ' should be located and 
their profiles obtained — by a survey if necessary, 

(6) At the bottom and top of all long grades (and perhaps at 
intermediate points if the grades are very long) spaces of known 



5^2^ RAILROAD CONSTRUCTION. § 536. 

length (preferably 293J feet) should be measured off and marked 
by flags, painted boards, or any other serviceable targets, 

(c) Provided with a stop-watch marking fifths of seconds . 
the observer should ride on the trains affected by these grades 
and note the exact interval of time required to pass these spaces.. 
If the space is 293-J feet, the velocity in miles per hour =200 -f- 
interval in seconds. In general, the velocity in miles per hour, 

distance in feet X 3600 



time in seconds X 5280' 



(d) Since these critical grades are those which require the 
greatest tax on the power of the locomotive, the conditions 
under which the locomotive is working must be known — i.e., 
the steam pressure, point of cut-off, and position of the throttle. 
Economy of 6oal consumption as well as efficient w^orking at 
high speeds requires that steam be used expansively (using an 
early cilt-off), and even that the throttle be partly closed; but 
when an engine is slowly climbing up a maximum grade with a 
full load it is not exerting its maximum tractive power unless 
it has its maxinium steam pressure, wide-open throttle, and is 
cutting off nearly at full stroke. These data must therefore 
be obtained so as to know whether the engine is developing 
at a critical place all th6 tractive force of which it is capable. 
The' condition of the track (wet and slippery or dry) and the 
approximate direction and force of the wind should be noted 
with sufficient accuracy to judge whether the test has been Inade 
under ordinary conditions rather than under conditions which 
are exceptionally favorable or unfavorable. 

(e) The train-loading should be obtained as closely as possible. 
Of course the dead weight of the cars is easily found, and the 
records of the freight department will usually give the live 
load with all sufficient accuracy. 

536. Use of the data obtained. A very brief inspection 
of the results, freed from refined calculations or uncertainties, 
wiU demonstrate the following truths: 

(ct) If, on a uniform grade, the velocity increases, it shows 
that, under those conditions of engine working, the load is less 
than the engine can handle on that grade 

(6) If the velocity decreases, it shows that the load is greater 
than the engine can handle on an indefinite length of such 



§536. IMPROVEMENT OF OLD LINES. 593 

grade. It shows that such a grade is being operated by momen- 
tum. Frciii the rate of decrease of velocity the maximum 
practicable length of such a grade (starting with a given velocity) 
may be easily computed. 

(c) By combining results under different conditions of grade 
but with practically the same engine working, the tractive 
power of the engine may be determined (according to the prin- 
ciples previously demonstrated) for any grade and velocity. 
For example: On an examination of the profile of a division 
of a road the maximum grade was found to be 1.62% (85.54 
feet per mile). At the bottom and near the top of this grade 
two lengths of 293' A" are laid off. The distance between the 
centers of these lengths is 6000 feet, A freight train moving 
up the grade is timed at 9f seconds on the lower stretch and 7f 

seconds on the upper. These times correspond to -—- r and =—z 

9.4 7.6 

or 21,3 and 26.3 miles per hour respectively. It is at once 

observed that the velocity has increased and that the engine 

could draw even a heavier load up such a grade for an indefmite 

distance. How much heavier might the load be? 

For simphcity we will assume that the conditions were 

normal, neither exceptionally favorable nor unfavorable, and 

that the engine was worked to its maximum capacity. The 

engine is a ''consolidation" weighing 128700 pounds, with 

112600 pounds on the drivers. The train-load behind the 

engine consists of ten loaded cars weighing 465 tons and eleven 

empties weighing 183 tons, thus making a total train- weight of 

712 tons. Applying Eq, 106, we find that the additional force 

which the engine has actually exerted per ton in increasing the 

velocity from 21.3 to 26.3 miles per hour in a distance of 6000 

feet is 

70 
P = r^(26.32- 21.32) =2,78 pounds per ton. 

The grade resistance on a 1.62% grade is 32.4 pounds per ton. 
The average train resistance may be computed from §§ 429 
and 439, 

Engine resistance, at say 8 m,p.h. (§ 429) =1615 lbs. 
Cars resistance, (648X2,2) + (21X 122) =3988 lbs. 



Total tractive resistance on level =5603 lbs. 



594 RAILROAD CONSTRUCTION. § 537. 

The average tractive resistance is therefore 5603 -f- 712 = 7.87 
pounds per ton. Adding the grade resistance (32.4) we have 
a total train resistance of 40.27 pounds per ton. But, com- 
puting from the increase in velocity, the locomotive is evidently- 
exerting a pull of 2.78 pounds per ton in exeess of the computed 
required pull on that grade, or a total pull of 43.05 pounds 
per ton. Therefore the train load might have been increased 
proportionately and might have been made 

^r-.o^/2.78+40.27 ^^, ^ 

712X ^Q^ — = 761 tons. 

This shows that 49 tons additional might have been loaded 
on to the train, or say, three more empties or one additional 
loaded car. 

A pull of 43.05 pounds per ton means a total adhesion at the 
drivers of 30,652 pounds, which is about 27% of the weight on 
the drivers — 112,600 pounds. This indicates average condi- 
tions as to traction, and as good as can be depended on for 
regular service. 

The above calculation should of course be considered simply 
as a " single observation." The performance of the same engine 
on the same grade (as well as on many other grades) on succeed- 
ing days should also be noted. It may readily happen that 
variations in the condition of the track or of the handling of the 
engine may make considerable variation in the results of the 
several calculations, but when the work is properly done it is 
always possible to draw definite and very positive deductions. 

537. Reducing the starting grade at stations. The resistance 
to starting a train is augmented from two causes: (a) the trac- 
tive resistances are usually about 20 pounds per ton instead of, 
say, 6 pounds, and (b) the inertia resistance must be overcome. 
The inertia resistance of a freight train (see § 435) which is 
expected to attain a velocity of 15 miles per hour in a distance 
of 1000 feet is (see Eq. 140) 

70 
P= :r7r^^(15^—0) = 15.8 pouuds per ton, 

which is the equivalent of a 0.79% grade. Adding this to a grade 
which nearly or quite equals the ruling grade, it virtually creates a 
new and higher ruling grade. Of course that additional force can 
be greatly reduced at the expense of slower acceleration, but even 



§537. 



IMPEOVEMENT OF OLD LINES. 



595 



this cannot be done indefinitely, and an acceleration to only 
15 miles per hour in 1000 feet is as slow as should be alloWed 
for. With perhaps 14 pounds per ton additional tractive 
resistance, we have about 30 pounds per ton additional — eqiiivSr 




Fig. 216. 



lent to a 1.5% grade. Instances are known where it has proven 
wise to create a hump (in what was otherwise a uniform grade) 
at a station. The effect of this on high-speed passenger trains 
moving up the grade would be merely to reduce their speed 
very slightly. No harm is done to trains moving down the 
grade. Freight trains moving up the grade and intending to 
stop at the station will merely have their velocity reduced as 
they approach the station and will actually save part of the 
wear and tear otherwise resulting from applying brakes. When 
the trains start they are assisted by the short down grade, 
just where they need assistance most. Even if the grade CD 
is still an up grade, the pull required at starting is less than that 
required on the uniform grade by an amount equal to 20 times 
the difference of the grade in per cent. 



CHAPTER XXV. 

STRESSES IN TRACK. 

538. Nature of the subject. The character and amount of 
the stresses in the rails, rail fastenings and ties, which make up 
'the track, and the intensity and distribution of the pressure 
which is transmitted by the ties through the ballast and embank- 
ment to the subsoil, have long been a subject of investigation by 
railroad engineers. The complexity of the subject is too great 
for a dependence on mere theoretical analysis. Even experi- 
mental work must be so elaborate that no one person or single 
individual railroad have hitherto obtained conclusive results, 
except upon isolated details. 

In 1913, a committee was appointed by the Amer. Rwy. Eng, 
Assoc, who cooperated with a similar committee appointed by 
the Amer. Soc, of Civil Engineers. Both societies appropriated 
money for the large expenses involved. Several railroads co- 
operated bj^ furnishing facilities for experimental work. Several 
steel-rail corporations contributed funds. Special instruments 
were designed for experimental use. After five years of work, 
a progress report, covering 184 pages, was made to the 1918 
convention of the Amer. Rwy. Eng. Assoc. The second pro-- 
gress report (170 pp.) was made to the 1920 convention. The 
investigation is not yet (1921) complete. But from these two 
voluminous reports, which indicate the magnitude of the prob- 
lem, the following very condensed summary has been compiled. 
The thoroughness of the investigation is indicated by the fact 
that the number of observations for rail strain only, made, read, 
recorded and reduced, and on which the fi.rst progress report 
was partly based, is more than 250,000. The conclusions, which 
can be drawn from the tests made, have already had their effect 
in modifying track construction, and will probably have still 
greater effect when the principles underlying the stresses in track, 
due to rapidly moving and very heavy rolling-stock, are more 
thoroughly comprehended and when these principles have crys- 
tallized into definite rules of practice. 

596 



§ 539. STRESSES IN TRACK. 597 

539. Action of track as ^n elastic structure. Wheel loads 
bear vertically, but usually with some horizontal component, 
on a rail. The rails are flexible beams, supported by flexible 
ties, which are supported by a more or less yielding but elastic 
ballast, which rests on a more or less yielding subsoil. For 
convenience, the term modulus of elasticity of rail support is 
used as a measure of the vertical stiffness of the rail support, 
and is defined as " the pressure per unit of length of each rail 
required to depress the track one imit." For example, a series 
of wheel loads, equivalent to 10,000 pounds per tie for each 
rail, depress the track an average of 0.3 inch. Then, on ther 
basis of proportionality of depression to pressure, 33,333 lbs. 
would produce one inch of depression, which for a tie spacing 
of 22 inches would require a pressure of 33,333 -f- 22 = 1515 lbs. 
per inch of length of rail per inch of depression. The elasticity 
and flexibility of these various materials affects the stresses to 
which they are subject. The spacing of wheels along a rail 
also affects very greatly the intensity and character of the 
stresses produced in the rails and ties. Although a purely 
theoretical solution is unsatisfactory and inadequate, a theoret- 
ical study makes it possible to limit the scope of the necessary 
experimental work. Theoretical analysis shows that the bend- 
ing moment of a rail wfll be comparatively large for a single 
concentrated load with no appreciable loads sufficiently near 
to hold the rail down and produce a negative bending moment 
at an adjacent point, thus reducing the positive moment immedi- 
ately under the concentrated load. But railroad loadings are 
always in groups. A heavy driver-load is almost invariably 
preceded and followed by a comparatively light truck-wheel 
load, if not by another driver. The variation in operating 
conditions as to spacing and intensity of wheel loads limits the 
use of precise calculations for purposes of generalization, but 
analysis (which is substantially confirmed by experimental tests) 
shows that " the assumption of a continuous elastic support 
under the rail is by far the most convenient, most easily applied 
and most comprehensive in its application to the questions 
involved in the work of the committee." 

540. Typical track depression profile for static load on one 
or two axles. See Fig. 217. Note that the depression for one 
axle extends for about ten tie spaces and that the rail is some- 
what raised above the normal height beyond a distance of about 



598 



RAILROAD CONSTRUCTION. 



§540. 



5 tie spaces from the load even if there is a comparatively- 
light wheel load on the rail at that point. The deflection is 
of course maximum directly under the load and it makes almost 



12'6^ 



One-ax 
> 



P 

le load 

■< 



12'6'^ 



^P^r^^jlpz^i?^^ 



V 



i.io 






.20 
30 
,40 


.10 
.20 
.30 
.40 



TRUCK 

■<r- 



X==^^^ 


*--«=9* 




























:;^ 


1 


















\ 


s^vj>- 


.^= 6000 lbs. 








^ 












X^ 




HWO 






> 


^/a 










p= 


= 1750i 


^<N> 


S^^ 






■'^ > 


y/ 










==25000- 


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yA 


f/ 






















y/ 










Loa 


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fe 


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-P=50(^ lbs. 








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n 


^"^^ 


"><^*- 









y 


/yY 










p= 


1750^ 


■^ 


^-^^^ 






^ A 


■y 












25000-1 


''x^ 




'^*. 




y// 


' 














N 


>V^ 




^ 


// 












.,„„„ 4 • 




^^ 




■*^ / 


• 






Loaa Den 


iveen t 


les 




Xy 




1 








\ 




*/ 









WHEEU 



-9'9"- 



-5'6"- 



Two-axle load 



9Vi- 



TRUC 



i^ 



WHEEL 



:^::.:l_);^j,;i;;,L^:.v-:.:LJ^ 




.10 

.20 
.30 
.40 
.50 



— *r; 


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i>=^ 


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S 


^ 














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-y 


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vO\ 




-~.,___ 


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-•":;^ 


// 














\ 




-— » — 




/ 








P = 


= load 


on ont 


J rail 




--«__ 




'vl^ 


L 


oads 


ver ti€ 


s 






1 




--.^ 




^ 











85-lb; 6x8x8 tie; 21 stone ballast; "before tamping' 



Fig. 217. — Teack Depression Profiles, Static Load on One and Two 

Axles. 



no difference whether the load is directly above a tie or between 
two ties. The curves are substantially identical, merely moved 
along as the load moves. The amount of the depression for a 
given load varies with the character of the ballast and the tamp- 



§540. 



STRESSES IN TRACK. 



599 



ing, whether recent or old, as was shown by the numerous other 
similar profiles given in the report. The effect of recent tamp- 
ing was investigated and it was shown that the depression under 
a load on recentlj'^ tamped track is nearly proportional to the 
loading, which implies a nearly constant " modulus of elasticity 
of rail support." On the other hand, if track has not been tamped 
for several months, there is a comparatively deep depression 
for the first 5000 lbs., proportionately less for the next additional 

25,000 




0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 
Rail depression in inches 

O— O Before Tamping o a After Tamping 

Fig. 218. — Tie Depression Diagram, Static Loads, 

6000 lbs., and perhaps still less for additional increments. This 
is also shown by Fig. 218, which shows the "after tamping" 
curve to be nearly a straight line; the " before tamping " curve 
is much more curved. It should be noted that the " before 
tamping " depression line is nearly a straight line after it is loaded 
to about 10,000 lbs. In later investigations this fact was utilized 
by producing this nearly straight line back to the line of zero 
pressure, as shown by the dotted line. The intercept on the 
line of zero wheel-load is a measure of the depression of the tie 
before it has its full bearing on the ballast. As a part 'of the 



600 RAILROAD CONSTRUCTION. § 54l. 

investigation on the stresses and the elastic curve of a tie undar 
load, the depression of a tie was very accurately measured at 
several points along the tie and for a regular series of light to 
heavy loads. For all cases where the tamping had not been 
recent (or " before tamping ") a curve, similar to those of Fig. 
218, was drawn for each point along the tie. Producing the 
depression line backward to the point of zero loading gives an 
intercept which is called " the initial position of the ballast bed 
with respect to the bottom of the tie for the compact condition 
of ballast existing in the track." Of course this does not mean 
that there is such an actual gap between the under side of the 
tie and the ballast, but such gap as may exist at some points 
along the tie will make up a large part of this initial depression. 
A comparison of similar curves for light and heavy rails proves 
what might have been predicted, that the depression under a 
heavy rail for a given load is less than that under a light load. 
The heavy rail, by its extra stiffness, distributes the loading 
over a greater number of ties and the one or two ties nearly 
under the load do not need to carry such a large proportion of 
the total. 

A broad general idea of the depressions due to track loading 
and df the proportions of the total depression due to rail, tie, 
ballast and sub-soil, may be obtained from the following figures, 
which, however, must be considered as very approximate and 
subject to great variation. 

Division of depressions of track under drivers of Mikado 
locomotive : 

1. Compression of tie under rail, plus effect of bending 

of tie to bring it to full bearing on the ballast along 

its length 0.05 in. 

2. Compression of 24!' of stone ballast immediately 

under the rail 0.15 in. 

3. Compression of roadway immediately under the rail. 0.15 in. 



0.35 in. 

Bending of 85-lb. rail between ties spaced 22" c.c. by a Mikado 
locomotive not more than 0.01." 

■ 541. Bending moment and depression in a rail due to a group 
of loads. Fig. 219 shows graphically the relative bending 
moments under each wheel of a Mikado locomotive. The 



§541. 



STRESSES IN TRACK. 



601 



Kght lines show the curves of moments due to each wheel; the 
heavier lines give the algebraic summation of the effects of all 
the wheels. Note (a) that the effect of each wheel is maximum 
directly under that wheel but the effect continues even beyond 
adjacent wheels; (b) a v/heel usually develops a negative 
moment under an adjacent wheel, which reduces the positive 
moment developed by the adjacent wheel; (c) as an example, 
calling the moment developed by the second driver (counting 



Mikado Locomotive. tOO-ib. rail 




:^ 



1.0 Mo 




Fia. 219. — Bending Moments and Depkessions. Combination of 

Loads. 



from right to left) under the second driver = +1.00, the effect 
of the first driver is —0.20; the effect of the third driver is 
— 0.20; that of the fourth driver is —0.05; the effect of the 
pilot is zero and likewise the effect of the trailer. The net effect 
is that the combination of wheels develops a moment imder 
the second driver of only 55% of that due to the second driver 
if it acted alone. Similarly the depression of rail produced by 
a wheel is maximum under that wheel, but it develops an up- 
ward force which may reduce the depression under some other 
wheel, although probably not the adjacent wheel. For example, 
calling the depression produced by the ;first driver = +1.00, the 



602 KAILROAD CONSTRUCTION. § 542. 

effect of the second driver is to cause a further depression under 
the first driver equal to +0.23; the third driver has an added 
effect of — 0.04; the effect of the pilot truck is neghgible. The 
net effect is a depression under the first driver which is 1.19 
times the depression which the first driver alone would cause. 
Note that, although there is depression under all the wheels, 
the depression between the fourth driver and the trailer is less 
than that under the trailer. Between the two trucks of a car, 
the depression is usually negative, i.e., the rails are curved 
upward above their normal position. 

542. Special instruments and devices for making tests. Tests 
were made to measure the depression of the rail, tie, ballast and 
roadbed, both for static loads and for moving loads. Static 



25 TO so TONS OF RAILS 




" T ' l U — TT ^ T 

Fig. 220. — Loading Device for Producing any desired One-axle ob 

Two-axle Loads. 

loads of any desired magnitude were produced by spotting a 
carload of 25 to 50 tons of rails over the track to be tested. Two 
H-beams (for two-axle loads) or one H-beam (for single-axle 
loads) were placed under the load of rails, each H-beam being 
supported by two struts having load-indicating screw jacks, 
which were carried on curved bearing blocks, placed on the track 
rails, the blocks having the same radius as car wheels but with- 
out coning. Since the bearing blocks were under the center 
of the car and were 12 to 15 feet in either direction from the car 
wheels, the effect of the car wheels was nearly negligible and was 
so considered. 

Unit rail stress. The stretching of the base of the rail under 
a static load was measured with a Berry strain gage just as any 
such stress in metal is measured in a testing laboratory. The 
Berry strain gage is not applicable for observing the rapidly 
changing stresses due to moving loads, which therefore require 
the use of a stremmatograph. The form used will record at 
any instant, on a revolving disk any minute variation in the 



§542. 



STRESSJ3S IN TRACK. 



603 



distance apart of a pair of gage points drilled in the base of the 
rail exactly 4" apart. 

Unit pressures. The unit pressure exerted at any depth of 
the ballast was measured by a pressure capsule. As shown in 
Fig. 221, the ballast bears on a circular bearing plate, having 
sq. in., which transmits the pressure to a thin 



an area 



of 5 





TOP VIEW 
Bearing plate removed 







•^ ^V,V^V \S K.N.V^S\\\\\\^ T^.4i^ Mh777k'y7?9. 



M 



SIDE VIEW-SECTION 




Pressuke Capsules. 



steel diaphragm. The movement of the diaphragm actuates 
a simple mechanism which pushes a rod enclosed in a pipe lead- 
ing to a dial located outside the ballast. The mechanism is 
calibrated by observing the readings for known pressures. 
Several of these capsules are inserted in the ballast almost im- 
mediately under the tie, and also at the bottom of the ballast 
just above sub-grade, as shown in Fig. 221. The simple dial 
form is used to measure the pressure produced by static loads^ 



604 



RAILROAD CONSTRUCTION. 



§542. 



Q 



rl 



,H1 



^ 



m 




For moving loads, the mechanism operates a stylus which makes 
a record on a revolving disk. 

Depression plugs. The actual depression of the ballast at 
any depth, or of the sub-soil at subgrade, was measured by 
locating a horizontal plate at the desired point. A vertical 
I" tube, enclosing a ys" rod, with a set screw for adjustment, 
(see Fig. 222) is attached to this plate. To avoid any binding 
action of the ballast through which it passes, 
the vertical rod and tube is surrounded by 
another f" tube. The top of the rod is ad- 
justed to be above the ballast and at a con- 
venient height for comparison of elevation 
with a fixed reference plug. Of course the 
plate will follow the strata in which it is 
placed in any change of elevation which may 
occur. The fixed plugs were located in the 
groimd far enough away from the track so 
that they would not be appreciably influ- 
enced by track depression, and at nearlj'- 
the same elevation as the tops of the 
vertical rods. The relative elevations were 
observed very accurately by means of a 
level-bar, a metal bar provided with a level 
bubble and a micrometer adjusting screw. 
Then, after the track had been loaded, 
minute changes of elevation, due to pressure, 
were observed. For measuring depressions 
directly under a tie, a double plug, having 
two vertical rods which would straddle a 
tie, were used, and the average reading of the 
two rods was taken. The level bar was used to observe depres- 
sions of rail, tie, ballast or subsoil, but only as to the effect 
of static loading. The depression of the rail under moving 
load was measured by a double exposure photograph. Pieces 
of black paper, with white crosses on them, having one line 
vertical, were pasted on the web of the rail. A camera was 
focused on the rail about 10 feet away. An initial exposure 
was taken of the unloaded rail. Then, without disturbance 
of the camera, the desired train load was run over the track 
at the desired speed. When the train (or locomotive) was at 
the desired point, it closed an electric circuit which operated 



rT- 



--I- 



FiG. 222. — Depres 
sioN Plug. 



§543. 



STEESSES IN TRACK. 



605 



the shutter for a .001 second exposure. The resultant photo- 
graph showed for each cross a double cross with one vertical 
and two horizontal lines whose distance apart represented, after 
suitable reduction, the depression of the rail. Using a magnify- 
ing micrometer microscope, and a computed constant multiplier, 
it was possible to measure from the photographic plate the 
actual deflection of the rail with a precision of about 0.01 inch. 
543. Pressure transmitted from tie to ballast. This subject 
was investigated both theoretically and experimentally. The 




30 

'27 24 21 18 15 12 9 6 3 3 6 9 12 15 18 21 24 27 

Distance from Center Line of Middle Tie-Inches 
Figures within diagram indicate percentages 

Fig. 223. — Lines of Equal Vebtical Pkessure in Ballast fob Equal 
Loads on Ties, Spaced 21" c.c. 



experimental work included not only track tests but also an 
extensive series of laboratory tests using sand ballast, pebbles, 
and broken stone. If ballast consists of absolutely clean spheres 
of perfectly elastic material, whose mutual actions and reactions 
are only pressure without friction, a definite theoretical solution 
as to distribution of pressure is possible, although complicated. 
The equation of pressure is a logarithmic equation which is 
chiefly useful in interpreting experimental results. Both theory 
and experimental tests demonstrate that: 

(a) " The bearing pressure of the tie varies in intensity from 
its edge to its middle* line." This is shown in Fig. 223, where 



606 RAILROAD CONSTRUCTION. § 543. 

the numbers within the diagram give percentages of the average 
tie pressure. The figure also shows that if the ballast is only 
6 inches thick the entire pressure on the subsoil is concentrated 
on a comparatively narrow area under the tie and that a con- 
siderable part of the subsoil between the ties carries but little 
pressure. The ballast must be nearly 24 inches deep (if the 
ties are spaced 21") before the load is distributed with sub- 
stantial uniformity. If the ties are spaced further apart, the 
depth for uniform pressure must be still greater. In a very 
approximate way, it may be said that the pressure becomes 
substantially uniform at a depth equal to the tie spacing. 

(6) " The pressures which react from the lower face of the 
tie act in other than vertical lines, the greatest variation from 
the vertical direction being at the edge of the tie." Fig. 223 
also shows this. 

(c) '' The variation in intensity of pressure in the ballast 
lengthwise of the tie (which is dependent upon size and stiff- 
ness of tie, quality of tamping, and condition of the bed on which 
the tie rests) becomes less and less with increase in depth and it 
may be expected that the variations will be smoothed out at 
a depth equal to the ordinary tie spacing, or a few inches below, 
where there will be fairly uniform pressure over the horizontal 
plane." 

{d) " For quiescent loading there is little difference in the 
manner and rate of transmission and distribution of pressure 
for broken stone, pebbles, and sand ballasts; that is, at a given 
depth the intensities of pressure v/ill be approximately the same, 
provided, of course, the ultimate carrying capacity of the ballast 
is not exceeded; and this conclusion may properly be extended 
to other non-cohesive materials. It will require less load to 
force the tie into sand ballast than into broken stone; the ulti- 
mate carrying capacity of the broken stone ballast under tie 
pressure is much greater than that of the sand ballast — the 
particles of sand ballast are more easily moved and rearrange 
themselves under lighter loads. For the different kinds of 
ballast there are great differences in the ultimate load which 
can be carried on a tie before ballast movement begins. Th.e 
ultimate carrying capacity depends upon size of particle, smooth^ 
ness of surface and degree of angularity. A material whose 
mobility under pressure is increased by the action of water or 
by mixture with other materials may thereby have its carrying 



§ 544. STRESSES IN TRACK. 607 

capacity decreased. For heavy loading the ultimate carrying 
capacity of a ballast material is especially important." 

(e) For quiescent loads the presence of ballast above the level 
of the bottom of the tie has little or no effect in increasing the 
maximum load which can be carried without forcing the ballast 
from under the tie and allowing the tie to settle. For moving 
loads which produce vibration, the presence of ballast up to the 
top of the tie, and particularly at the tie ends, increases con- 
siderably the resistance to lateral displacement. The greater 
the velocity of trains, the greater the necessity for such lateral 
reinforcement. 

544. Transverse stresses in the tie. The character and dis- 
tribution of transverse stresses in the tie depend very largely 
on the tamping. If the tamping were absolutely uniform 
throughout the length of the tie, the upward pressure would 
be uniform and there wouldT be a maximum positive moment 
under each rail, a maximum -negative moment in the center of 
the. tie, and points of inflection between the center and each rail. 
If the tie is very strongly tamped under the center and tamped 
very little if at all under the ends, making it " center-bound," 
there will be a severe negative moment in the center, and little 
or none under the rails. Concentrating the tamping for a short 
space on each side of each rail, and leaving the center almost 
clear of ballast, relieves the center of any transverse stress and 
even minimizes that under the rails. From the standpoint of 
stress in the tie, it is desirable, but it makes an undesirable 
concentration of pressure on the ballast and roadbed. Prob- 
ably the ballast would soon crush down under such a concen- 
trated pressure. The best method of tamping is that which 
makes the tamping firmest on either side of each rail, with 
enough tamping in the center to give good support and yet 
not so much that a negative moment would be developed which 
would be in excess of the positive moment under the rail. Since 
the amounts of these moments depend on the tamping and since 
the effect of the tamping may be more or less altered with the 
passage of each train, due to a slight settlement of the ballast, 
any attempt at precise quantitative computation of m.oment 
is fruitless. Nevertheless tests were made to determine the 
moments under a variety of conditions (center-bound ties, end- 
bound ties, etc.) so as to determine maximum and minimum 
values for the moments under the rails and in the center. Fig. 



608 



RAILBOAD CONSTRUCTION. 



§545. 



224 is a composite of the deflections of three ties on Class A 
track on the Ch. M. & St. P. Rwy. The vertical scale is 500 
times the horizontal scale. The curve represents the depres- 
sion of the tie and may also be considered to represent the 
deformed neutral axis and that tbe curvature indicates the 
character of the bending. The curve shows the usual case of 
a negative moment in the center and positive moments under 
each rail. Static tests under a truck load of 100,000 lbs., on 
poorly ballasted track, showed a negative bending moment in 
the center of as much as —4.5 W inch-pounds, in which TF = the 
load in pounds carried by one tie. This was observed to be 
about 15,000 lbs. 4.5X15,000 = 67,500 in. lbs. For a 6" X 8" 
tie, a moment of 67,500 in. lbs. means a maximum unit stress 



m 

•TO, 

<u 

H 

o 0, 

a 
o 

'to 



0.06 



—, 



Fig. 224. — Composite Diagkam of Tie Flextjke. 



of 1406 lbs. per sq. in. But this stress was produced by a static 
load. The effect of speed and dynamic augment (see § 413) 
would largely increase this figure and perhaps make it exceed 
the safe working stress for even an oak tie. On the other hand, 
for track in good condition, a negative bending moment of 
—2.0 W in the center is as much as should be expected. 

545. Effect of counterbalancing. In § 413 there is given an 
elementary explanation of the necessity for counterbalancing 
and some of the rules for accomplishing it. It was also explained 
that perfect counterbalancing is necessarily impossible and that 
there is always an unbalanced dynamic augment which pro- 
duces an increased pressure on the rails at some part of the 
revolution of the driver, or a racking of the locomotive frame 
at each half-stroke of each piston. The dynamic augment 
increases as the square of the velocity, and its effect is therefore 



§545. 



STRESSES m TRACK. 



609 



very great and serious at high speeds. It is sometimes found 
impracticable to make the counterweight on the main driver 
sufficiently large and heavy to balance the effect of the very 
great weight of the side rods, main rod, etc., of a very heavy 
locomotive. In such a case, the driver is said to be under-^ 
balanced and then the greatest stress in the rail may occur 
when the counterweight is up rather than when it is down. 
The underbalance of the main driver is made up by overbalancing 




0.2 0.3 0.4 0.5 0.6 0.7 0.8 

Position of counterweight in parts of a revolutioa 

Fig. 225. — Counterbalancing Stress in Rail, under Main Driver, 
DURING one Revolution. 



the other drivers, and this increases the pressure under them 
when the counterweight is down. Although the dynamic aug- 
ment may be computed numerically, as illustrated in § 413, its 
real effect on the rail is modified by the action of the equalizing 
levers and also by the effect that a change of pressure by the 
other drivers has on the rail and on the reaction of the rail on 
the driver considered. 
Another item which increases the pressure exerted by the 



610 RAILROAD CONSTRUCTION. § 545. 

main driver on the rail is the vertical component of the pull 
(or push) of the main rod. This component acts doAvnward 
both when the crank pin is up and when it is down. For one 
case this was computed to be about 12% of the cylinder pres- 
sure at mid-stroke. This is a very significant addition to the 
rail pressure. It was not included in the fi^gures observed in 
the tests since steam was shut off when the locomotive passed 
over the test track. 

In Fig. 225 are shown plotted results of tests with a loco- 
motive of the Santa Fe type (2-10-2). The diagram shows 
only the stresses under the main driver — which carries the crank 
pin. Corresponding diagrams for the other drivers showed very 
different results. The position of the counterweight is shown, 
indicating that, since the main driver was underbalanced, the 
maximum stress in the rail occurred when the crank pin was 
down and the counterweight up. Note that the minimima 
stress in the base of the rail was 14,000 lbs. per sq. in. for a speed 
of 50 m.p.h. The pressures for 5 m.p.h. were so nearly uniform 
that a mean average line to represent them was drawn at about 
10,700. The mean value of the ordinates for 50 m.p.h. indicated 
a mean stress of about 26,000 lbs. per sq. in. The difference 
between 10,700 and 26,000, or 15,300 lbs. per sq. in., is considered 
to represent the mean value of the effect of speed alone, or the 
effect of increasing the velocity from 5 to 50 m.p.h. without 
reference to counter-weight effects. Similar tests with a Pacific 
type locomotive (4-6-2) showed that the effect of speed is to 
increase the rail stress 1.95 and 2.25 times by ineyeasing the 
speed from 5 m.p.h. to 45 and to 60 m.p.h. respectively. But 
thes3 figures are of academic interest chiefly, since the critical 
figure is the maximum actual stress in the rail at high speed. 
The average maximum shown by the tests was 44,700 lbs. per 
sq. in. during each revolution of the drivers. One observed 
stress was as high as 52,000 lbs. per sq. in. 

It should also be noted that, for the type of rail used in this 
test, the maximum stress in the head of the rail is about 10% 
greater than that in the base for vertical loads. In these tests 
the strain measurements were taken in the base of the rail 
since, the lateral stresses are greater there than in the head and 
are " great enough to be significant." 

Trailer. These tests developed some very unexpected results 
with respect to the stresses under the trailers — the compara- 



§ 545. STRESSES IN TRACK. 611 

tively small loose wheels under, the firebox and next behind the 
drivers. These wheels are presumably perfectly balanced and 
normally carry a definite proportion of the load of the engine. 
It might have been expected that the rail pressure would be 
substantially uniform- and that any variation in pressure would 
be due to some accidental unevenness in the track. On tho 
contrary the variations were quite marked, especially at high 
speeds, and the positions of maximum stress seem to bear a 
definite relation to the position of the counterweight on the 
drivers. If this relation were constant for all locomotives, its 
analysis would be simplified. For a locomotive of the Santa F6 
type, the maximum stress occurred when the counterweight 
was at a position from 0.6 to 0.8 of a revolution after the low 
position. For a locomotive of the Pacific type, the maximum 
effect occurred at about 0.4 of a revolution after the low position. 
In each case the various observations of the tests were so consist- 
ent that the conclusions are indisputable. The differences in 
results for different locomotives shows that it depends on the 
relative weights on the wheels and on the equalizer system. 
The systematic variation from uniformity shows the effect of 
variable pressure of adjacent drivers, acting through the equaliz- 
ing levers, and also through the rails, to modify what would 
otherwise be a uniform pressure. It also helps to explain cer- 
tain apparent inconsistencies in the results for the driver pres- 
sures. Evidently there is a large field for future investigation, 
and it is to be expected that succeeding reports from this com- 
mittee will throw more light on this phase of the subject. 



APPENDIX. , 

THE ADJUSTMENTS OF INSTEUMENTS. : 

The accuracy of instrumental work may be vitiated by any 
one of a large number of inaccuracies in the geometrical relations 
of the parts of the instruments. Some of these relations are so 
apt to b*^ pltered by ordinary usage of the instrument that the 
makers have provided ad justing -screws so that the inaccuracies] 
may be readily corrected. There are other possible defects, 
which, however, will seldom be found to exist, provided the 
instrument was properly made and has never been subjected tc 
treatment sufficiently rough to distort it. Such defects, when 
found, can only be corrected by a competent instrument-maker 
or repairer. 

A WARNING is necessary to those who would test the accuracy 
of instruments, and especially to those whose experience in such 
work is small. Lack of skill in handling an instrument will 
often indicate an apparent error of adjustment when the real 
error is very different or perhaps non-existent. It is always a 
safe plan when testing an adjustment to note the amount of the 
apparent error ; then^ beginning anew, make another independent 
determination of the amount of the error. When two or more 
perfectly independent determinations of such an error are made 
it will generally be found that they differ by an appreciable 
amount. The differences may be due in variable measure to 
careless inaccurate manipulation and to instrumental defects 
which are wholly independent of the particular test being made. 
Such careful determinations of the amounts of the errors are 
generally advisable in view of the next paragraph. | 

Do NOT DISTURB THE ADJUSTING-SCREWS ANY MORE THAN 

NECESSARY. Although metals are apparently rigid, they are 
really elastic and yielding. If some parts of a complicated | 
mechanism, which is held together largely by friction, are sub- 
jected to greater internal stresses than other parts of the mech- 

612 



APPENDIX, 613 

anism, the jarring resulting from handling will frequently cause 
a slight readjustment in the parts which will tend to more nearly 
equalize the internal stresses. Such action frequently occurs 
with the adjusting mechanism of instruments. One screw may 
be strained more than others. The friction of parts may pre- 
vent the opposing screw from immediately taking iip an equal 
stress. Perhaps the adjustment appears perfect under these 
conditions Jarring diminishes the friction between the parts, 
and the unequal stresses tend to equalize. A motion takes place 
which, although microscopically minute, is sufficient to indicate 
an error of adjustment. A readjustment made by unskillful 
hands may not make the final adjustment any more perfect. 
The frequent shifting of adjusting-screws wears them badly, 
and when the screws are worn it is still more difficult to keep 
them from moving enough to vitiate the adjustments. It is 
therefore preferable in many cases to refrain from disturbing the 
adjusting-screws, especially as the accuracy of the work done is 
not necessarily affected by errors of adjustment, as may be 
illustrated : 

(a) Certain operations are absolutely unaffected by certain 
errors of adjustment. 

(6) Certain operations are so slightly affected by certain small 
errors of adjustment that their effect may properly be neglected. 

(c) Certain errors of adjustment may be readily allowed for 
and neutralized so that no error results from the use of the" 
unadjusted instrument. Illustrations of all these cases will be 
given under their proper heads. 

ADJUSTMENTS OF THE TRANSIT. 

1. To have the plate-bubbles in the center of the tubes when the 
axis is vertical. Clamp the upper plate and, with the lower 
clamp loose, swing the instrument so that the plate-bubbles are 
parallel to the lines of opposite leveling-screws. Level up until 
both bubbles are central. Swing the instrument 180°. If the 
bubbles again settle at the center, the adjustment is perfect. If 
either bubble does not settle in the center, move the leveling- 
screws until the bubble is half-way back to the center. Then, 
before touching the adjusting-screws, note carefully the position 
of the bubbles and observe whether the bubbles always settle at 
the same place in the tube, no matter to what position the in- 



614 RAILROAD CONSTRUCTION. 

strumerit may be rotated. AVhen the instrument is so leveled,! 
the axis is tnily vertical and the discrepancies between thig 
constant position of the bubbles and the centers of the tubes 
measure the errors of adjustment. I^y means of the adjusting- 
screws bring each bubble to the center of the tube. If this is 
done so skillfully that the true level of the instrument is not 
disturbed, the bubbles should settle in the center for all positions 
of the instrument. Under unskillful hands, two or more such 
trials may be necessary. 

When the plates are not horizontal, the measured angle is greater than 
the true horizontal angle by the difference between the measured angle 
and its projection on a horizontal plane. When this angle of inclination 
is small, the difference is insignificant. Therefore when the plate-bubbles 
«ire very nearly in adjustment, the error of measurement of horizontal 
angles may be far within the lowest unit of measurement used. A small 
>rror of adjustment of the plate-bubble perpendicular to the telescope will 
affect the horizontal angles by only a small proportion of the error, which 
will be perhaps imperceptible. Vertical angles will be affected by the 
Bame insignificant amount. A small error of adjustment of the plate- 
bubble parallel to the telescope will affect horizontal angles very slightly, 
but will affect vertical angles by the full amount of the error. 

All error due to unadjusted plate-bubbles may be avoided by noting in 
what positions in the tubes the bubbles will remain fixed for all positions 
of azimuth and then keeping the bubbles adjusted to these positions, for 
Ihe axis is then truly vertical. It will often save time to work in this way 
temporarily rather than to stop to make the adjustments. This should 
especially be done when accurate vertical angles are required. 

When the bubbles are truly adjusted, they should remain stationary 
regardless of whether the telescope is revolved with the upper plate loose 
and the lower plate clamped or whether the whole instrument is revolved, 
the plates being clamped together. If there is any appreciable difference, 
it shows that the two vertical axes or "centers" of the plates are not con- 
centric. This may be due to cheap and faulty construction or to the exces- 
sive wear that may be sometimes observed in an old instrument originally 
well made. In either case it can only be corrected by a maker. 

2. To make the revolving axis of the telescope perpendicular to 
the vertical axis of the instrument. This is best tested by using 
a long plumb-line, so placed that the telescope must be pointed 
upward at an angle of about 45° to sight at the top of the plumb- 
line and downward about the same amount, if possible, to 
sight at the lower end. The vertical axis of the transit must 
be made truly vertical. Sight at the upper part of the line» 
clamping the horizontal plates. Swing the telescope down 
and see if the crossr-wire again bisects the cord. If so, the 
adjustment is probably perfect (a conceivable exception will be 



APPENDIX. 615 

noted later) ; if not, raise or lower one end of the axis by m*. ana 
of the adjusting-screws, placed at the top of one of the standards, 
until the cross-wire will bisect the cord both at top and bottom. 
The plumb-bob may be steadied, if necessary, by hanging it 
in a pail of water. As many telescopes cannot be focus<^d 
on an object nearer than G or 8 feet from the telescope, this 
method requires a long plumb-line swung from a high point, 
which may be inconvenient. 

Another method is to set up the instrument about 10 feet 
from a high wall. After leveling, sight at some convenient 
mark high up on the wall. Swing the telescope down and make 
a mark (when working alone some convenient natural mark may 
g3neraUy be found) low down on the wall. Plunge the telescope 
and revolve the instrument about its vertical axis and again sight 
at the upper mark. Swing down to the lower mark. If the 
wire again bisects it, the adjustment is perfect. If not, fix a 
point half-way between the two positions of the lower mark. 
The plane of this point, the upper point, and the center of the 
instrument is truly vertical. Adjust the axis to these upper and 
lower points as when using the plumb-line. 

3. To make the line of collimation 'perpendicular to the revolving 
axis of the telescope. With the instrument level and the telescope 
nearly horizontal point at some well-defined point at a- distance 
of 200 feet or more. Plunge the telescope and establish a point 
in the opposite direction. Turn the whole instrument about the 
vertical axis until it again points at the first mark. Again 
plunge to "direct position" (i.e., with the level-tube under 
the telescope). If the vertical cross-wire again points at the 
second mark, the adjustment is perfect. If not, the error is 
one-fourth of the distance between the two positions of the 
second mark. Loosen the capstan screw on one side of the 
telescope and tighten it on the other side until the vertical 
wire is set at the one-fourth mark. Turn the whole instrument 
by means of the tangent screw until the vertical wire is midway 
between the two positions of the second mark. Plimge the 
telescop<3. If the adjusting has been skillfully done, the cross- 
wire should come exactly to the first mark. As an "erecting 
eyepiece" reinverts an image already inverted, the ring carrying 
the cross-wires must be moved in the same direction as tho 
apparent error in order to correct that error. 



616 EAILROAD CONSTRUCTION. 

The necessity for the third adjustment lies principally in the practice 
of producing a Line by plunging the telescope, but when this is required to 
be done with great accuracy it is always better to obtain the forward point 
by reversion (as described above for making the test) and take the mean 
of the two forward points. Horizontal and vertical angles are practically 
unaffected by small errors of this adjustment, unless, in the case of hori- 
zontal angles, the vertical angles to the points observed are very different. 

Unnecessary motion of the adjusting-screws may sometimes be avoided 
by carefully establishing the forward point on line by repeated reversions 
of the instrument, and thus determining by repeated trials the exact amount 
of the error. Differences in the amount of error determined would be 
evidence of inaccuracy in manipulating the instrument, and would show 
that an adjustment based on the first trial would probably prove unsatis- 
factory. 

The 2d and 3d adjustments are mutually dependent. If either adjust- 
ment is badly out, the other adjustment cannot be made except as follows! 

(a) The second adjustment can be made regardless of the third when 
the lines to the high v^oint and the low point make equal angles with the 
horizontal. 

(6) The third adjustment can be made regardless of the second when 
the front and rear points are on a level with the instrument. 

When both of these requirements are nearly fulfilled, and especially 
when the error of either adjustment is small, no trouble will be found in 
perfecting either adjustment on account of a small error in the other ad- 
justment. 

If the test for the second adjustment is made by means of the plumb- 
line and the vertical cross- wire intersects the line at all points as the tele- 
scope is raised or lowered, it not only demonstrates at once the accuracy 
of that adjustment, but also shows that the third adjustment is either 
perfect or has so small an error that it does not affect the second. 

4. To have the bubble of the telescope-level in the center of the 
tube when the line of collimation is horizontal. The line of colli- 
mation should coincide with the optical axis of the telescope. 
If the object-glass ^nd eyepiece have been properly centered, 
the previous adjustment will have brought the vertical cross- 
wire to the center of the field of view. The horizontal cross- 
fire should also be brought to the center of the field of view, 
and the bubble should be adjusted to it. 

a. Peg method. Set up the transit at one end of a nearly 
level stretch of about 300 feet. Clamp the telescope with its 
bubble in the center. Drive a stake vertically under the eye- 
piece of the transit, a,nd another about 300 feet away. Observe 
the height of the center of the e3^cpiece (the telescope being 
level) above the stake (caUing it a); observe the reading of the 
rod when held on the other stake (calling it b) ; take the instru- 
ment to the other stake and set it up so that the eyepiece is 



APPENDIX. 617 

vertically over the stake, observing the height, c ; take a reading 
on the first stake, calling it d. If this adjustment is perfect^ 
then 

a—d=h—Cf 
or (a-^)-(6-c)=0. 
Call (a-d)-{h-c)=2m. 
When m is positive, the line points downward} 
" m " negative, " " " upward. 

To adjust: if the line points up, sight the horizontal cross- 
wire (by moving the vertical tangent screw) at a point which is 
m lower, then adjust the bubble so that it is in the center. 

By taking several independent values for a, b, c, and d, a mean vaJuS 
for m is obtained, which is more reliable and which may save much ion- 
necessary working of the adjusting-screws. 

b. Using an auxiliary level. When a carefully adjusted level 
is at hand, this adjustment may sometimes be more easily 
made by setting up the transit and level, so that their lines of 
collimation are as nearly as possible at the same height. If a 
point may be found which is half a mile or niore away and 
which is on the horizontal cross-wire of the level, the horizontal 
cross-wire of the transit may be pointed directly at it, and the 
bubble adjusted accordingly. Any slight difference in the 
heights of the lines of collimation of the transit and level (say 
\") may almost be disregarded at a distance of | mile or more, 
or, if the difference of level would have an appreciable effect^ 
even this may be practically eliminated by making an estimated 
allowance when sighting at the distant point. Or, if a distant 
point is not available, a level-rod with target may be used at a 
distance of (say) 300 feet, making allowance for the carefully 
determined difference of elevation of the two lines of collimation. 

5. Zero of vertical circle. When the line of collimation is truly 
horizontal and the vertical axis is truly vertical, the reading 
of the vertical circle should be 0°. If the arc is adjustable, 
it should be brought to 0°. If it is not adjustable, the index 
error should be observed, so that it may be applied to aU readings 
of vertical angles. 

ADJUSTMENTS OF THE WYE LEVEL. 

1. To make the line of collimation coincide with the tenter of 
the rings. Point the intersection of the cross-wires at some 



618 RAILROAD CONSTRUCTION". 

f^rell-defmed point which is at a considerable distance. The in- 
strument need not be level, which allows much greater liberty 
in choosing a convenient point. The vertical axis should be 
clamped, and the clips over the wyes should be loosened and 
raised. Rotate the telescope in the wyes. The intersection of 
the cross-wires should be continually on the point. If it is not^ 
it requires adjustment. Rotate the telescope 180° and adjust 
one-half of the error by means of the capstan-headed screws that 
move the cross-wire ring. It should be remembered that, with, 
an erecting telescope, on account of the inversion of the image, 
the ring should be moved in the direction of the apparent error. 
A.djust the other half of the error with the leveling-screws. 
Then rotate the telescope 90°. from its usual position, sight 
accurately at the point, and then rotate 180° from that position 
and adjust any error as before. It may require several trials, 
but it is necessary to adjust the ring until the intersection of 
the cross-wires will remain on the point for any position of 
rotation. 

If such a test is made on a very distant point and again on a point only 
10 or 15 feet from the instrument, the adjustment may be found correct 
for one point and incorrect for the other. This indicates that the object- 
slide is improperly centered. Usually this defect can only be corrected by 
an instrument-maker. If the difference is very small it may be ignored, 
but the adjustment should then be made on a point which is at about the 
mean distance for usual practice — say 150 feet. 

If the whole image appears to shift as the telescope is rotated, it indi- 
cates that the eyepiece is improperly adjusted. This defect is likewise 
usually corrected only by the maker. It does not interfere with instru- 
mental accuracy, but it usually causes the intersection of the cross-wirea 
to be eccentric with the field of view. 

2. To make the axis of the level-tube parallel to the line of collv- 
mation. Raise the clips as far as possible. Swing the level 
so that it is parallel to a pair of opposite leveling-screws and 
clamp it. Bring the bubble to the middle of the tube by ineans 
of the leveling-screws. Take the telescope out of the wyes and 
replace it end for end, using extreme care that the wyes are not 
"jarred by the action. If the bubble does not come to the center, 
correct one-half of the error by the vertical adjusting-screws at 
one end of the bubble. Correct the other half by the leveling- 
screws. Test the work by again changing the telescope end for 
end in the wyes. 

Care should be taken while making this adjustment to see 



appendix/ 619 

that the level-tube is vertically under the telescope. With the 
bubble in the center of the tube, rotate the telescope in the wyes 
for a considerable angle each side of the vertical. If the first 
half of the adjustment has been made and the bubble moves, it 
shows that the axis of the wyes and the axis of the level-tube 
are not in the same vertical plane although both have been made 
horizontal. By moving one end of the level-tube sidewise by 
means of the horizontal screws at one end of the tube, the two 
axes may be brought into the same plane. As this adjustment 
is liable to disturb the other, both should be alternately tested 
until both requirements are complied with. 

By these methods the axis of the bubble is made parallel to 
the axis of the wyes; and as this has been made parallel to the 
lines of collimation by means of the previous adjustment, the 
axis of the bubble is therefore parallel to the line of collimation. 

3. To make the line of collimation 'perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling carefully 
over one pair of screws, revolve the telescope. 180°. If it is not 
level, adjust half of the error by means of the capstan-headed 
screw under one of the wyes, and the other half by the leveling- 
screws. Reverse again as a test. 

When the first two adjustments have been accurately made, good level- 
ing may always be done by bringing the bubble to the center by means of 
the leveling-screws, at every sight if necessary, even if the third adjust- 
ment is not made. Of course this third adjustment should be made as a 
matter of convenience, so that the line of collimation may be always level 
no matter in what direction it may be pointed, but it is not necessary to 
stop work to make this adjustment every time it is found to be defective. 

ADJUSTMENTS OF THE DUMPY LEVEL. 

\. To make the axis of the level-tube perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of levehng-screws. Then, after levehng care- 
fully over one pair of screws, revolve the telescope 180°. If 
it is not level, adjust one-half of the error by means of the adjust- 
ing-screws at one end of the bubble, and the other half by 
means of the leveling-screws. Reverse again as a test. 

2. To make the line of collimation perpeyidiculor to the vertical 
axis. The method of adjustment is identical with that for 
the transit (No. 4, pi. 505) except that the cross-wire must be 



620 RAILROAD CONSTRUCTION. 

adjusted to agree with the level-bubble rather than vice versa, an 
is the case with the corresponding adjustment of the transit; 
i.e., with the level-bubble in the center, raise or lower the hori- 
zontal cross-wire until it points at the mark known to be on 
a level with the center of the instrument. 

If the instrument has been well made and has not been dis- 
torted by rough usage, the cross-wires will intersect at the 
center of the field of view when adjusted as described. If they 
do not, it indicates an error which ordinarily can only be cor- 
rected by an instrument-maker. I'he error may be due to any 
one of several causes, which are 

(a) faulty centering of object-slide; 

ib) faulty centering of eyepiece; 

(c) distortion of instrument so that the geometric axis of 
the telescope is not perpendicTilar to the vertical axis. If the 
error is only just percepliblej it will not probably pause ^ny 
MTor i^ the work. 



AZIMUTH. 

The azimuth of a line on the surface of the earth is its ^ngle 
with a true meridian through a point on the line. It is the 
true bearing as distinguished from " magnetic bearing." Fed- 
eral law requires that all surveys of government lands shall be 
P^ade by " Solar Observations " (rather than with the magnetic 
needle) so as to obtain true bearings. . .jM 

Solar Azimuth may be obtained in two general ways, (a) by 
direct observation on the sun with an ordinary " complete " 
transit, provided with a colored glass shade, and (6) by the use 
of a '' solar attachment " or a solar compass. The first method 
only requires as special equipment a colored glass shade costing 
but a few dollars, but it requires the separate solution of a for-, j 
mula for each observation made. Even the colored glass shade 
is not always necessary — as when the disc of the sun is just seen 



APPENDIX. 



621 



through thin clouds and is not too bright to be observed with the 
naked eye. The " colored glass shade " may be merely a piece 
of colored glass fitted over the eye-piece, or the glass may be 
set into a frame very similar to the object glass cover and readily 
taken off and put on. In the latter case the glass must be 
" optically perfect/' i.e., with the sides perfectly plane and 
parallel, so that there shall be no refraction of the image, or 
such glass as is used for the sun shade of a sextant. 

The second method (h) does not require any calculation of a 
formula; the true meridian is given directly but it requires the 
use of a special instrument, whose adjustments must be made 
with great care or the resulting azimuth will often be in error by 
a much larger amount than the error in the adjustment. A 
proper appreciation of either method requires an understand- 
ing of certain astronomical relatione 




Fig. 1. 

Fig. 1 represents the orthographic projection of the celestial 
sphere, projected on the plane of the meridian of the observer. 
H P Z E represents the meridian of the observer. 

Z = the zenith. 
CP = the polar axis of the earth. 
CE = the plane of the equator. 

^ = the position of the sun. 
EZ = the latitude of the observer = (f>. 



ZP = 9O°-</> = co0. 



h. 



SG = the true altitude of the sun 

SZ = 90° -h = coh. 

ST = the declination of the sun, north or south of the equator 

= 5. 
SP = -90 ° — 5 = CO 5, also called p = polar distance. 



6^2 RAILROAD dONS'TRtJCTiON. 

The essential sign of 5 must be considered. If the sun is 
south of the equator (as it is from about September 21 to March 
21), 5 is negative and if the declination is (say) S 20°, 5= —20°. 
Then CO 5 = 90°- 5 = 90° -(-20°) = 110°. 

Z = the angle from the position of the sun to the true north = 
the spherical angle SZP. A is its supplement = 180° —Z. 

Of several possible formulae, the U. S. Coast and Geodetic 
Survey prefer the following : 



'4 



CoHA -l^^(S-^)siniS-h) 



cos S cos (S—p) 

in which S = ^{(})+h-\-p). 

The sun describes each day a path which is approximately 
parallel with the equator, the change in declination being very 
smaU during June and December and fastest when the sun is 
crossing the equator in March and September, the greatest rate 
of change being about 59 seconds of arc per hour. The declina- 
tion of the sun must be known for the time of observation. This 
is obtainable from the Nautical Almanac or Ephemeris. 

Example. — Declination for Philadelphia, Feb. 20, 1914, at 
8:10 A. M., standard time, 75th meridian. Since " standard 
time " is a definite time interval from Greenwich mean local 
time, we may use it here regardless of precise longitude or mean 
local time, 8:10 A. M. on the 75° meridian is 1:10 P. M. mean 
time, at Greenwich. 1.17/iX53".64 = 62".58 = 1' 2".6 and 
-11° 7' l".l+0° r 2".6= -11° 5' 58".5 which is south decli- 
nation. 

Refraction. Refraction causes the sun to appear higher than 
it actually is. Therefore when the altitude of the sun is observed, 
the computed refraction should be subtracted from the apparent 
altitude to obtain the true altitude. The amount of the refrac- 
tion is a very complicated function of the temperature and of 
the barometric pressure. For refined astronomical work, large 
refraction tables should be used, making due allowance for 
temperature and pressure, but for such work as may be done 
with an ordinary transit the values given in the following table 
will suffice. 

Angular diameter of sun. The sun's angular diameter is 
about 0° 32'. With the comparatively high power telescopes 
now generally used on transits, this fills a l9,rge part of the field 
of view and it is impossible to accurately bisect such a large 



APPENDIX. 



623 



MEAN REFRACTIONS — ^[bESSEL] TRUE FOR BAROMETER AT 29". 6, 

TEMP. 48° F. 



Alt. 


Refr. 


Alt. 


Refr. 


Alt. 


Refr. 


0° 0' 


34' 54" 


1° 30' 


20' 51" 


5° 0' 


9' 46" 


10 


32 49 


40 


19 52 


30 


9 02 


20 


30 52 


50 


18 58 


6 


8 23 


30 


29 03 


2 


18 09 


30 


7 49 


40 


27 23 


30 


16 01 


7 


7 20 


50 


25 50 


3 


14 15 


30 


6 53 


1° 


24 25 


30 


12 48 


8 


6 30 


10 


23 07 


4 


11 39 


30 


6 08 


20 


21 56 


30 


10 40 


9 


5 49 



Alt. 


Refr. 


Alt. 


Refr. 


Alt. 


Refr. 


9° 30' 

10 

11 

12 

13 

14 

15 

16 

17 


5' 32" 
5 16 
4 48 
4 25 
4 05 
3 47 
3 32 
3 19 
3 07 


18° 

19 

20 

21 

22 

23 

24 

26 

28 


2' 56" 
2 46 
2 37 
2 29 
2 22 
2 15 
2 09 
1 58 
1 48 


30° 

35 

40 

45 

50 

60 

70 

80 

90 


1' 40" 
1 22 
1 09 
58 
48 
33 
21 
10 




P.M 



A.M. 




^ ^x^;^ 

>^-> 



angular width especially as the apparent motion of the sun across 
the field of view is very rapid. It therefore becomes advisable 
(when sighting directly at the sun with the transit telescope) to 
sight the cross wires on the edges of the 
sun, as shown in Fig. 2, and make due 
allowance for the semi-diameter of the 
sun. The effect of this is to obtain an 
altitude which differs from the true alti- 
tude by the angular value of the semi- 
diameter. The observed azimuth differs 
from the true azimuth by the semi- 
diameter -^ cos h. When the sun is at 
the horizon, cos /i = 1, and the allowance equals the semi-diameter 
both for altitude and azimuth. For higher altitudes the allow- 
ance for azimuth is much larger than the semi-diameter, since 
the divisor (cos h) is small. If several observations are taken 
within a short interval, the change in this allowance for azimuth 
during this short interval may be too small for notice and one 
value may be sufficiently accurate for all the observations. 

There is a slight variation in the semi-diameter as is shown in 
the accompanying tabular form, giving average values, which 



Fig. 2. 



62^i 



RAILROAD CONSTRUCTION. 



may be used by interpolation, if a closer value than the nearest 
minute is desired. 



Time. 


Semi-diam. of the Sun 
in minutes of arc. 


Jan. 1 . . . . 
Ap-il 1 . . . 

July 1 

Oct. 1 


16'.30 (max) 
16.03 

15 .76 (min) 

16 .01 



Latitude. If the latitude of the place of observation is not 
known to the nearest minute, it may readily be obtained by 
observing the altitude of the sun at culmination at noon. The 
horizontal cross wire should be sighted at the upper (or the lower) 
edge of the disc of the sun. 

li d = angular diameter of sun r = refraction 

4> = latitude 6 = declination 

h' = observed angle of elevation 

then 4> = 90° - [h' -r-8±^d] 

§ii which I (? is 4 for an observation on the lower edge, 
and I d is — for an observation on the Upper edge. 

Set up the transit several minutes before noon, taking sufficient 
time to level up with the utmost care. Set the horizontal cross 
wire on the upper (or lower) edge of the sun and with the tangent 
screw follow the motion of the sun. As the required angle is 
found at culmination, the motion of the telescope should cease 
when the highest altitude is obtained and the sun begins to 
descend. 

Azimuth by an Observation with the transit telescope. Set 
up the transit at a convenient station from which an unobstructed 
view of the sun may be obtained at all times and from which a 
convenient permanent azimuth mark (e.g., a distant steeple or 
chimney) may be observed. Point at the azimuth mark with 
the horizontal plates reading zero. With the upper plate loose, 
point at the sun observing the time, altitude and the horizontal 
angle from the azimuth mark*. Three or more such observations 
are generally advisable, especially as they are so easily and 
quickly taken and are such a valuable check on each other. A 
single observation may be vitiated by some inaccuracy or blunder 



APPENDIX. 



625 



in manipulation or reading which would not be discovered unless 
more than one observation is taken, in which case the error 
would hardly be precisely repeated both in nature and amount. 
Finally, point at the azimuth mark to test whether the lower 
plate has slipped. The reading on the azimuth mark should 
be 0°. 

Reducing the Observations. Compute the declinations for 
the given times of observation. If several observations are 
taken, it is generally best to compute the declinations for the 
times of the first and last observations and interpolate for the 
others. The observations may most readily be reduced by using 
a regular form as given below. The six observations quoted 
were taken in 15 minutes by one of the author's students. 













1 


Semi- 






Apparent 










diam. 


True Azi. 


Time 


Altitude 


a 


h 


5 


1 Z 

\ 


cos.ap. 
alt. 


of Mark. 


4:50 


22° 48'.5 


287° 41' 


22° 30'. 3 


14° 45'. 6 


89° 16'. 6 


17'. 2 


213° 19'.6 


4:53 


22 12 .5 


238 11 


21 54 .3 


45 .6 


88 46 .0 


17 .2 


19 .6 


4:55 


21 44 .5 


238 34 


21 26 .2 


45 .6 


'88 23 .3 


17 .1 


19 .8 


4:. 58 


21 19 .0 


238 55 


21 .7 


45 .7 


88 02 .4 


17 .1 


19 .7 


5:00 


20 49 .5 


239 19 .5 


20 31 .1 


45 .7 


87 38 .0 


17 .0 


19 .5 


5:03 


20 28 .0 


239 38 .0 


20 9 .5 


14 45 .7 


p 19 .9 

1 


17 .0 


213 19 .1 



Mean = 213°19'.55. 

Observations taken Apr. 29, 1897: Semi-diam. of Sun 15'.9, 
Sun observed in lowei* left-hand corner. 

a = horizontal angle to azimuth mark, the angle being measured 

to the right. 
/i = app. alt.— refraction — semi-diam. of sun; semi-diam. is 

-fwhen sun is above hor. cross wire, —when below. 
6 ~ declination, and Z = computed angle (as illustrated below). 



1 c* 1 o Semi-diam. 

True azimuth of mark = 540 d= — ±Z— a, m which 

cos. app. alt. 

r T^ ^T 11 Semi-diam. . 
Z 18+ for A. M. and— for P. M, and the — — -is + when tha 



COS. app. alt. 



sun is on the left of the middle wire (as above); - 



Semi-diam. 



is— when the sun is on the right of the middle wire. 



cos. app. alt. 



626 RAILROAD CONSTRUCTION. 

As a numerical specimen of the reduction :— App. decl. Green- 
wich mean noon Apr. 29, 1897, 14° 38'.0; hourly change +077; 
diff. of time between Greenwich and Philadelphia 5.0 hours; 
5 P. M. at Philadelphia = 10 P. M. at Greenwich; therefore 5 
for 5 P. M. at Philadelphia = 14° 38'.0 + 10X0'.77 = 14° 45'.7. 
Using the equation 



4 



cot ^A = 
in which S = ^ (0+^+p)- 

^= 39° 5S'.0 s-0 = 

h= 22° 30'. 3 s-h = 

p= 75° 14'.3 s-p=- 6° 23'.0, 



sin (S—<j>) sin (S — h) 



cos S cos (S — p) 



<i>= 39° 5S'.0 s-0= 28° 53'. 3, sin = 9. 68404 

h= 22° 30'. 3 s-h= 46° 21'.0, sin = 9. 85948 



137° 42'.6 

8= 68° 51'.3 cos 68° 51'. 3 = 9. 55718 
cos- 6° 23'.0 = 9.99730 



9 . 54352 



9.55448 9 . 55448 

^A=45°21'.7 _2^| 9.98904 

^=90° 43'. 4 9.99452= cot 45° 21'.6 

Z=89° 16'. 6 

Semi-diam. Sun 15.9 



— = 17'.2 
COS. app. alt. cos 22° 48' 

540°+17'.2= 540° 17'.2 
-Z-a = -89° 16'.6-237° 41' = -326° 57'.6 



213° 19'. 6 =true azimuth of mark. 

The instrument used had a vertical circle reading 30" directly 
and could be estimated to 15". 



EXPLANATORY NOTE ON THE USE OF THE TABLES, 

The loGjarithms here given are "five-place," but the last 
figure sometimes has a special mark over it {e.g., 6) v/hich indi- 
cates that one-half a unit in the last place should be added. 
For example 



the value 
.69586 
.69586 



includes all values between 
•6958575000 + and .6958624999. 
.6958625000 + and ^958674999. 



The maximum error in any one value therefore does not 
exceed one-quarter of a fifth-place unit. 

When adding or subtracting such logarithms allow a half-unit 
for such a sign. For example 

.69586 .69586 .69586 

.10841 .10841 .10841 

.12947 .12947 .1294? 



.93372 .93375 .93375 

All other logarithmic operations are performed as usual and 
are supposed to be understood by the student, 

627 









TABLE 


I.— RADII OF CURVES. 








Deg 


0° 




1° 


2° 


3° 


Deg 


Min 


Radiu 

00 


s. Log li 


Radius. 


.Log JB 


Radius. 
2864.9 


LogiJ 


Radius. 


Logi? 


Min 





00 


5729-6 


3-75813 


3.45711 


1910.1 


3. 28105 





1 


343775 


5-53627 


5635-7 


-75095 


2841.3 


.45351 


1899-5 


.27864 


1 


2 


171887 


5-23524 


5544-8 


•74389 


2818.0 


.44993 


1889-1 


.27625 


2 


3 


114592 


5.05915 


5456-8 


•73694 


2795.1 


.44639 


1878-8 


.27387 


3 


4 


85944 


4.93421 


5371-6 


•73010 


2772-5 


.44287 


1868-6 


.27151 


4 


5 


68755 
57298 


4. 83730 


5288-9 


•72336 


2750-4 


•43939 
3^43593 


1858-5 


.26915 


5 
1 


6 


4-75812 


5208-8 


3-71673 


2728-5 


1848-5 


3-26681 


7 


49111 


.69117 


5131-0 


•71020 


2707.0 


•43249 


1838-6 


-26448 


7 


8 


42972 


•63318 


5055-6 


•70377 


2685.9 


•42909 


1828-8 


.26217 


8 


9 


38137 


.58203 


4982-3 


•69743 


2665.1 


•42571 


1819-1 


.25986 


9 


10 


3*1377 
31252 


.53627 


4911-2 


•69118 


2644-6 


-42235 


1809-6 


-25757 


10 
11 


11 


4.49488 


4842-0 


3-68502 


2624.4 


3^41903 


1800-1 


3-25529 


12 


28648 


.45709 


4774-7 


-67895 


2604-5 


•41572 


1790-7 


-25303 


12 


13 


26444 


.42233 


4709-3 


-67296 


2584-9 


•41245 


1781-5 


.25077 


13 


14 


24555 


.39014 


4645-7 


-66705 


2565.6 


•40919 


1772-3 


.24853 


14 


15 


22918 


.36018 


4583-8 


-66122 


2546.6 


.40597 


1763-2 


•24629 


15 
18 


16 


21486 


4.33215 


4523-4 


3-65547 


2527-9 


3.40276 


1754-2 


3 •24407 


17 


20222 


.30582 


4464-7 


•64979 


2509.5 


.39958 


1745-3 


•2418Q 


17 


18 


19099 


.28100 


4407-5 


-64419 


2491.3 


.39642 


1736-5 


•23967 


18 


19 


18093 


-25752 


4351.7 


-63865 


2473-4 


.39329 


1727-8 


.23748 


19 


20 


17189 


.23524 


4297.3 


-63319 


2455.7 


•3901? 
3-38708 


1719-1 


.23530 


20 
21 


21 


16370 


4.21405 


4244.2 


3-62780 


2438-3 


1710-6 


3.23314 


22 


15626 


.19385 


4] 92- 5 


-62247 


2421.1 


•38401 


1702-1 


•23098 


22 


23 


14947 


.17454 


4142.0 


-61720 


2404. 2 


•38097 


1693-7 


•22884 


23 


24 


14324 


.15606 


4092.7 


-61200 


2387-5 


•37794 


1685-4 


.22670 


24 


25 


13751 
13222 


-13833 


4044 . 5 
3997-5 


-60686 


2371-0 


.37.494 


1677-2 
1669-1 


.22458 


25 
26 


.26 


4-12130 


3-60178 


2354-8 


3.37195 


3-22247 


27 


12732 


-10491 


3951-5 


-59676 


2338-8 


.36899 


1661-0 


.22037 


27 


28 


12278 


-08911 


3906-6 


-59180 


2323-0 


.36604 


1653-0 


.21827 


28 


29 


11854 


•07387 


3862-7 


-58689- 


2307-4 


.36312 


1645-1 


.21619 


29 


-30 


11459 
11090 


•05915 
4- 04491 


3819-8 
3777-9 


-58204 


2292.0 


•36021 
3-35733 


1637-3 


.21412 


30 
31 


31 


3-57724 


2276-8 


1629-5 


3-21206 


S2 


10743 


■03112 


3736-8 


-57250 


2261-9 


•35446 


1621-8 


.21000 


32 i 


33 


10417 


-01776 


3696-6 


-56780 


2247-1 


.35162 


1614-2 


.20796 


33 


34 


10111 


4-00479 


3657-3 


-56316 


2232-5 


.34879 


1606-7 


-20593 


34 


35 


9822 


.2 3-99221 


3618-8 


-55856 


2218-1 
2203-9 


.34598 
3.34318 


1599-2 


-20300 


35 
36 


36 


9549 


•3 397997 


3581-1 


3-55401 


1591-8 


3-20189 


37 


9291 


-3 -96807 


3544-2 


•54951 


2189-8 


•34041 


1584-5 


.19988 


37 


38 


9046 


•7 -95649 


35^8-0 


•54506 


2176-0 


•33765 


1577-2 


•19789 


38 


39 


8814 


-8 -94521 


3472-6 


•54065 


2162-3 


•33491 


1570-0 


.19590 


39 


40 


8594 


.4 -93421 


3437- 9 


•53629 


2148-8 


.33219 
3.32949 


1562-9 


•19392 


40 
41 


41 


8384 


-8 3-92349 


3403-8 


3-53197 


2135-4 


1555-8 


^•19195 


42 


8185 


-2 -91302 


3370-5 


-52769 


2122-3 


.32680 


1548-8 


.18999 


42 


43 


7994 


8 .90281 


3337-7 


-52345 


2109-2 


.32412 


1541-9 


.18804 


43 


44 


7813 


1 -89282 


3305-7 


•51925 


2096.4 


.32147 


1535-0 


.18610 


44 


45 


7639 


5 -88306 


3274-2 


•51510 


2083.7 


.31883 


1528-2 
1521-4 


•18417 


45 
46 


46 


7473 


4 3-87352 


3243-3 


3^51098 


2071.1 


3.31621 


3-18224 


47 


7314 


4 -86418 


3213-0 


•50691 


2058.7 


•31360 


1514-7 


•18032 


47 


48 


7162 


-85503 


3183-2 


•50287 


2046.5 


.31101 


1508-1 


•17842 


48 


49 


7015 


9 -84608 


3154-0 


•49883 


2034.4 


.30843 


1501-5 


•17652 


49 


50 


6875 


6 -83731 


3125-4 


•49490 


2022.4 
2010 • 6 


.30587 


1495-0 


•17462 


50 
51 


51 


6740 


7 3-82871 


3097-2 


3^49097 


3.30332 


1488-5 


3.17274 


52 


6611 


1 -82027 


3069-6 


.48707 


1998-9 


.30079 


1482-1 


.17087 


52 


53 


6486 


4 ^81200 


3042-4 


.48321 


1987^3 


.29827 


1475-7 


.16900 


53 


54 


6366. 


3 •80388 


3015-7 


.47939 


1975^9 


.29577 


1469-4 


.16714 


54 


55 


6250. 
6138. 


5 .79591 


2989-5 


.47559 


1964.6 


•29328 
3 • 29081 


1463-2 


.16529 


55 
56 


56 


9 3-78809 


2963-7 


3.47183 


1953.5 


1457-0 


3-16344 


57 


6031. 


2 -78040 


2938-4 


.46811 


1942.4 


•28835 


1450.8 


.16161 


57 


58 


5927. 


2 -77285 


2913-5 


•46441 


1931.5 


.28590 


1444.7 


•15978 


58 


59 


5826. 


8 .76542 


2889-0 


.46075 


1920.7 


.28347 


1438.7 


.15796 


59 


60 


5729. 


•6 .75813 


2864-9 


.45711 


1910.1 


.28105 


1432.7 


.15615 


60 










62 


8 



















TABLE I.- 


-RADII OF CURVES. 










Deg 


4° 


5° 


6° 




7" 




Deg 


Min 


Radius. L 


.ogi? 


Radi 


US. 1 


.Og-B 
05929 


Radius. L 


.ogM 
98017 


Radius. L 


OgM 
91329 


Min 





1432.7 3 


15615 


1146 


3 3 


955-37 2 


819-02 2- 





1 


142G 


7 


15434 


1142 


5 


05784 


952 


72 


97896 


817- 


08 - 


91226 


1 


2 


1420 


8 


15255 


1138 


7 


05640 


950 


09 


97776 


815. 


14 - 


91123 


2 


3 


1415 





15076 


1134 


9 


05497 


947 


48 


97657 


813- 


22 - 


91021 


3 


4 


1409 


2 


14897 


1131 


2 


05354 


944 


88 


97537 


811. 


30 - 


90918 


4 


5 


1403 


5 


14720 


1127 


5 


0506§ 


942 


29 


97418 


809. 


40 - 


90816 


5 


6 


1397 


8 3 


14543 


1123 


8 3 


939 


72 2 


97300 


807. 


50 2- 


90714 


6 


7 


1392 


1 


14367 


1120 


2 


04928 


937 


16 


97181 


805. 


61 


90612 


7 


8 


1386 


5 


14191 


1116 


5 


04787 


934 


62 


97063 


803. 


73 


90511 


8 


9 


1380 


9 


14017 


1112 


9 


04646 


932 


09 


96945 


801. 


86 - 


90410 


9 


10 


1375 


4 


13843 
13669 


1109 


3 


04506 


929 


57 


96828 


800. 


00 


90309 


10 


11 


1389 


9 3 


1105 


8 3 


04366 


927 


07 2 


96711 


798. 


14 2- 


90208 


11 


12 


1334 


5 


13497 


1102 


2 


04227 


924 


58 


96594 


796. 


30 - 


90107 


12 


13 


1359 


1 


13325 


1098 


7 


04088 


922 


10 


96478 


794. 


46 - 


90007 


13 


14 


1353 


8 


13154 


1095 


2 


03949 


919 


64 


96361 


792. 


63 - 


89907 


14 


15 


1348 


4 


12983 


1091 


7 


03811 


917 


19 


96246 


790. 


81 


89807 


15 


16 


1343 


2 3 


12813 


1088 


3 3 


03674 


914 


75 2 


96130 


789. 


00 2. 


89708 


16 


17 


1338 





12644 


1084 


8 


03537 


912 


33 


96015 


787. 


20 . 


89608 


17 


18 


1332 


8 


12475 


1081 


4 


03400 


909 


92 


95900 


785. 


41 . 


89509 


18 


19 


1327 


6 


12307 


1078 


1 


03264 


907 


52 


95785 


783. 


62 . 


89410 


19 


20 


1322 


5 


12145 


1074 


7 


03128 


905 


13 


95671 


781. 


84 


89312 


20 


21 


1317 


5 3 


11974 


1071 


3 3 


02992 


902 


76 2 


95557 


780. 


07 2. 


89213 


21 


22 


1312 


4 


11808 


1068 





02857 


900 


40 


95443 


778. 


31 


89115 


22 


23 


1307 


4 


11642 


1064 


7 


02723 


898 


05 


95330 


776. 


55 . 


89017 


23 


24 


1302 


5 


11477 


1061 


4 


02589 


895 


71 


95217 


774. 


81 


88919 


24 


25 


1297 


6 


11313 


1058 


2 


02455 


893 


39 - 


95104 


773. 


07 


88821 

88724 


25 


26 


1292 


7 3 


11150 


1054 


9 3 


02322 


891 


08 2 


94991 


771. 


34 2. 


26 


27 


1287 


9 


10987 


1051 


7 


02189 


888 


78 


94879 


769. 


61 


88627 


27 


28 


1283 


1 


10825 


1048 


5 


02056 


886 


49 


94767 


767. 


90 ■ 


88530 


28 


29 


1278 


3 


10663 


1045 


3 


01924 


884 


21 - 


94655 


766. 


19 - 


88433 


29 


30 


1273 


6 


10502 


1042 
1039 


1 
3 


01792 
01661 


881 


95 - 


94544 


764. 


49 


88337 


30 


31 


1268 


9 3 


10341 


879 


69 2- 


94433 


762. 


80 2. 


88241 


31 


32 


1264 


2 


10182 


1035 


9 


01530 


877 


45 ■ 


94322 


761- 


11 . 


88145 


32 


33 


1259 


6 


10022 


1032 


8 


01400 


875 


22 


94212 


759- 


43 . 


88049 


33 


34 


1255 





09864 


1029 


7 


01270 


873 


00 - 


94101 


757- 


76 . 


87953 


34 


35 


1250 


4 


09705 


1026 
1023 


6 


01140 


870 


80 - 


93991 


756- 


10 


87858 


35 


36 


1245 


9 3 


09548 


5 3 


01010 


868 


60 2. 


93882 


754. 


44 2. 


87762 


36 


37 


1241 


4 


09391 


1020 


5 


00882 


866 


41 - 


93772 


752. 


80 


87668 


37 


38 


1236 


9 


09234 


1017 


5 


00753 


864 


24 - 


93663 


751. 


16 . 


87573 


38 


39 


1232 


5 


09079 


1014 


5 


00625 


862 


07 - 


93554 


749. 


52 - 


87478 


39 


40 


1228 


1 


08923 


1011 


5 


00497 


859 


92 - 


93446 


747. 


89 


87384 


40 


41 


1223 


.7 3 


08789 


1008 


6 3 


00370 


857 


78 2- 


93337 


746- 


27 2. 


87290 


41 


42 


1219 


4 


08614 


1005 


6 


00242 


855 


65 - 


93229 


744- 


66 . 


87198 


42 


43 


1215 


1 


08461 


1002 


7 3 


00116 


853 


53 


93122 


743- 


06 . 


871G2 


43 


44 


1210 


8 


08308 


999 


76 2 


99989 


851 


42 - 


93014 


741. 


46 . 


87008 


44 


45 


1206 


6 


08155 


996 


87 


99863 


849 


32 


92907 


739. 


88 . 


86915 
86822 


45 


4B 


1202 


4 3 


08003 


993 


99 2 


99738 


847 


23 2 


92800 


738. 


28 2. 


46 


47 


1198 


2 


07852 


991 


13 


99613 


845 


15 


92693 


736. 


70 


86729 


47 


48 


1194 





07701 


988 


28 


99488 


843 


08 


92587 


735. 


13 


86636 


48 


49 


1189 


9 


07550 


985 


45 


99363 


841 


02 


92480 


733 


58 


86544 


49 


50 


1185 


8 


07400 


982 


64 


99239 


838 


97 


92374 


732 


01 


86451 


50 


51 


1181 


7 3 


07251 


979 


84 2 


99115 


836 


93 2 


92269 


730 


45 2 


86359 


51 


52 


1177 


7 


07102 


977 


03 


98992 


834 


90 


92163 


728 


91 


86267 


52 


53 


1173 


6 


06954 


974 


29 


98869 


832 


89 


92058 


727 


37 


86175 


53 


54 


1169 


7 


06806 


971 


54 


98746 


830 


88 


91953 


725 


84 


86084 


54 


55 


1165 


7 


06658 


968 


81 


98624 


8?8 


88 


91849 


724 


Rl 


85992 


55 


56 


1161 


8 3 


06511 


966 


09 2 


98501 


828 


89 2 


-91744 


722 


-79 2 


.85901 


56 


57 


1157 


9 


06385 


963 


39 


98380 


824 


91 


-91640 


721 


-28 


.8581(1 


57 


58 


1154 





06219 


960 


70 


98258 


822 


93 ( 


.91536 


719 


-77 


.85719 


58 


59 


1150 


1 


06074 


958 


03 


98137 


820 


9V 


-91433 


718 


.27 


-85629 


59 


60 


1146 


3 


05929 


955-37 


98017 


819-02 


.91329 


716.78 


•85538 


60 



629 







_. 


„.. 




TABLE 


1. 


— HADII OF CURVES. 












beg. 


8° 


9° 


10° 


11° 


Deg 


Min. 


Radius. 


logB 


Radius. 


logH 


Radius. 


log a 


Radius. 


logH 


Min 




1 

2 
3 
4 
5 


716 
715 
713 
712 
710 
709 


■ 78 

• 29 
81 

• 34 

■ 87 
.40 


2 


■85538 
.85448 
.85358 
.85268 
.85178 
.85089 


637 
636 
634 
633 
632 
631 

630 
629 
627 
626 
_625 

624" 

623 

622 

621 

620 


■ 27 

■ 10 

■ 93 

■ 76 

■ 60 
.44 

29 
14 
99 
85 
71 

58 

45 
32 
20 
09 


2 


80432 
80352 
80272 
80192 
80113 
80033 


573 
572 
571 
570 
569 
568 


69 
73 
78 
84 
90 
96 


2 


75867 
75795 
75723 
75651 
75579 
75508 


521 
520 
520 
519 
518 
517- 


67 
88 
10 
32 
54 
76 


2 


71739 
71674 
71608 
71543 
71478 
71413 



1 
2 
3 
4 
5 


6 
7 
8 
9 
10 


707 
706 
705 
703 
702 


95 
49 
05 
61 
17 


2 


85000 
84911 
84822 
84733 
84645 


2 


79954 
79874 
79795 
79716 
79637 


568 
567 
566 
565 
564 


02 
09 
16 
23 
31 


2 


75436 
75365 
75293 
75222 
75151 


516 
516 
515 
514 
513 


99 
21 
44 
68 
91 


2 

"2. 


71348 
71283 
71218 
71153 
71088 

71024 
70959 
70895 
70831 
70767 


6 
7 
8 

9 
10 


11 
12 
13 
14 
15 


700 
.699 
697 
696 
695 


75 
33 
91 
50 
09 


2 


84556 
84468 
84380 
84292 
84204 


2 


79558 
79480 
79401 
79323 
79245 


563 
562 
561 
560 
559 


38 

47 
55 
64 
73 


2 


75080 
75009 
74939 
74868 
74798 


513. 

512 

511 

510. 

510 


15 
38 
63 
87 
11 


11 
12 
13 
14 
15 


16 
17 
18 
19 
20 


693 
692 
G90 
689 
G88 


70 
30 
91 
53 

IG 


2 

y 

2" 


84117 
84029 
83942 
83855 
83768 

83682 
83595 
83509 
83423 
83337 

83251 
83166 
83080 
82995 
82910 


618 
617 
616 
615 
614 


97 
87 
76 
66 
56 


2 

y 


79167 
79089 
79011 
78934 
78856 

78779 
78702 
78625 
78548 
78471 


558 
557 
557 
556 

555 


82 
92 
02 
12 
23 


2 


74727 
74657 
74587 
74517 
74447 


509 

508 

507. 

507. 

506. 


36 
61 
86 
12 
38 


2. 


70702 
70638 
70575 
70511 
7C447 


16 
17 
18 
19 
20 


21 
22 
23 
24 
25 


GOG 
G85 
684 
002 
G81 


78 
42 
06 
70 
35 


613 
612 
611 
610 
609 

608 
606 
605 
604 
603 


47 
38 
30 
21 
14 

06 
99 
93 
86 
80 


554 
553 
552 
551 
550 


34 
45 
56 
68 
80 


2 


74377 
74307 
74238 
74168 
74099 


505. 

504. 

504 

503 

502 


64 
90 
16 
42 
69 


2 


7C383 
70320 
70257 
70193 
70130 


21 
22 
23 
24 
25 


26 
27 
28 
29 
30 


C30 
G78 
G77 
G76 
074 


01 
67 
34 
01 
69 


2 


78395 
78318 
78242 
78165 
78089 


549 
549 
548 
547 
546 


92 
05 
17 
30 
44 


2 


74030 
73961 
73892 
73823 
78754 


501. 

501 

500 

4G9 

499 


96 
23 
51 
78 
06 


2 


70067 
700C4 
69941 
6S878 
69815 


26 
27 
28 
29 
30 


31 
32 
33 
34 
35 


673 
672 
670 
669 
638 


37 
06 
75 
45 
15 


2 


82825 
82740 
82656 
82571 
82487 


602 
601 
600 
599 
598 


75 
70 
65 
61 
57 


2 


78013 
77938 
77862 
77786 
77711 


545 
544 
543 
543 
542 


57 
■ 71 
86 
00 
15 


2 


73685 
73617 
73548 
73480 
73412 


4S8 
497 
496 
496 
495 


34 
62 
91 
19 
48 


2 


69752 
69690 
69627 
69565 
69503 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


666 
665 
664 
663 
681 


86 
57 
29 
01 
74 


2 


82403 
82319 
82235 
82152 
82068 


597 
596 
595 
594 
593 


53 
50 
47 
44 
42 


2 


776.36 
77561 
77486 
77411 
77336 


541 
540 
539 
538 
537 


■ 30 
.45 
.61 
.76 
.92 


2 


73343 
.73275 
•73207 
.73140 

73072 


494 
494 
493 
492 
491 


77 
07 
36 
66 
96 


2 


69440 
69378 
69316 
.69254 
69192 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


660 
659 
657 
656 
655 


47 
21 
95 
69 
45 


2 


81985 
81902 
81819 
8173G 
81653 


592 
591 
590 
589 
588 


40 
38 
37 
36 
36 


2 

y 


77261 
77187 
77112 
77038 
76964 

76890 
76816 
76742 
7G669 
76595 


537 
536 
535 
534 
533 


.09 
25 
42 
59 
77 


2 


.73004 
.72937 
.72869 
.72802 
.72735 


491 
4C0 
4S9 
489 
488 


26 
56 
86 
17 
48 


2 


69131 
.69069 

69007 
.68946 

68884 


41 
42 
43 
44 
45 


46 
47 
48 
49 
50 


654. 
652. 
651. 
650. 
G49. 


20 
96 
73 
50 
27 


2 


81571 
81489 
81406 
81324 
81243 


5C7 
586 
585 
584 
583 


36 
36 
36 
37 
38 


532 
532 
531 
530 
529 


94 
12 
30 
49 
67 


2 


72668 
72601 
72534 
.72467 
72401 


4C7 
487 
486 
485 
485 


79 
• 10 
42 
73 
05 


2 


68823 
.68762 
68701 
68640 
C8r.79 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


648. 
646. 
645. 
644. 
643 


05 
84 
63 

42 
22 


2 


81161 
81079 
80998 
80917 
80836 


582 
581 
580 
579 
_57_8. 
577 
576 
575 
574 
573 


40 
42 
44 
47 
49 

53 
56 
60 
64 
69 


2 


76522 
76449 
76376 
76303 
76230 


528 
528 
527 
526 
525 


86 
05 
25 
44 
64 


2 


72334 
72267 
72201 
72135 
72069 


484 
483 
483 
482 
481 


37 
69 
02 
84 
67 


2 

y 


68518 
68457 
683DC 
68335 
68275 

68214 
68154 
68094 
68033 
67973 


51 
52 
53 
54 
55 


56 
57 
58 
')9 
dO 


642 
640 
639 
638 
637 


02 
83 
-64 
'45 
27 


2. 


80755 
80674 
80503 
80513 
80432 


2 


76157 
76084 
76012 
75939 
75867 


524 

524. 

523 

522 

£21 


84 
05 
25 
46 
67 


2 


72003 
71937 
71871 
71805 
71739 


481 
480 
479 
479 
478 


00 
33 
67 
00 
34 


56 
57 
58 
59 
60 



630 



• 










TABLE I. 


—RADII OF CURVES. 










Deg. 


Radius. 


Log-R 


Deg. 


Radius. 


Log JJ 


Deg. 


Radius. 


log It 


Deg. 

31° 

10 

20 
30 
40 
50 

33° 
10 
20 
30 
40 
50 

33° 

10 
20 
30 
40 
50 

34° 
10 
20 
30 
40 
50 

35° 
30 

36° 
30 

37° 
30 

38° 
30 

39° 

30 

30° 

30 

31° 

33 

33 

34 

35 

36 
37 

38 
39 
40 

41 
43 
43 
44 
45 

46 

47 
48 
49 
50 

53 
54 
56 
58 
60 


Radius 


Log 18 


12° 

2 
4 
6 
8 


478 
477 
475 
474 
473 


34 
02 
71 
40 
10 


2 


.67973 
67853 
67734 
67614 
67495 


14° 

2 
4 
8 
8 

10 
12 
14 
18 
18 
20 
22 
24 
26 
28 

30 
32 
34 
36 
38 

40 
42 
44 
46 
48 

50 
52 
54 
56 
58 

15° 

2 
4 
8 
8 

10 
12 
14 
16 
18 

20 
22 
24 
26 
28 

30 
32 
34 
36 
38 

4C 
42 
44 
46 
48 

50 
52 
54 
56 
58 

16° 


410 
409 
408 
407 
406_ 

405 
404 
403 
402 
401 


28 
31 
34 
38 

_42 

47 
53 
58 
65 
71 


2 


.81307 
81205 
61102 
6100G 
6089S 


16° 

5 
10 
15 

20 
■ 25 

30 
35 
40 
45 
50 
55 

17° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

18° 
5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

19° 

5 
10 
15 
20 
25 

30 
35 

40 
45 
50 
55 

30° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

31° 


359 
357 
355 
353 
351 
350 


.28 
42 
59 
77 
98 
21 


2 


.55541 
•55317 
.55094 
.54872 
.54652 
54432 


274 
272 
270 
268 

2G6 
264 


37 

• 23 
13 
06 
02 
02 


2. 43835 

43494 

.43157 

.42823 

.42492 


10 


471 
470 
469 
467 
466 


81 
53 

25 
98 

72 


2 


67376 
67258 
67140 
67022 
66905 


2 


60796 
60694 
60593 
60492 
60391 


■42163 


12 
14 
16 
18 


348 
346 
344 
343 
341 
339 


45 
71 
99 
29 
80 
93 


2 


54214 
53997 
53780 
53565 
53351 
53138 


262 
280 
258 
256 

254 
252 


04 
10 
18 
29 
43 
60 


2-41837 
.41513 
.41192 
.40873 


20 
22 


465 
464 
462 
461 
460 

459 
458 
456 
455 
451 

453 
452 
450 
449 
448 


48 
21 
97 
73 
50 

28 
08 
85 
65 

45 

28 
07 
89 
72 
56 


2 


66788 
68671 
66555 
66439 
66323 


400 
399 
398 
398 
397 


78 
86 
94 
02 
11 


2 


60291 
60190 
60090 
59990 
59891 


.40557 
.40243 


24 
26 
28 


338 
338 
335 
333 
331 
330 

328 

327 

325. 

324. 

322. 

321. 


27 
64 
01 
41 
82 
24 

88 
13 
60 
09 
59 
10 


2 


52927 
52716 
52506 
52297 
5209C 
51883 


250. 
249. 
247- 
245. 
243. 
242. 


79 
01 
26 
53 
82 
14 


2.39931 
.39822 
.3931F 


30 
32 
34 


2 


66207 
66092 
65977 
65863 
65748 


396 
395 
394 
393 
392 

391- 

390 

389. 

389 

388 


20 
30 
40 
50 
61 
12 
84 
96 
08 
21 


2 


59791 
59692 
59593 
59494 
59396 


.3901S 

•38707 

38407 


36 
38 


2 


51677 
51472 
51269 
51066 
50864 
50663 


240. 
238. 
237. 
235. 
234. 
232. 


49 
85 
24 
65 
08 
54 


2.38109 
.37813 


40 
42 
44 
46 


2 


65634 
65521 
65407 
65294 
65181 


2 


59298 
59199 
59102 
59004 
58907 


37519 
.37227 
■36937 
.36849 


48 


319. 

318 

316 

315 

313 

312 

311 
309 
308 
306 
305 
304 

302 
301 
300 
299 
297 
296 


62 
16 
71 
28 
86 
45 

06 
67 
30 
95 
80 
27 

94 
63 
33 
04 
77 
50 


2 


50464 
50265 
50067 
49869 
4967§ 
49478 


231. 
226. 
222. 
218. 


01 
55 
27 
15 


2.36363 


50 
52 
54 


447 
446 
445 
443 
442 


40 
24 
09 
95 
81 


2 


65069 
64957 
84845 
64733 
64622 


387 
388 
385 
384 
383 


34 
48 
62 
77 
91 


2 


58809 
58713 
58618 
58519 
58423 


.35517 
•34688 
.33875 


56 
S8 


214. 
210. 
206. 
203. 


18 
36 
88 
13 


2. 33078 
•32296 


13° 

9, 


441 
440 
439 
438 

437 


68 
56 
44 
33 

22 


2 


64511 
64400 
64290 
64180 
64070 


383 
382 
381 
380 
379- 


06 
22 
38 
54 
71 


2 


58327 
58231 
58135 
58040 
57945 


2 


49284 
49090 
48898 
48706 
48515 
48325 


•31529 
.30776 


4 
6 
8 


199. 
196. 
193. 
190^ 

187. 
181. 
176. 
171. 
168. 

161. 
157. 
153. 
149. 
146. 


70 
38 
19 
09 

10 
40 
05 
02 
28 
80 
58 
58 
79 
19 


2.30037 
•2931S 
.2859Z 


10 
12 
14 
16 
18 


436 
435 
433 
432 
431 


12 
02 
93 
84 
76 


2 


63960 
63851 
63742 
63633 
63524 


378 
378 
377 
376 
375 


88 

05 
23 
41 
60 


2 


57850 
57755 
57661 
57566 
57472 


•27896 


2 


48136 
47948 
47760 
47573 
47388 
47203 


2-27207 
•25863 
•24563 
•23303 


20 


430 
429 
428 
427 

426 

425 
424 
423 
422 
421 

420 
419 
418 
417 
416 

415 
414 
413 
412 
4U 


69 
62 
56 
50 
44 

40 
35 
32 
28 
_26 
23 
22 
20 
19 
19 

^9 

.20 

21 

23 

.25 


2 


63416 
63308 
63201 
63093 

62936 


374 
373 
373 
372 
371 


79 
98 
17 
37 
57 


2 


57378 
57284 
57191 
57097 
57004 


-2J?083 


22 
24 
26 
28 

30 
32 
34 
36 
38 

40 
42 
44 
46 
. 48 

50 
52 
54 
56 
58 

14° 


2-20899 
.19749 
•18631 
•17547 


295 
294 
292 
291 
290 
289 


25 
00 
77 
55 
33 
13 


2 


47018 
46835 
46852 
46471 
46289 
46109 


2 


62879 
62773 
62666 
62560 
.62454 


370 
369 
369 
388 
367 
366 
368 
365 
364 
363 

363 
362 
361 
360 
360 


78 
99 
20 
42 
64 

86 
09 
31 
55 
78 

02 
26 
51 
78 
01 


2 


56911 
56819 
56726 
56634 
56542 


•16492 


142. 

139. 

136 

133 

130_ 

127 
125 
122 
120 
118 


77 
52 
43 
47 
86 

97 
39 
93 
57 
31 


2.15464 
.14464 
.13489 
.12539 


287 
288 
285 
284 
283 
282 


94 
76 
58 
42 
27 
12 


2 


4593C 
45751 
45573 
45396 
45219 
45044 


2 


.62349 
.82243 
.62138 
.62034 
.61929 


2 


58450 
56358 
58266 
56175 
56084 


.11613 


2.10709 
.09827 
.08965 
.08124 


280 
279 
278 
277 
276 
275 


99 
86 

75 

64 

54 

.45 


2 


44869 
44694 
.44521 
.44348 
-44178 
.44004 


2 


61825 

61721 

61617 

.61514 

.61410 


2 


.55993 
.55902 
.55812 
.55721 
•55631 


.07302 


114 
110 
106 
103 
100 


06 
13 
50 
13 
00 


2.05713 
.04192 
.02736 
.01340 


274 


.37 


2 


.43833 


410 


.28 


2 


.61307 


359 


.28 


2 


.55541 


2. 00000 



631 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A i'^ CURVE. 



A 


"■"^l^- 


Ext. 
Dist. 


Chord 
LC. 


A 

11° 

10 
20 
30 
40 
50 

12° 

10 
20 
30 
40 
50 

13° 

10 
20 
30 
40 
50 

14° 

10 
20 
30 
40 
50 

15° 

10 
20 
30 
40 
50 

16° 

10 
20 
30 
40 
50 

17° 
10 
20 
30 
40 
50 

18° 
10 
20 
30 
40 
50 

19° 

10 
20 
30 
40 
50 

30° 

10 
20 
30 
40 
50 

31° 


Tanff. 


Ext. 
Dist. 


Long 

Chord 

LC. 


A 

31° 

10 

20 
30 
40 
CO 

33° 
10 
20 
30 
40 
50 

33° 

10 
20 
30 
40 
50 

34° 
10 
20 
30 
40 
50 

35° 
10 
20 
30 
40 
50 

36° 
10 
20 
30 
40 
50 

37° 
10 
20 
30 
40 
50 

38° 
10 
20 
30 
40 
50 

39° 
10 
20 
30 
40 
50 

30° 

10 
20 
30 
40 
50 

31° 


Tang. 


Ext. 
Dist. 


Long 
Chord 


1° 

10' 

20 

30 

40 

50 


50 
58 
66 
75 
83 
91 


.00 

• 34 
.67 
.01 
.34 

• 68 










.218 
.297 

388 
.491 
.606 

733 


100 
116 
133 
150 
166 
183 


• OC 
.67 
.33 
.00 

• 66 
.33 


551 
560 
568 
576 
585 
593 


.70 
• 11 
.53 
.95 
.36 
.79 


26 
27 
28 
28 
29 
30 


.500 
• 313 
.137 
.974 
.824 
.686 


1098 
1114 
1131 
1148 
1164 
1181 


• 3 
.9 

.5 
.1 
•7 

r 


1061 
1070 
1079 
1087 
1096 
1105 


.9 
6 
.2 
.8 
.4 
• 1 


97 
99 
100 
102 
103 
105 


58 

.15 

• 75 

• 35 
97 

.60 


2088.3 
2104.7 
2121.1 
2137.4 
2153.8 
2170.2 


2° 
10 
20 
30 
40 
50 


100 
108 
116 
125 
133 
141 


.01 

35 

• 68 

02 
3G 

7C 





873 

.024 

• 188 
.364 

• 552 
752 


199 
216 
233 
249 
266 
283 


.99 
.66 
32 
98 
65 
31 


602 
610 
619 
627 
635 
644 


.21 
.64 
.07 
.50 
.93 
• 37 


31 
32 
33 
34 
35 
36 


.561 

.447 

347 

• 259 
.183 

• 120 


1197 
1214 
1231 
1247 
1264 
1280 


• £ 

• 4 

• C 

• 5 

• 1 
•7 


1113 
1122 
1131 
1139 
1148 
1157 


.7 
.4 
.0 
.7 
.4 
.0 


107 
108 
110 
112 
113 
115 


.24 
• 90 
.57 

.25 

.9!: 

.6f 


2186.5 
2202.9 
2219.2 
2235. & 
2251.9 
2268.3 


3° 

10 
20 
30 
40 
50 


150 
158 
166 
175 
183 
191 

200 
208 
216 
225 
233 
241 


04 
38 
72 
06 
40 
74 
08 
43 
77 
12 
47 
81 


1 

2 
2 
2 
2 
3 


964 
188 
425 
674 
934 
207 


299 
316 
333 
349 
366 
383 


97 
6S 
29 
95 
61 
27 


652 
661 
669 
678 
686 
695 


.81 
.25 
• 70 
.15 
60 
06 


37 
38 
39 
39 
40 
42 


.069 
• 031 
.006 
.993 
.992 
.004 


1297 
1313 
1330 
1346 
1363 
1380 


.2 

• 8 

• 3 

• 9 

• 4 

• 


1165 
1174 
1183 
1191 
1200 
1209 


.7 

.4 

.1 

8 

5 

2 


117 
119 
120 
122 
124 
126 


38 

.12 
87 

.63 
41 
2C 


2284-6 
2301.0 
2317-3 
2333.6 
2349.9 
2366.2 


4° 
10 
20 
30 
40 
50 


3 
3 
4 
4 
4 
5 


492 
790 
099 
421 
755 
100 


399 
416 
433 
449 
466 
483 


92 
58 
24 
89 
54 
20 


703 
711 
720 
728 
737 
745 


.51 
.97 

44 
.90 
.37 

85 


43 
44 
45 
46 
47 
48 


.029 
.066 
.116 
.178 
.253 
.341 


1396 
1413 
1429 
1446 
1462 
1479 


• 5 

• 1 

• 6 

• 2 
•7 
.2 


1217 
1226 
1235 
1244 
1252 
1261 


9 
6 
3 

8 
5 


128 
129 
131 
133 
135 
137 

139 
141 
142 
144 
146 
148 


CC 
82 
65 
5C 
36 
_23 

11 
01 
93 
85 
79 
75 


2382.5 
2398.8 
2415.1 
2431.4 
2447-7 
2464.0 


5° 
10 
20 
30 
40 
50 


250 
258 
266 
275 
283 
291 

300 

308 

316 

325 

333- 

342. 

350. 
358. 
367. 
375. 
383. 
392. 


16 
51 
36 
21 
57 
92 

28 
64 
99 
35 
71 
08 
44 
81 
17 
54 
91 
28 


5 
5 
6 
6 
7 
7 


459 
829 
211 
606 
013 
432 


499 
516 
533 
549 
566 
583 

599 

616 

633 

649 

666. 

682. 


85 

50 
15 
80 
44 
09 

73 
3C 
02 
66 
3C 
94 


754 
762 
771 
779 
788 
796 


32 

go 

29 
77 
26 
75 


49 
50 
51 
52 
53 
55 


.441 
554 

• 679 
818 
969 
132 


1495 
1512 
1528 
1545 
1561 
1578 


• 7 
3 
8 
3 
8 
3 


1270 
1279 
1287 
1296 
1305 
1314 


2 

7 
5 
3 



2480.2 
2496-5 
2512.8 
2529-0 
2545-3 
2561-5 


6° 

10 
20 
30 
40 
50 


7 
8 
8 
9 
9 
10 


863 
307 
762 
230 
710 
202 


805 
813 
822 
830 
839 
847 


25 
75 
25 
76 
27 
78 


56 
57 
58 
59 
61 
62 


309 
498 
699 
914 
141 
381 


1594 
1611 
1627 
1644 
1660 
1677 


8 
3 
8 
3 
8 
3 


1322 
1331 
1340 
1349 
1358 
1366 


8 
6 
4 
2 

8 


150 
152 
154 
156 
158 
160 


71 
69 
69 
70 
72 
76 


2577.8 
2594.0 
2610-3 
2626-5 
2642.7 
2658.9 


r 

10 
20 
30 
40 
50 


10. 
11. 
11. 
12. 
12. 
13- 


707 
224 
753 
294 
847 
413 


699. 
716. 
732. 
749. 
766. 
782. 


57 
21 
84 
47 
10 
73 


856 

864 

873. 

881. 

890. 

898. 


30 
82 
35 
88 
41 
95 


63 

64 

66 

67. 

68. 

70. 


634 
90C 
178 
47C 
774 
091 


1693 

1710 

1726 

1743 

1759. 

1776. 


8 
3 
8 

2 
7 
2 


1375 
1384 
1393 
1402 
1410 
1419 


6 
4 
2 

9 
7 


162 
164 
166 
169 
171 
173 


81 
87 
95 
04 
15 
27 


2675.1 
2691.3 
2707.5 
2723-7 
2739-9 
2756-1 


8° 
10 
20 
30 
40 
50 


400. 
409. 
417. 
425. 
434. 
442. 


66 
03 
41 
79 
17 
55 


13. 
14. 
15. 
15. 
16. 
17. 


991 
582 
184 
799 
426 
066 


799. 
815. 
832. 
849. 
865. 
882. 


36 
99 
61 
23 
85 
47 


907. 
916. 
924. 
933. 
941. 
950. 


49 
03 
58 
13 
69 
25 


71. 
72. 
74. 
75. 
76. 
78. 


421 
764 
119 
488 
869 
264 


1792. 
1809. 
1825. 
1842. 
1858. 
1874. 


6 
1 

5 
C 
4 
9 


1428. 
1437- 
1446. 
1455. 
1464. 
1472. 


6 
4 
3 

1 

9 


175 

177. 

179. 

181. 

184. 

186. 


41 
55 
72 
89 
08 
29 


2772-3 
2788-4 
2804.6 
2820-7 
2836-9 
2853-0 


9° 
10 
20 
30 
40 
50 


450. 
459. 
467. 
476. 
484. 
492. 

501. 

509. 

518 

526 

534 

543 


93 
32 
71 
10 
49 
88 

28 
68 
08 
48 
89 
29 


17. 
18. 
19. 
19. 
20. 
21. 


717 
381 
058 
746 
447 
161 


899. 
915. 
932. 
948. 
965. 
982. 


00 
7C 
31 
92 
53 
14 


958. 
967. 
975. 
984. 
993. 
1001. 


81 
38 

96 
53 
12 
70 


79- 
81. 
82. 
83. 
85. 
86 


671 
092 
525 
972 
431 
904 


1891. 
1907. 
1924. 
1940. 
1957- 
1973. 


3 
8 

2 
6 
1 
5 


1481. 
1490. 
1499. 
1508. 
1517- 
1526. 


8 
7 
6 
5 
4 
3 


188- 
190. 
192. 
195. 
197. 
199. 


51 
74 
9S 
25 
53 
82 


2869-2 
2885-3 
2901-4 
2917-6 
2933-7 
2949-8 


10° 

10 
20 
30 
40 
50 


21. 

22. 

23 

24 

24 

25 


886 
624 
375 
138 
913 
700 


998. 
1015. 
1031. 
1048. 
1065 
1081 


74 
35 
95 
54 
14 
73 


1010 

1018. 

1027 

1036. 

1044 

1053 


29 
89 
49 
09 
70 
31 


88 
89 
91 
92 
94 
96 


389 
888 
399 
924 
462 
013 


1989 • 
2006 • 
2022. 
2039. 
2055. 
2071. 

2088. 


9 
3 
7 
1 
5 
9 

3 


1535. 
1544. 
1553. 
1562. 
1571. 
1580. 


8 

2 
1 
1 

C 



202. 
204. 
206. 
209. 
211 • 
213- 


12 
44 
77 
12 
48 
86 


2965-» 
2982-0 
2998-1 
3014-2 
3030-2 
3046-3 


11° 


551 


.70.26 


500 


1098 


33 


1061 


93 


97 


577 


1589. 


0^ 


216. 


25 


3062.4, 



632 



TABLE II —TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A 1° CURVE. 



A 


"•"^"-g- 


Ext. 
Dist. 


Long 
Chord 


A 


^f,S' 


Ext. 
Dist. 


Long 
Chord 


A 


Tang. 


Ext. 
Dist. 

JiJ. 


Long 
Chord 


31° 

10' 
20 
30 
40 
50 


1589 
1598 
1606 
1615 
1624 
1633 



• 

y 

9 

9 
& 


216 
218 
221 

223 
225 
2i!8^ 

230 
233 
235 
238 
240 
243 


■ 25 

■ 66 

■ 08 

■ 51 
96 
42 

90 
39 
9C 

43 
96 

52 


3062 
3078 
3094 
3110 
3126 
3142 

3158 
3174 
3190 
3206 
3222 
3238 


4 
4 
5 
5 
6 
6 

6 
6 
6 
6 
6 
6 


41° 

10 
20 
30 
40 
50 

43° 
10 
20 
30 
40 
50 

43° 

10 
20 
30 
40 
50 

44° 

10 
20 
30 
■ 40 
50 

45° 
10 
20 
30 
40 
50 

46° 

10 
20 
30 
40 
50 


2142 
2151 
2161 
2170 
2180 
2189 


2 387 

7 390 

2 394 

8 397 

3 400 

9 404 


38 4013 

■ 71 4028 

■ 06 4044 

■ 43 4059 
82 4075 

■ 22 4091 


1 

■ 7 

■ 3 

■ 9 
5 
1 


51° 
10 
20 
30 
40 
50 

53° 
10 
20 
30 
40 
50 

53° 

10 
20 
30 
40 
50 

54° 

10 
20 
30 
40 
50 

55° 
10 
20 
30 
40 
50 

56° 

10 
20 
30 
40 
50 

57° 
10 
20 
30 
40 
50 

58° 
10 
20 
30 
40 
50 

59° 

10 
20 
30 
40 
50 

60° 

10 
20 
30 
40 
50 

61° 


2732 
2743 
2753 
2763 
2773 
2784 


9 618 
1 622 
4 627 
7 631 
9'6S6 
2:640 


3S 
81 
24 
69 
16 
66 


4933 
4948 
4963 
4978 
4993 
5008 


4 
4 
4 
4 
4 
4 


33° 

10 
20 
30 
40 
50 


1643 
1652 
1661 
1670 
1679 
1688 






c 

1 
1 


2199 
2209 
2218 
2228 
2237 
2247 


4 

6 
1 
7 
3 


407 
411 
414 
417 
421 
424 


64 4106 
07 4122 
52 4137 
99 4153 
48 4168 
98 4184 


6 
2 
7 
3 
8 
3 


2794 
2804 
2815 
2825 
2835 
284,6 


5 
9 
2 
6 
9 
3 


645 
649 
654 
658 
663 
668 

672 
677 
681 
686 
691 
696 


17 
7C 
25 
83 
42 
03 

66 
32 
99 
68 
4C 
13 


5023 

5038. 

5053 

5068. 

5083 

5098 


4 
4 
4 
3 
3 
2 


33° 

10 
20 
30 
40 
50 


1697 
1706 
1715 
1724 
1733 
1742 


2 
3 
3 
4 
5 
6 


240 
24G 
251 
253 
256 
259 


08 
66 
26 
87 
50 
14 


3254 
3270 
3286 
3302 
3318 
3334 


6 
6 
6 
5 
5 
4 


2257 
2266 
2276 
2285 
2295 
2305 



6 
2 
9 
6 
2 


428 
432 
435 
439 
442 
446 

449 
453 
457 
460 
464 
468 


50 
04 
59 
16 
75 
35 

98 
62 
27 
95 
64 
35 


4199 
4215 
4230 
4246 
4261 
4277 


8 
3 
8 
3 
8 
3 


2856 
2867 
2877 
2888 
2898 
2908 


7 
1 
5 

4 
9 


5113 
5128 
5142 
5157 
5172 
5187 


1 

9 
8 
7 
6 


34° 

10 
20 
30 
40 
50 


1751 
1760 
1770 
1779 
1788 
1797 


7 
8 



1 
2 
4 


261 
264 
267 
269 
272 
275 


80 
47 
16 
86 
58 
31 


3350 
3366 
3382 
3398 
3414 
3430 


4 
3 
2 
2 
1 



2314 
2324 
2334 
2344 
2353 
2363 


9 
6 
3 
1 
8 
5 


4292 
4308 
4323 
4339 
4354 
4369 


7 
2 
6 

5 
9 


2919 
2929 
2940 
2951 
?Q61 

29 y^ 
2982 
2993 
3003 
3014 
3025 
3035 


4 
9 
4 

5 
1 

7 
3 
9 
5 
2 
8 


700 
705 
710 
715 
720 
724 


89 
66 
46 
28 
11 
97 


5202 
5217 
5232 
5248 
5261 
5276 


4 
3 
1 
9 
7 
5 


35° 

10 
20 
30 
40 
50 


1806 
1815 
1824 
1834 
1843 
1852 


6 
7 
9 
1 
3 
5 


278 
280 
283 
286 
289 
292 


05 
82 
60 
39 
20 
02 


3445 
3461 
3477 
3493 
3509 
3525 


9 
8 

7 
5 
4 
3 


2373 
2333 
2392 
2402 
2412 
2422 


3 
1 
8 
6 
4 
3 


472 
475 
479 
483 
487 
490 


08 4385 
82 4400 
59 4416 
37 4431 
16 4446 
98 4462 


3 
7 
1 
4 
8 
2 


729 
734 
739 
744 
749 
754 


85 
76 
68 
62 
5S 
57 


5291 
5306 
5320 
5335 
5350 
53f'5 


3 
1 
9 
6 
4 
1 


36° 

10 
' 20 
30 
40 
50. 


1881 
1870 
1880 
1889 
1898 
1907 


7 
9 
1 
4 
6 
9 


294 
297 
300 
303 
306 
309 


86 

72 
59 
47 
37 
29 


3541 
3557 
3572 
3588 
3604 
3620 


1 

8 
6 
5 
3 


2432 
2441 
2451 
2461 
2471 
2481 

2491 
2501 
2511 
2521 
2531 
2541 

2551 
2561 
2571 
2581 
2591 
2601 


1 
9 
8 
7 
5 
4 

3 

2 
2 
1 
1 







1 
1 


494 
498 
502 
506 
510 
514 


82 4477 
67 4492 
54 4508 
42 4523 
33 4538 
25 4554 


5 
8 
2 
5 
8 
1 


3046 
3057 
3067 
3078 
3089 
3100 


5 
2 
9 
7 
4 
2 


759 
764 
769 
774 
779 
784 


58 
61 
66 
73 
83 
94 


5379 

0394 

5409. 

5423- 

5438 ■ 

5453 


8 
5 
2 
9 
6 
3 


37° 

10 
20 
30 
40 
50 


1917 
1926 
1935 
1945 
1954 
1963 


1 
4 
7 

3 
6 


312 
315 
318 
321 
324 
327 


22 
17 
13 
11 
11 
12 


3636 
3651 
3667 
3683 
3699 
3715 


1 
9 
7 
5 
3 



47° 
10 
20 
30 
40 
50 

48° 
10 
20 
30 
40 
50 

49° 
10 
20 
30 
40 
50 

50° 
10 
20 
30 
40 
50 

51" 


518 
522 
526 
530 
534 
538 


20 
16 
13 
13 
15 
18 


4569 
4584 
4599 
4615 
4630 
4645 


4 
7 
9 
2 
4 
7 


3110 
3121 
3132 
3143 
3154 
3165 


9 
7 

6 
4 
2 
1 


790 
795 
800 
805 
810 
816 


08 
24 
42 
62 
85 
10 


5467 ■ 

5482 

5497 

5511 

5526 

5541 


9 
5 
2 
8 
4 



38° 
10 
20 
30 


1972 
1982 
199] 
2000 
2010 
2019 


9 

2 
f 


2 
6 


330 
333 
338 
339 

342 
345 


15 
19 
25 
32 
41 
52 


3730 
3746 
3762 
3778 
3793 
3809 


8 
5 
3 

8 
5 


542 
546 
550 
554 
558 
562 


23 
30 
39 
50 
63 
77 


4660 
4676 
4691 
4706 
4721 
4736 


9 
1 
3 
5 
7 
9 


3176 
3186 
3197 
3208 



9 
8 
8 


821 
826 
831 
837 
842 
848 
853 
858 
864 
869 
875 
880 


37 
66 
98 
31 
67 
06 

46 
89 
34 
82 
32 
84 


5555 
5570 
5584 
5599 
5613 
5628 

5642 
5657 
5671 
5686 
5700 
5715 


6 
2 
7 
3 


40 
50 


3219 
3230 


7 
7 


8 
3 


39° 

10 
20 
30 
40 
50 


2029 
2038 
2047 
2057 
2066 
2076 



4 
8 
2 
6 



348 
351 
354 
358 
361 
364 


64 
78 
94 
11 
29 
50 


3825 
3840 
3856 
3872 
3888 
3903 


2 
9 
6 
3 

6 


2611 
2621 
2631 
2641 
2651 
2661 


2 
2 
3 
4 
5 
6 


566 
571 
575 
579 
583 
588 


94 
12 
32 
54 
78 
04 


4752 
4767 
4782 
4797 
4812 
4827 


1 
3 
4 
5 
7 
8 


3241 
3252 
3263 
3274 
3285 
3296 


7 
7 
7 
8 
8 
9 


8 
3 
8 

g 
8 

2 


40° 

10 
20 
30 
40 
50 


2085 
2094 
2104 
2113 
2123 
2132 


4 
9 
3 
8 
3 
7 


387 
370 
374 
377 
380 
384 


72 
95 
20 
47 
76 
06 


3919 
3935 
3950 
3966 
3981 
3997 


3 

6 
3 
9 
5 


2671 
2881 
2692 
2702 
2712 
2722 


8 
9 

1 
3 
5 
7 


592 
596 
600 
605 
609 
614 

618 


32 
62 
98 
27 
62 
00 

39 


4842 
4858 
4873 
4888 
4903 
4918 


9 


1 
2 
2 
3 


3308 
3319 
3-330 
3341 
3352 
3363 



1 
3 
4 
6 
8 


886 
891 
897 
903 
908 
914 


38 

95 
54 
15 
79 
45 


5729 
5744 
5758 
5772 
5787 
5801 


7 

1 
5 
■ 9 
3 
7 


41° 


2142 


2 


387 


38 


4013 


1 


2732 


_9 


4933 


■ 4 


3375 


■ 


920 


14 


08] 6 


._Q 



633 



TABLE II. — TANGENTS, EXTERNAL DISTANCES, AND LONG 
CHORDS FOR A 1° CURVE. 



A 


Tang. 
T. 


Ext. 
Dist. 
E. 


Long 
Chord 
LC. 


A 


Tang. 


Ext. 
Dist. 
E. 


Long 
Chord 
LC. 


A 


Tang. 
T. 


Ext. 
Dist. 
E. 


Long 
Chord 
LC. 


61° 

10' 

20 

30 

40 

50 


3375 
3386 
3397 
3408 
3420 
3431 



3 
5 
8 
1 
4 


920 
925 
931 
937 
943 
948 


14 
85 
58 
34 
12 
92 


5816 
5830 
5844 
5859 
5873 
5887 



4 
7 

1 
4 
7 


68° 

10' 

20 

30 

40 

50 


3864 
3876 
3889 
3901 
3913 
3925 


7 
8 

2 
4 
6 


1181 
1188 
1195 
1202 
1208 
1215 


6 
4 
2 

9 
8 


6408 
6421 
6435 
6449 
6463 
6476 



8 
6 
4 
1 
9 


75° 
10' 
20 
30 
40 
50 


4396 
4409 
4423 
4436 
4449 
4463 


5 
8 
1 
4 
7 
1 


1492 
1500 
1508 
1516 
1524 
1533 


4 
5 
6 
7 
9 
1 


6976 
6989 
7002 
7015 
7028 
7041 



2 
4 
6 
8 
9 


63° 

10 
20 
30 
40 
50 


3442 
3454 
3465 
3476 
3488 
3499 


\ 

4 
8 
2 
7 


954 
960 
966 
972 
978 
984 


75 
60 
48 
39 
31 
27 


5902 
5916 
5930 
5944 
5959 
5973 



3 
5 
8 

3 


69° 

10 
20 
30 
40 
50 

70° 
10 
20 
30 
40 
50 

71° 
10 
20 
30 
40 
50 


3937 
3950 
3962 
3974 
3987 
3999 


9 
2 
5 
8 
2 
5 


1222 
1229 
1236 
1243 
1250 
1257 


7 
7 
7 
7 
8 
9 


6490 
6504 
6518 
6531 
6545 
6559 


6 
4 
1 
8 
5 
1 


76° 

10 
20 
20 
40 
50 


4476 
4489 
4503 
4516 
4530 
4544 


5 
9 
4 
9 

t 


1541 
1549 
1558 
1566 
1574 
1583 


4 
7 

3 
7 
1 


7055 
7068 
7081 
7094 
7107 
7120 



2 
3 
4 
5 
5 


63° 

10 
20 
30 
40 
50 


3511 
3522 
3534 
3545 
3557 
3568 


1 
6 
1 
6 
2 
7 


990 
996 
1002 
1008 
1014 
1020 


24 

24 

3 

3 

4 

5 


5987 
6001 
6015 
6030 
6044 
6058 


5 
7 
9 

2 
4 


4011 
4024 
4036 
4049 
4061 
4074 


9 
4 
8 
3 
8 
4 


1265 
1272 
1279 
1286 
1293 
1300 



1 
3 
5 
7 
9 


6572 
6586 
6600 
6613 
6627 
6640 


8 
4 
1 
7 
3 
9 


77° 
10 
20 
30 
40 
50 

78° 
10 
20 
30 
40 
50 


4557 
4571 
4584 
4598 
4612 
4626 


6 
2 
8 
5 
2 



1591 
1600 
1608 
1617 
1625 
1634 


6 
1 
6 
1 
7 
4 


7133 
7146 
7159 
7172 
7185 
7198 


6 
6 
6 
6 
6 
6 


64° 

10 
20 
30 
40 
50 


3580 
3591 
3603 
3615 
3626 
3638 


3 

9 

5 

• 1 

.8 

.5 


1026 
1032 
1039 
1045 
1051 
1057 


6 
8 

2 
4 
7 


6072 
6086 
6100 
6114 
6128 
6143 


5 
6 
7 
8 
9 



4086 
4099 
4112 
4124 
4137 
4150 


9 
5 
1 
8 
4 
1 


1308 
1315 
1322 
1330 
1337 
1345 


2 
5 
9 
3 
7 
1 


6654 
6668 
6681 
6695 
6708 
6722 


4 

6 
1 
6 
1 


4639 
4653 
4667 
4681 
4695 
4709 


8 
6 
4 
3 
2 
2 


1643 
1651 
1660 
1669 
1678 
1686 



7 
5 
2 
1 
9 


7211 
7224 
7237 
7250 
7263 
7276 


6 
5 
4 
4 
3 
1 


65° 

10 
20 
30 
40 
50 


3650 
3661 
3673 
3685 
3697 
3709 


.2 
9 
7 
4 
2 



1063 
1070 
1076 
1082 
1089 
1095 


9 
2 
6 
9 
3 
7 


6157 
6171 
6185 
6199 
6213 
6227 


1 
1 
2 
2 
2 
2 


73° 
10 
20 
30 
40 
50 


4162 
4175 
4188 
4201 
4214 
4226 


8 
6 
4 
2 

8 


1352 
1360 
1367 
1375 
1382 
1390 


6 
1 
6 
2 
8 
4 


6735 
6749 
6762 
6776 
6789 
6802 


6 
1 

5 

4 
8 


79° 
10 
20 
30 
40 
50 


4723 
4737 
4751 
4765 
4779 
4793 


2 
2 
2 
3 
4 
6 


1695 
1704 
1713 
1722 
1731 
1740 


8 
7 
7 
7 
7 
8 


7289 
7301 
7314 
7327 
7340 
7353 



9 
7 
5 
3 
1 


66° 

10 
20 
30 
40 
50 


3720 
3732 
3744 
3756 
3768 
3780 


9 
7 
6 
5 
5 
4 


1102 
1108 
1115 
1121 
1128 
1134 


2 
6 
1 
7 
2 
8 


6241 
6255 
6269 
6283 
6297 
5310 


2 
2 
1 
1 

9 


73° 
10 
20 
30 
40 
50 


4239 
4252 
4265 
4278 
4291 
4304 


7 
6 
6 
5 
5 
6 


1398 
1405 
1413 
1421 
1429 
1436 



7 
5 
2 

8 


6816 
6829 
6843 
6856 
6869 
6883 


3 
6 

4 
7 
1 


80° 
10 
20 
30 
40 
50 

81° 
10 
20 
30 
40 
50 

83° 


4807 
4822 
4836 
4850 
4864 
4879 


7 

2 
5 
8 
2 


1749 
1759 
1768 
1777 
1786 
1796 


9 

2 
4 
7 



7365 
7378 
7391 
7404 
7416 
7429 


9 
7 
4 
1 
8 
5 


67° 

10 
20 
30 
40 

50 


3792 
3804 
3816 
3828 
3840 
3852 


4 

4 
4 
4 
5 
6 


1141 
1148 
1154 
1161 
1168 
1174 


4 

7 
3 
1 
8 


6324 
6338 
6352 
6366 
6380 
6394 


8 
7 
6 
4 
3 
1 


74° 
10 
20 
30 
40 
50 


4317 
4330 
4343 
4356 
4370 
4383 


6 
7 
8 
9 
1 
3 


1444 
1452 
1460 
1468 
1476 
1484 


6 
5 
A 

4 
4 
4 


6896 
6909 
6923 
6936 
6949 
6962 


4 
7 

2 
5 
8 


4893 
4908 
4922 
4937 
4951 
4966 


6 

5 

5 
1 


1805 
1814 
1824 
1833 
1843 
1852 


3 
7 
1 
6 
1 
6 


7442 
7454 
7467 
7480 
7492 
7505 


2 
9 
5 
2 
8 
4 


68° 


3864 


7 


1181 


6 


6408 





75° 


4396 


5 


1492 


4 


6976 





4980 


7 


1862 


2 


7518 












Correction Table (always additive) 














Degree of curve. 


A 


5 


10° 


15° j 


20° 




T 


E 


LC 


T 


E 


LC 


T 


E 


... 
LC 


T 


E 


LC 


10° 


.03 


.001 


06 


.06 


• 003 


.13 


.10 


• 004 


.17 


13 


.006 


.25 


20 


.06 


• 005 


12 




13 


• Oil 




25 




19 


• 017 




38 


26 


.022 




51 


30 


, .09 


012 


18 




19 


025 




37 




29 


.038 




56 


39 


051 




75 


40 


1 .13 


.022 


24 




26 


046 




49 




40 


070 




74 


53 


093 


1 


00 


■ 50 


i .16 


.036 


30 




34 


• 075 




61 




51 


112 




92 


68 


151 


1 


23 


60 


.20 


• 054 


35 




42 


.111 




72 




63 


.168 


1 


09 


84 


.225 


1 


46 


70 


.24 


077 


40 




50 


.159 




83 




76 


.240 


1 


25 1 


02 


321 


1 


67 


80 


.29 


107 


45 




60 


.220 




93 




91 


• 332 


1 


40 1 


22 


.455 


1 


87 


90 


.35 


.145 


49 




72 


298 


1 


02 ll 


09 


.451 


1 


54 1 


46 


.603 


2 


06 



634 



TABLE IIA. EXCESS LENGTH OF SUB CHORDS. SEE § 48. 



4> 



lo 
11 

12 
13 

14 
15 
16 

17 
18 
19 

20 
21 
22 

23 

24 
25 
26 
27 
28 

29 
30 



Nominal length of sub chord. 



10 

003 
005 
006 

008 
010 
013 

015 
018 
021 

025 
028 
032 

036 
041 
045 
050 
056 
061 

067 
073 
079 

085 
092 
099 

107 
114 



20 



006 
009 
012 

016 
020 
024 

029 
035 
041 
048 
055 
063 

071 
079 
088 

098 
108 
118 

129 
141 
153 
166 
179 
192 

207 
221 



30 



009 
012 
017 
022 
028 
035 
042 
050 
059 
068 
079 
089 
100 
113 
125 

139 
153 
168 

184 
201 
218 

236 
254 
273 

293 
314 



40 



Oil 
015 
021 

027 
035 
043 

052 
062 
072 

084 
097 
109 
123 
139 
154 

171 
189 
207 
226 
247 
268 
290 
313 
337 
361 
387 



45 



Oil 
016 
022 

029 
037 
046 

055 
066 
077 
090 
103 
117 
132 
148 
165 

183 

202 
221 

242 
264 
286 
310 
334 
359 

386 
413 



50 



012 
017 
023 

030 
038 
048 

058 
069 
080 

094 
108 
122 

138 
155 
172 

191 
211 
231 
253 
275 
299 
324 
349 
375 

403 
431 



55 



012 
018 
024 

031 
039 
049 

059 
070 
082 

096 
110 
125 
141 
158 
176 

195 
215 
237 

259 
282 
306 
331 
357 
384 

412 
441 



60 



012 
018 
024 

031 
039 
049 

059 
070 
083 

096 
110 
125 
141 
158 
177 

196 
216 
237 

259 
282 
306 

331 
357 
384 

412 
442 



65 



012 
017 
023 
030 
039 
048 

058 
069 
081 
094 
108 
122 

138 

155 
172 

191 
211 
231 

253 
276 
299 

324 
349 
376 

403 
432 



70 



Oil 
016 
022 

029 
037 
045 

055 
066 
077 

089 
103 
116 

131 
147 
164 

182 
200 
220 

241 
262 
284 

308 
332 
357 

383 
410 



75 



010 
015 
020 

027 
034 
042 

051 
060 
071 

082 
094 
107 
120 
135 
151 

167 
184 
202 

221 
241 
261 

283 
305 
328 
352 
377 



80 



009 
013 
018 
023 
030 
037 
044 
053 
062 

072 
083 
094 

106 
119 
132 

147 
162 
177 

194 
211 
229 
248 
268 
288 

309 
331 



85 



007 
Oil 
015 
019 
024 
030 

036 
043 
051 

059 
068 
077 

087 
097 
108 

120 
132 
145 

159 
173 
188 
203 
219 
236 

253 
271 



90 



005 
008 
Oil 

014 
018 
022 

026 
031 
037 
043 
049 
056 

063 
070 
079 

037 
096 
105 

115 
125 
136 

147 
159 
171 

183 

196 



95 

003 
004 
006 

008 
010 
012 
014 
017 
020 
023 
027 
030 

034 
038 
043 
047 
052 
057 
062 
068 
074 
080 
086 
093 

099 
109 



TABLE III.— SWITCH LEADS AND DISTANCES. 
A. TRIGONOMETRICAL FUNCTIONS OF THE FROG ANGLES. 



Frog 

No. 
in) 


Frog Angle 
{F). 


Nat. 
sin F. 


Nat. 
cos F. 


Log 
sinF. 


Log 
cos F. 


Log 
cot F. 


Log 
vers F. 


Frog 

No. 

in) 


5 
6 
7 
8 


11° 25' 16" 
9 31 38 
8 10 16 
7 09 10 


.19802 
.16552 
.14213 
•12452 


.98020 
•98621 
•98985 
■99222 


9^29670 
•21884 
•15268 
•09522 


9.9913T 
.99397 
•99557 
•99660 


10.69461 
•77513 
•84288 
•90138 


8-29670 

-13966 

8-00655 

7-89110 


5 
6 
7 
8 


9 
10 
11 
12 


6 21 35 
5 43 29 
5 12 18 
4 46 19 


•11077 
•09975 
•09072 
•08319 


•99385 
•99501 
•99588 
•99653 


9 • 04442 

8^99891 

•95770 

•92007 


•99732 
•99783 
•99820 
•99849 


•95289 
10^99892 
11^04050 

•07842 


•78915 
•69787 
•61527 
-53986 


9 
10 
11 
12 


14 
15 
16 


4 05 27 
3 49 06 
3 34 47 


•07134 
•06659 
.06244 


•99745 
•99778 
•99805 


.8533T 
•82343 
.79543 


•99889 
•99903 
•99915 


•14557 
.17560 
.20370 


•40616 
-34631 
.29028 


14 
15 
16 


18 
20 


3 10 56 
2° 51' 51" 


.05551 
.04997 


•99846 
.99875 


• 74438 
8.69869 


•99933 
9-99945 


•25494 
11^30076 


.18807 
7-09663 


18 

20 



635 



TABLE III.— SWITCH LEADS AND DISTANCES— Con^wtied 
B. THEORETICAL LEADS, USING STRAIGHT POINT-RAILS AND 



STRAIGHT FROG RAILS; 


GAUGE 4' BY'. 


See §§ 305 and 313. 


• 




CD 

m 

(3 




Fr 


og. 


Switch Rail. 


Switch Dimensions. 


" 




ft 








Degree 


^it 


O 


3 

O 


Toe leng 
to theo] 
pt. of fr 


Heel to 
theoret 
of frog. 


■*^ 

PI 
(P 
1-1 


Angle. 


Radius. 


of 

lead 

curve. 


Ac. pt. o 
sw. rail 
ac. pt. f 


(«) 




(TF) 


{K) 


iS) 


(a) 


(r) 


(-D) 


i.L') 




ft. 


ft. in. 


ft. in. 


ft. in. 


o in 


ft. 


O t II 


ft. 


5 


0.21 


3 4 


5 8 


11 


2 36 19 


185-59 


31 15 28 


43-15 


6 


0.25 


3 6 


6 6 


11 


2 36 19 


280 


48 


20 32 14 


48 


66 


7 


029 


4 .5 


7 7 


16 6 


1 44 11 


364 


88 


15 47 19 


62 


23 


8 


0:33 


4 9 


8 3 


16 6 


1 44 11 


488 


71 


11 44 40 


67 


80 


9 


0-37 


6 


10 


16 6 


1 44 11 


616 


27 


9 18 27 


72 


61 


10 


0.42 


6 


10 6 


16 6 


1 44 11 


790 


25 


7 15 18 


11 


93 


11 


0.46 


6 


11 


22 


1 18 


08 


940 


21 


6 05 48 


92 


52 


12 


0-50 


6 5 


12 1 


22 


1 18 


08 


1136 


34 


5 02 38 


97 


75 


14 


0.58 


7 3 


14 3 


22 


1 18 


08 


1600 


73 


3 34 48 


107 


74 


15 


062 


7 8 


14 10 


30 


57 18 


1764 


69 


3 14 50 


126 


49 


16 


0.67 


8 


16 


30 


57 


18 


2082 


74 


2 49 08 


131 


82 


18 


0-75 


8 10 


17 8 


30 


57 


18 


2632 


76 


2 10 35 


141 


93 


20 


083 


9 8 


19 4 


30 


57 


18 


3334 


16- 


1 43 06 


151 


60 



C. PRACTICAL LEADS, USING STRAIGHT POINT-RAILS AND STRAIGHT 

FROG rails; GAUGE 4' 8|". See §§ 305-307. 









-1-5 


-p 

a . 

% 2 












Radius 


Degree 




03 Jh 


'-' c5 n 








6 

1 


of 

center 

line. 


of 

lead 

curve. 


c3 o 


^ o 
o 

H 


pi p: o 
< 


Closure for 

straight 

rail. 


Closure for 

curved 

rail. 


(n) 


(r) 


{D) 


(r,) 


(T/) 


{L') 










ft. 


o r II 


ft. 


ft. 


ft. 








5 


175-40 


33 07 28 


0-00 


0-97 


42.54 


1-28 . 




1-28.31 


6 


254-00 


22 42 20 


0-00 


2-00 


47-50 


1-32 . 75 


1-33 




7 


361-69 


15 53 30 


0-00 


0-22 


62-08 


1-26 1-14-87 


1-26 


1-15 - 12 


8 


487.37 


11 46 36 


0.32 


0-00 


68-00 


1-30 1-16-42 


1-30 


1-16-58 


9 


605-18 


9 28 42 


0-00 


0-57 


72-28 


1-33 1-16-41 


1-33 


1-16- 59 


10 


779-82 


7 21 08 


1-56 


0-00 


78-75 


1-28 1-27-83 


2-28 




11 


922-65 


6 12 47 


2-99 


0-00 


94-31 


1-33 1-32.85 


2-33 




12 


1098-73 


5 12 59 


5. 33 


0-00 


100-80 


2-24 1-23 . 88 


3-24 




14 


1512-14 


3 47 23 


0-00 


2-84 


106-27 


2-30 1-16.44 


2-30 


1-16-56 


15 


1748-29 


3 16 40 


0-00 


0-51 


126-19 


2-30 1-27-90 


2-30 


1-28 


16 


2019-18 


2 50 16 


0-00 


0-40 


131-56 


2-30 1-32-90 


2-30 


1-33 


18 


2380.47 


2 24 26 


0-00 


6-38 


138-50 


2-33 1-32-92 


3-33 




20 


3322-13 


1 43 29 


0-00 


0-27 


151-46 


2-33 1-30 

1-14-96 


2-33 


1-30 
1-15 - 02 



The lengths of switch rail used with each frog are the same as those 
specified for theoretical leads. 

636 



TABLE IV.— FUNCTIONS OF THE TEN-CHORD SPIRAL. 
Part A. — Coefficients of ai for deflection angles to chord points. 



Deflection 






Transit at chord-point i 


aumber. 


angle to 


















chord-point 



T. S. 




















10 
S. C. 


number. 


1 


2 


3 


' 


5 


6 


7 


8 


9 


T. S. 





2 


8 


18 


32 


50 


72 


98 


128 


162 


200 


1 


1 





5 


14 


27 


44 


65 


90 


119 


152 


189 


2 


4 


4 





8 


20 


36 


56 


80 


108 


140 


176 


8 


9 


10 


7 





11 


26 


45 


68 


95 


126 


161 


4 


16 


18 


16 


10 





14 


32 


54 


80 


110 


144 


5 


25 


28 


27 


22 


13 





17 


38 


63 


92 


125 


6 


36 


40 


40 


36 


28 


16 





20 


44 


72 


104 


7 


49 


54 


55 


52 


45 


34 


19 





23 


50 


81 


8 


64 


70 


72 


70 


64 


54 


40 


22 





26 


56 


9 


81 


88 


91 


90 


85 


76 


63 


46 


25 





29 


10 S. C. 


100 


108 


112 


112 


108 


100 


88 


72 


52 


28 






























Part B. — Values of 



--. and -f. 

Li Li 



<f> 


U 
L 


V 
L 


<}> 


U 
L 


V 
L 


0° 

1 

2 


666 667 
666 678 
666 710 


333 333 

333 343 
333 372 


23° 

24 

25 


672 423 

672 943 

673 486 


338 586 

339 061 
339 559 


4 
5 


666 763 
666 838 
666 935 


333 421 
333 490 
333 578 


26 
27 
28 


674 054 

674 645 

675 261 


340 078 

340 619 

341 183 


6 
7 
8 


667 053 
667 193 
667 354 


333 685 
333 812 
333 959 


29 
30 
31 


675 901 

676 566 

677 256 


341 769 

342 378 

343 Oil 


9 
10 
11 


667 537 
667 742 
667 968 


334 126 
334 313 
334 519 


32 
33 
34 


677 971 

678 712 

679 478 


343 667 

344 346 

345 050 


12 
13 
14 


668 216 
668 487 
668 779 


334 746 

334 992 

335 259 


35 
36 
37 


680 270 
681089 
681935 


345 777 

346 529 

347 307 


15 
16 
17 


669 094 
669 431 
669 790 


335 546 

335 853 

336 181 


38 
39 
40 


682 808 

683 708 

684 636 


348 109 

348 937- 

349 791 


18 

19 
20 


670 172 
670 576 
671003 


336 529 

336 899 

337 289 


41 

42 
43 


685 592 

686 577 

687 590 


350 671 
351578 
352 513 


21 

22 


671453 
671926 


337 700 

338 132 


44 
45 


688 633 

689 706 


353 474 

354 464 



Table IV, of which Part C is condensed, was computed by the Track 
Committee of the American Railway Engineering Association and is taken 
from the Proceedings of the Association. 

637 



TABLE IV.— FUNCTIONS OF THE TEN-CHORD SPIRAL. 

Part C. 



Total spiral 




A 


C 


X 


F 


angle, <i> 




^X 


L 


L 


L 


0"= 


0' 


0^ 


00' 00" 


1.000 000 


1.000 000 


.000 000 




30 





10 00 


.999 997 


.999 993 


.002 909 


1 








20 00 


.999 987 


.999 970 


.005 818 




30 





30 00 


.999 970 


.999 932 


.008 726 


2 








40 00 


.999 947 


.999 879 


.011635 




30 





50 00 


.999 916 


.999 811 


.014 542 


3 







00 00 


.999 880 


.999 727 


• 017 450 




30 




10 00 


.999 836 


.999 629 


• 020 357 


4 


GO 




20 00 


.999 786 


.999 515 


.023 263 


., , 


30 




30 00 


.999 729 


• 999 387 


• 026169 


5 


00 




40 00 


.999 666 


.999 243 


.029 073 


, 


30 




50 00 


.999 596 


.999 084 


.031977 


6 


00 




59 59 


.999 519 


.998 910 


.034 880 




30 


2 


09 59 


.999 435 


.998 721 


• 037 781 


7 


00 


2 


19 59 


.999 345 


.998 517 


.040 681 




30 


2 


29 59 


.999 248 


.998 298 


.043 581 


8 


00 


2 


39 58 


.999 145 


.998 063 


• 046 478 




30 


2 


49 58 


.999 035 


.997814 


• 049 374 


9 


00 


2 


59 58 


.998 918 


.997 549 


• 052 269 




30 


3 


09 57 


.998 794 


.997270 


.055 162 


10 


00 


3 


19 57 


.998 664 


.996 975 


.058 053 




30 


3 


29 57 


.998 527 


.996 666 


.060 942 


11 


00 


3 


39 56 


.998 384 


.996341 


.063 829 




30 


3 


49 55 


.998 233 


.996 002 


.066 714 


12 


00 


3 


59 55 


.998 077 


.995 647 


.069 598 




30 


4 


09 54 


.997913 


.995 278 


• 072 478 


13 


00 


4 


19 53 


.997 743 


.994 893 


.075 357 




30 


4 


29 53 


.997566 


.994 494 


.078 233 


14 


00 


4 


39 52 


.997383 


.994 079 


• 081106 




30 


4 


49 51 


.997192 


.993 650 


• 083 977 


15 


00 


4 


59 50 


.996 996 


.993 206 


• 086 846 




30 


5 


09 49 


.996 792 


.992 747 


.089 711 


16 


00 


5 


19 48 


.996 582 


.992 273 


.092 574 




30 


5 


29 47 


.996366 


.991785 


.095 433 


17 


00 


5 


39 45 


.996142 


.991281 


.098 290 




30 


5 


49 44 


.995 912 


.990 763 


.101143 


18 


00 


5 


59 43 


.995 676 


.990 230 


.103 993 




30 


6 


09 41 


.995 432 


.989 682 


• 106 840 


19 


00 


6 


19 40 


.995 183 


.989 120 


• 109 683 




30 


6 


29 36 


.994 926 


.988 543 


• 112 523 


20 


00 


6 


39 36 


.994 663 


.987 951 


.115 360 




30 


6 


49 34 


.994393 


.987 344 


.118 192 


21 


00 


6 


59 32 


.994117 


.986 723 


.121021 




30 


7 


09 30 


.993 834 


.986 088 


• 123 846 


22 


00 


7 


19 28 


.993 545 


.985 437 


.126 667 


22** 


30' 


7° 


29' 26" 


.993 248 


.984 772 


• 129 483 



638 



TABLE IV.— FQNCTIONS OF THE TEN-CHORD- SPIRAL. 

Part C.—Con. 



Total spiral 


A 


C 


X 


y 


angle, <A 


L 


L 


L 


22° 30' 


7° 29' 26" 


.993 248 


• 984 772 


.129 483 


23 00 


7 39 24 


.992 946 


.984 093 


• 132 296 


30 


7 49 21 


.992 636 


.983 399 


• 135 105 


24 00 


7 59 19 


.992 321 


• 982 691 


.137 909 


30 


8 09 16 


.991998 


.981968 


• 140 708 


25 00 


8 19 14 


.991669 


.981231 


. 143 504 


30 


8 29 11 


.991333 


.980 479 


.146 294 


26 00 


8 39 08 


.990 991 


.979 714 


.149 080 


30 


8 49 05 


.990 642 


.978 933 


.151861 


27 00 


8 59 02 


.990 287 


.978 139 


. 154 638 


30 


9 08 58 


• 989 925 


.977 330 


.157 409 


28 00 


9 18 55 


.989 557 


.976 508 


.160176 


30 


9 28 51 i 


.989 182 


.975 670 


.162 937 


29 00 


9 38 48 ' 


• 988 800 


• 974 819 


.165 693 


30 


9 48 44 ^ 


• 988 412 


• 973 954 


.168 444 


30 00 


9 58 40 


.988 018 


•973 074 


.171189 


30 


10 08 36 


• 987 617 


.972 181 


.173 929 


31 00 


10 18 32 


• 987209 


.971 273 


.176 664 


30 


10 28 27 


• 986 795 


.970 352 


.179 392 


32 00 


10 38 23 


• 986375 


.969 417 


.182 116 


30 


10 48 18 


• 985 948 


•968 468 


.184 833 


33 00 


10 58 13 


.985 514 


.967 504 


.187 544 


30 


11 08 08 


• 985 074 


.966 528 


.190 250 


34 00 


11 18 03 


.984 627 


.965 537 


• 192 949 


30 


11 27 58 


.984174 


.964 532 


.195 643 


85 00 


11 37 53 


.983 715 


.963 515 


.198 330 


30 


11 47 47 


.983 249 


.962 483 


.201010 


36 00 


11 57 41 


.982 777 


.961 438 


.203 685 


30 


12 07 36 


.982^298 


.960 379 


.206 353 


37 00 


12 17 30 


• 981813 


•959 306 


.209 014 


30 


12 27 23 


• 981321 


•958 221 


.211669 


38 00 


12 37 17 


• 980 823 


•957 121 


.214317 


30 


12 47 11 


.980318 


•956 009 


.216 959 


39 00 


12 57 04 


• 979 807 


•954 883 


.219 593 


30 


13 06 57 


• 979 290 


.953 744 


.222 221 . 


40 00 


13 16 50 


• 978 766 


•952 591 


.224 841 


30 


13 26 43 


• 978 236 


• 951426 


.227 455 


41 00 


13 36 35 


.977 700 


• 950 247 


.230 061 


30 


13 46 28 


.977 157 


.949 055 


.232 660 


42 00 


13 56 20 


.976 608 


.947 850 


.235 252 


30 


14 06 12 


.976 053 


.946 632 


• 237 836 


43 00 


14 16 04 


.975 491 


.945 402 


.240 413 


30 


14 25 56 


.974 923 


• 944 158 


.242 982 


44 00. 


14 35 47 


• 974 348 


• 942 901 


.245 544 


30 


14 45 38 


.973 768 


.941 632 


.248 098 


45*' 00' 


14° 55' 29" 


• 973131 


• 940 350 


• 250 644 



639 









TABLE V 


.—LOGARITHMS OF NUMBERS. 












II 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


100 


00 000 


043 


087 


130 


173 


216 


260 


303 


346 


389 


























43 43 42 41 


101 


432 


475 


518 


561 


604 


646 


689 


732 


775 


817 


• 1 


4.3 


4.3 


4.2 


4.1 


102 


860 


902 


945 


987 


*030 


*072 


*114 


*157 


*199 


*241 


■ 2 


8 


7 


8 


6 


8 


4 


8.2 


103 


01 283 


326 


368 


410 


452 


494 


536 


578 619 


661 


3 


13 





12 


9 


12 


6 


12.3! 


104 


703 


745 


787 


828 


870 


911 


953 


994,*CS6 


*C77 


.4 


17 


4 


17 


2 


16 


8 


16.4 


105 


02 119 


160 


201 


243 


284 


325 


366 


407 


448 


489 


• 5 


21 


7 


21 


5 


21 





20.5 


106 


53U 


57l 612 


653 


694 


735 


775 


816 


857 


898 


e 


26 


1 


25 


8 


25 


.2 


24. P! 


107 


938 


97ii*019 


*06G 


*100 *141 


*181 


*221 


*2e2 


*302 


■7 


30 


.4 


30 


1 


29 


4 


28.? 


108 


03 342 


382 


422 


463 


503 543 


583 


623 


665 


703 


8 


34 


8 


34 


4 


33 


6 


32.fi 
36-0 


109 


742 


782 


822 


862 


901 941 
297 336 


951 
375 


='=020 


*cec 


*10C 


.9 


39 


1 


38 


7 


37 


8 


110 


04 139 


178 


218 


257 


415 


454 


493 


40 40 39 38 




















111 


532 


571 


610 649 


688 


727 


766 


805 


844 


883 


• 1 


4.0 


4 


.0 


3-9 


3.8 


112 


922 


960 


9991*038 


=1=076 


*116 


*154 


*192 


*231 


*269 


• 2 


8.1 


8 


• 


7 


8 


7.6 


113 


05 308 


346 


384 


423 


461 


499 


538 


576 


614 


652 


3 


12.1 


12 


■ 


11 


• 7 


11.4 


114 


690 


728 


766 


804 


842 


8&0 


918 


956 


994 


*032 


• 4 


16-2 


16 


■ 


15 


6 


15.2 


115 


06 070 


107 


145 


183 


220 


258 


296 


333 


371 


408 


• 5 


20.2 


20 


■ C 


19 


■ 5 


19.0 


116 


446 


483 


520 


558 


595 


632 


670 


707 


744 


781 


6 


24-3 


24 


• C 


23 


.i 


22.8 


117 


81§ 


355 


893 


930 


967 


*004 


*040 


*077 


*114 


*151 


• 7 


28.3 


28 


■ C 


27 


3 


26.6 


118 


07 188 


225 


261 


298 


335 


372 


408 


445 


481 


518 


■ 8 


32. 4 


32 


C 


SI 


2 


30.4 


119 


554 


591 


627 


664 


700 


737 
*098 


773 
*134 


809 


845 


882 


• 9 


36-4 


36 


.0 


35 


• 1 


34.2 


120 


918 


954 


990 


*026 


*062 


*170 


*206 


*242 


























ov o 


/ , *5^ tJO 


121 


08 278 


314 


350 


386 


422 


457 


493 


529 


564 


600 


• 1 


3.7 


3 


7 


3-6 


3.5 


122 


636 


671 


707 


742 


778 


813 


849 


884 


92C 


955 


.2 


7 


.5 


7 


4 


7 


2 


70 


123 


990 


*026 


*061 


*09t 


*131 


*166 


*202 


*237 


*272 


*307 


• 3 


11 


.2 


11 


1 


10 


8 


10.5 


124 


09 342 


377 


412 


447 


482 


517 


552 


586 


621 


656 


• 4 


15 





14 


8 


14 


4 


14.0 


125 


691 


725 


760 


795 


830 


864 


899 


933 


968 


*002 


■ 5 


18 


7 


18 


5 


18 


C 


17.5 


126 


10 037 


071 


106 


140 


174 


209 


243 


277 


312 


346 


e 


22 


5 


22 


2 


21 


e 


21.0 


127 380 


414 


448 


483 


517 


551 


585 


619 


653 


687 


• 7 


26 


2 


25 


9 


25 


2 


24.5 


128 721 


755 


789 


822 


856 


890 


924 


958 


991 


*025 


8 


30 





29 


6 


28 


8 


28.0 


129 


11 059 
394 


092 


126 


160 
494 


193 
528 


227 
561 


260 


294 


327 


361 


• 9 


33 


7 


33 


3 


32 


4 


31.5 


130 


427 


461 


594 


627 


661 


694 


o^ o >• oo ork 


131 


727 


760 


793 


826 


85§ 


892 


925 


958 


991 


*024 


.1 


34 


3 4 


Od 

33 


0/5 

3.2 


132 


12 057 


090 


123 


156 


189 


221 


254 


287 


320 


352 


• 2 


6 


9 


6 


8 


6 


6 


6.4 


:i33 385 


418 


450 


483 


515 


548 


580 


613 


645 


678 


3 


10 


.1 


10 


2 


9 


9 


9.6 


134 710 


743 


775 


807 


84C 


872 


904 


937 


969 


*001 


.4 


13 


IJ 


13 


6 


18 


2 


12.8 


135 


13 033 


065 


097 


130 


162 


194 


226 


258 


290 


322 


5 


17 


f 1 

i' 


17 


C 


16 


5 


16. 


136 


354 


386 


417 


449 


48] 


513 


545 


577 


608 


640 


■ 6 


20 


'i 


20 


4 


19 


8 


19.2 


137 


672 


703 


785 


767 


798 


830 


862 


893 


925 


956 


• 7 


24 


1 


23 


8 


23 


1 


22.4 


138 


988 


*019,*051 


*082 


*118 


*145 


*176 


*207 


*239 


*270 


8 


27 


6 


27 


2 


26. 


4 


25.6 


139 


14 301 
613 


332 


364 


395 


426 


457 


-488 


519 


550 


582 


9 


31 


6 


30. 


6 


29. 


7 


28.8 


140 


644 


675 


706 


736 


767 


798 


82Q 


860 


891 




141 


922 


952 


983,*014 


*045 


*G75*10g 


*137 


*167 


*198 


.1 


ox 

3 1 


3.1| 


30 


2.9 


142 


15 229 


259 


290: 320 


351 


38l 


412 


442 


473 


503 


• 2 


6 


3 


(5. 


2 


6. 


C 


58 


143 


533 


564 


5941 624 


655 


685 


715 


745 


776 


806 


• 3 


9 


4 


9. 


3 


9. 





8.7 


144 


83P 


866' 896^ 926 


956 


987 *017 *047 


*077 


*107 


• 4 


12 


6 


12. 


4 


12. 


c 


11.6 


145 


16 137 


1661 196i 226 


256 


286 


316 346 


376 


405 


• 5 


15 


7 


15- 


5 


15. 


c 


14.5 


146 


435 


465 494! 524 


554 


584 


61§ 643 


672 


702 


■ 6 


18 


9 


18. 


6 


18. 





17.4 


147 


731 


7611 791; 820 


849 


879 


908! 938 


967 


997 


• 7 


22 





21. 


7 


21. 


c 


20.3 


148 


17 026 


055 


085 


114 


143 


172 


202| 231 


260 


289 


.8 


25 


2 


24. 


8 


24. 





23.2 


149 


318 
609 




348 


377 


406 


435 


464 


493 522 


551 


580 


■ 9 


28 


3 


27- 


« 


i:7. 





26.1 


150 


638 

1 


&67 

2 


696 
3 


725 

4 


753 
5 


782 


811 


840 
8 


869 
9 




N. 


6 7 








P. 


P 


• 







640 









TABLE v.— LOGARITHMS 


OF NUMBERS 


, 










N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


150 


17 609 


638 


667 


696 


725 


753 


782 


811 


840 


869 


<^^~^ <^ C~k rf^ M4 
























29 28 27 


151 


897 


926 


955 


984 


*012 


*04l 


*070 


*098 


*127 


*156 


.1 


2- 


9 28 


27 


152 


18 104 


213 


241 


270 


298 


327 


355 


384 


412 


445 


• 2 


5. 


8 5 


• 6 


5 


4 


153 


469 


497 


526 


554 


582 


611 


639 


667 


695 


724 


• 3 


8. 


7 8 


.4 


8 


1 


154 


752 


780 


808 


836 


864 


893 


921 


949 


977 


*005 


.4 


11. 


6 11 


.2 


10 


8 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


256 


284 


.5 


14. 


5 14 


• 


13 


5 


158 


312 


340 


368 


396 


423 


451 


479 


507 


534 


562 


.6 


17- 


4 16 


8 


16 


2 


157 


590 


617 


645 


673 


700 


728 


755 


783 


810 


838 


• 7 


20. 


3 19 


6 


18 


9 


158 


865 


893 


920 


948 


975 


*003 


*030 


*057 


*085 


*112 


.8 


23- 


2 22 


.4 


21 


6 


159 


20 139 


1G7 


194 


221 


249 
520 


276 
547 


303 
574 


330 
601 


357 
628 


385 


.9 


26. 


1 25 


.2 


24 


3 


160 


412 


439 


466 


493 


655 


26 26 








_ 
















161 


682 


709 


736 


763 


790 


817 


844 


871 


898 


924 


• 11 


2.6 


2-6 


162 


951 


978 


*005 


*032 


*058 


*085 


*112 


*139 


*165 


*192 




.2 


5.3 


5 


.2 


163 


21 219 


245 


272 


298 


325 


352 


378 


405 


431 


458 




.3 


7-9 


7 


8 


164 


484 


511 


537 


564 


590 


616 


643 


669 


695 


722 




.4 10.6 


10 


4 


165 


748 


774 


801 


827 


853 


880 


906 


932 


958 


984 




.5 13.2 


13 


.0 


166 


22 Oil 


037 


063 


089 


115 


14l 


167 


193 


219 


245 




• 6 15.9 


15 


-6 


167 


271 


297 


323 


349 


375 


401 


427 


453 


479 


505 




.7 18. 5 


18 


2 


168 


531 


557 


582 


608 


634 


660 


686 


711 


737 


763 




.8 21.2 


20 


8 


169 


7a8 


814 


840 


865 


891 


917 


942 


968 


994 


*019 




.9 23.8 


23 


•4 


170 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 


tfn'r* £"% rr c% M 


171 


299 


825 


350 


375 


401 


426 


451 


477 


502 


527 


■ 1 


2.' 


3 2.5 


/5» 

2.4 


172 


553 


578 


603 


628 


653 


679 


704 


729 


754 


779 


• 2 


5. 


L 5 


.0 


48 


173 


804 


829 


855 


880 


905 


930 


955 


980 


*005 


*030 


• 3 


7. 


3 7 


.5 


7.2 


174 


24 055 


080 


105 


129 


154 


179 


204 


229 


254 


279 


• 4 


10.: 


2 10 





9-6 


175 


304 


328 


353 


378 


403 


427 


452 


477 


502 


526 


■ 5 


12.' 


1 12 


.5 


12-0 


176 


55l 


576 


600 


625 


650 


674 


699 


723 


748 


773 


■ 6 


15.; 


3 15 


.0 


14.4 


177 


797 


822 


846 


871 


895 


920 


944 


968 


903 


*017 


■ 7 


17.! 


ill 


.5 


16-8 


178 


25 042 


066 


091 


115 


139 


1G4 


188 


212 


237 


261 


• 8 


20.^ 


120 


.0 


19-2 


179 


285 
527 


309 
55l 


334 
575 


358 

599 


382 

623 


406 
647 


430 
672 


455 


479 


503 


.9 


22. i 


5 22.51 


21.6 


180 


696 


720 


744 


tf^o" f\'r* 


181 


768 


792 


816 


840 


863 


887 


911 


935 


959 


983 


.1 i 


2.3 


2.3 


182 


26 007 


031 


055 


078 


102 


126 


150 


174 


197 


221 


■ 2 ^ 


17 


4 


6 


183 


245 


2G9 


292 


316 


340 


363 


387 


411 


434 


458 


.8 ' 


^o 


6 


9 


184 


482 


505 


529 


552 


576 


599 


623 


646 


670 


693 


.4 { 


).4 


9 


2 • 


185 


717 


740 


764 


787 


811 


834 


858 


881 


904 


928 


.51] 


L.7 


11 


5 


186 


951 


974 


998 


*021 


*044: 


*068 


*091 


*114 


*137 


*161 


.6 1^ 


l.l 


13 


8 


187 


27 184 


207 


230 


254 


277 


300 


323 


346 


369 


392 


• 7 It 


5.4 


16 


1 


188 


416 


439 


462 


485 


508 


531 


554 


577 


600 


623 


.8 1{ 


{•8 


18 


4 


189 


646 


669 


692 
921 


715 
944 


738 
966 


761 


784 


806 


829 


852 


.9 2] 


..1 


20.7 


190 


875 


898 


989 


*012 


*035 


*058 


*080 


C^O Of» 07 


191 


23 103 


126 


149 


17l 


194 


217 


239 


262 


285 


307 


.1 


2.2 


I 2.21 2.1 


192 


330 


352 


375 


398 


420 


443 


465 


488 


510 


533 


.2 


4. J 


: 4 


■ i' 4.3 


193 


555 


578 


600 


623 


645 


668 


690 


713 


735 


758 


.3 


6.' 


1 6 


6 6.4 


194 


780 


802 


825 


847 


869 


892 


914 


936 


959 


981 


■ 4 


9.( 


: 8 


• 8 8-6 


195 


29 003 


025 


048 


070 


092 


114 


137 


159 


181 


203, 


.5 


11. i 


? 11 


.0,10-7 


196 


225 


248 


270 


292 


314 


336 


358 


380 


402 


424 


• 6 


13. f 


5 13 


.2:12.9 


197 


446 


468 


490 


512 


534 


556 


578 


600 


622 


644 


• 7 


15.' 


( 15 


■41 


15.0 


198 


666 


638 


710 


732 


754 


776 


798 


820 


84l 


863 


• 8 


18. ( 


) 17 


.6! 


17.2 


199 


885 


907 


929 
146 


950 


972 


994 


*016 


*038 


*G59 


*08l 


.9 


20.2 


I 19.8119.3 


300 


30 103 



124 


168 
3 


190 
4 


2ll 
5 


233 
6 


254 

7 


276 
8 


298 
9 




^■fl^sa 




N. 


1 


2 


P.P. 



641 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 

124 


3 

146 


3 

168 


4 

190 


5 

211 


6 

233 


7 
254 


8 
276 


9 

298 


P.P. 


300 


30 103 


























33 31 


201 


319 


341 


363 


384 


406 


427 


449 


470 


492 


513 




.1 


2.21 


2.1 


202 


535 


556 


578 


599 


621 


642 


664 


685 


707 


728 




• 2 


4. 


4 


4.2 


203 


749 


771 


792 


813 


835 


856 


878 


899 


920 


941 




.3 


6. 


fi 


6.3 


204 


963 


984 


*005 


*027 


*048 


*069 


*090 


*112 


*133 


*154 


.4 


8- 


8 


8.4 


205 


31 175 


196 


217 


239 


260 


281 


302 


323 


344 


365 


.5 


11. 





10.5 


208 


386 


408 


429 


450 


471 


492 


513 


534 


555 


576 


.6 


IS- 


2 


12-6 


207 


597 


618 


639 


660 


681 


702 


722 


743 


764 


785 


.7 


IS- 


4 


14-7 


208 


806 


827 


848 


869 


890 


910 


931 


952 


973 


994 


.8 


17- 


6 


16-8 


209 


32 014 
222 


035 
242 


056 


077 


097 


118 


139 


160 


180 


201 


.9 


19- 


8 


18.9 


310 


263 


284 


304 


325 


346 


366 


387 


407 
















30 30 


211 


428 


449 


469 


490 


510 


531 


551 


572 


592 


613 


.1 


2-0 


2.0 


212 


633 


654 


674 


695 


715 


736 


756 


776 


797 


817 


.2 


4 


1 


4.0 


213 


838 


858 


878 


899 


919 


940 


960 


980 


*001 


*021 


.3 


6 


1 


6-0 


214 


33 041 


061 


082 


102 


122 


142 


163 


183 


203 


223 




• 4 


8 


2 


8-0 


215 


244 


264 


284 


304 


324 


344 


365 


385 


405 


425 




.5 


10 


2 


10.0 


216 


445 


465 


485 


505 


525 


546 


566 


586 


606 


626 




.6 


12 


3 


12.0 


217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


825 




.7 


14 


3 


14.0 


218 


845 


865 


885 


905 


925 


945 


965 


985 


*004 


*024 




.8 


16 


4 


16.0 


219 


34 044 
242 


064 
262 


084 
28l 


104 
301 


123 
321 


143 


163 
360 


183 
380 


203 


222 


.9 


18 


4 


18.0 


330 


341 


400 


419 


19 19 












221 


439 


459 


478 


498 


518 


537 


557 


576 


596 


615 


.1 


1-9 


1.9 


222 


635 


655 


674 


694 


713 


733 


752 


772 


791 


811 


.2 


3 


-9 


3-8 


223 


830 


850 


869 


889 


908 


928 


947 


966 


986 


*CC5 


.8 


5 


-8 


5.7 


224 


35 025 


044 


063 


083 


102 


121 


141 


160 


179 


199 


.4 


7 


8 


7-6 


225 


218 


237 


257 


276 


295 


314 


334 


353 


372 


391 


.5 


9 


-7 


9-5 


226 


411 


430 


449 


468 


487 


507 


526 


545 


564 


583 


.6 


11 


-7 


11-4 


227 


602 


621 


641 


660 


679 


698 


717 


736 


755 


774 


.7 


13 


-6 


13.3 


228 


793 


812 


831 


850 


869 


888 


907 


926 


945 


964 


.8 


15 


-6 


15.2 


229 


983 


*002 


*021 
210 


*040 


*059 
248 


*078 


*097 


*116 


*135 


*154 


.9 


17 


• 5 


17.1 


330 


36 173 


191 


229 


267 


286 


305 


323 


342 


18 18 
















231 


361 


380 


399 


417 


436 


455 


474 


492 


511 


530 


.1 


1.8 


1.8 


232 


549 


567 


586 


605 


623 


642 


661 


679 


698 


717 




2 


3 


7 


3.6 


233 


735 


754 


773 


791 


810 


828 


847 


866 


884 


903 




3 


5 


-5 


5.4 


234 


92l 


940 


958 


977 


996 


*014 


*033 


*051 


*070 


*C88 




4 


7 


• 4 


7.2 


235 


37 107 


125 


143 


162 


180 


199 


217 


236 


254 


273 




5 


9 


-2 


9.0 


236 


291 


309 


328 


346 


364 


383 


401 


420 


438 


456 




6 


11 


-1 


10.8 


237 


475 


493 


511 


530 


548 


566 


584 


603 


621 


639 




7 


12 


-9 


12.6 


238 


657 


676 


694 


712 


730 


749 


767 


785 


803 


821 




8 


14 


-8 


14-4 


239 


840 
38 021 


858 
039 


876 
057 


894 
075 


912 
093 


930 
111 


948 
129 


967 
147 


985 


*CC3 




9116 


-6116.2 


340 


165 


183 


17 17 
























241 


201 


219 


237 


255 


273 


291 


309 


327 


345 


363 


.1 


1-7 


1-7 


242 


381 


399 


417 


435 


453 


471 


489 


507 


525 


543 


.2 


35 


3-4 


243 


560 


578 


596 


614 


632 


650 


667 


685 


703 


721 


.3 


5-2 


5-1 


244 


739 


757 


774 


792 


810 


828 


845 


8L3 


881 


899 


.4 


7-C 


6-8 


245 


916 


934 


952 


970 


987 


*005 


*023 


*040 


*058 


*076 


.5 


8-7 


8-5 


246 


39 093 


111 


129 


146 


164 


181 


199 


217 


234 


252 


.6 


10-5 


10-2 


247 


269 


287 


305 


322 


340 


357 


375 


392 


410 


427 


.7 


12-2 


11-9 


248 


445 


462 


480 


497 


515 


532 


550 


567 


585 


602 


-8 


14-0 


13-6 


249 


620 


637 


655 


672 


689 


707 


724 
898 

6 


742 
915 

7 


759 
933 

8 


776 


• 91 


15.7 


15.3 


350 


794 


8ll 
1 


828 
3 


846 
3 


863 
4 


881 
5 


950 
9 




N. 











P. 


P 


• 



642 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 



250 



O 



39 794 



251 
252 
253 
254 
255 
253 
257 
258 
259 

360 

261 
262 
263 
264 
265 
266 
267 
268 
269 

270 

271 
272 
273 
274 
275 
276 
277 
278 
279 

280 

281 
282 
283 
284 
285 
286 
287 
288 
289 

290 

291 
292 
293 
294 
295 
296 
297 
298 
299 

300 



N. 



40 



41 



967 
140 



811 



984 
157 



828 



312, 329 
483 500 
654 671 
8241 841 
993,*010 
162, 179 
330 346 



497 



*002 
174 
346 
517 
688 
858 

*027 
195 
383 



846 



514 530 



42 



664' 680 
8301 846 
995*012 
160 177 
3241 341 
488' 504 



651 
813 
975 



43 136 



44 



297 
457 
616 
775 
933 
091 
248 
404 
560 



716 



45 



46 



870 
025 
178 
332 
484 
636 
788 
939 
090 



240 



47 



389 
538 
687 
834 
982 
129 
27^ 
421 
567 

712 







687 
829 
991 



152 



313 
473 
632 
791 
949 
106 
263 
420 
576 



73l 



886 
040 
194 
347 
499 
652 
803 
954 
105 



255 



404 
553 
701 
849 
997 
144 
290 
436 
581 

726 



697 
863 

*028 
193 
357 
521 
683 
846 

*007 



*019 
191 
363 
534 
705 
875 

*044 
212 
380 



863 

*036 
209 
380 
55l 
722 
892 

*061 
229 
397 



547 564 



714' 730 
880 896 

*045 *031 
209, 226 
373| 390 
537| 553 
700 716 
862 878 

*023 *040 



168 184 200 



329 345 

489 505 

648 664 

806 822 

965 980 

122 138 

279 295 

435 451 

591 607 



7471 762 



901i 
055 
209 
382. 
515 
667, 
818 
969 
120 



917 
071 



361 
520 
680 
838 
996 
154 
310 
467 
622 



778 



932 

086 



2241 240 



377 
530 
682 
833 
984 
135 



269 284 



419' 434 
568' 583 
716; 731 
864' 879 
*011 *026 
158: 173 
305 319 
451 465 
596i 610 

741 755 



393 

545 
697 
848 
999 
150 



299 



449 
597 
746 
894 
*041 
188 
334 
480 
625 

770 



6 7 8 



881 

*054 
226 
398 
5691 
739| 
908 

*077 
246 
413 

581 



n * 



747 
913 

*078 
242 
406 
569 
732 
894 

*O50 



898 

*071 
243 
415 
586 
756 
925 
094 
263 
430 

597 



764 
929 

*094 
259 
423 
586 
748 
910 

*072 



216! 233 



377 
536 
695 
854 
*012 
169 
326 
482 
638 



393 
552 
7ll 
870 
*028 
185 
342 
498 
653 



793 809 



948 

102 
255 
408 
560 
712 
864 
*014 
165 



963 
117 
270 
423 
576 
727 
879 
*029 
180 



314 



464 
612 
761 
908 
*055 
202 
348 
494 
639 



329 



479 
627 
775 
923 
*070 
217 
363 
509 
654 



784 799 



643 



915 

*088 
260 
432 
603 
773 
942 

*111 
279 
447 



614 



780 
946 

*111 
275 
439 
602 
765 
927 

*088 

249 

409 
568 
727 
886 
*043 
201 
357 
513 
669 



824 



978 
132 
286 
438 
591 
743 
894 
^044 
195 



344 



493 
642 
790 
938 
^085 
232 
378 
523 
668 

813 



933 

*105 
277 
449 
620 
790 
959 

*128 
296 
464 



950 



631 

797 
962 

*127 
292 
455 
618 
781 
943 

*104 

265 



425 
584 
743 
901 
*059 
216 
373 
529 
685 



839 



994 
148 
301 
454 
606 
758 
909 
*059 
210 



359 



508 
657 
805 
952 
*100 
246 
302 
538 
683 

828 



*123 
295 
466 
637 
807 
976 

*145 
313 
480 



647 

813 
979 

*144 
308 
472 
635 
797 
959 

*120 



281 



441 
600 
759 
917 
^075 
232 
389 
545 
700 



855 



*009 
163 
316 
469 
621 
773 
924 

*075 
225 



374 



523 
672 
820 
967 
*114 
261 
407 
552 
697 

842 



P. P. 



17_ 

1-71 
3-5 
5.2 
70 
8-7 
10-5 
.7!l2.2 11.9 

8 14.0 13.6 

9 15.7 15-3 



17 

1.7 
3-4 
5-1 
6.8 
8.5 
10.2 



16_ 16 

6 

■ 2 

• 8 

• 4 
.0 
.6 
.2 

8 

■ 4 



.1 


1 


6 


1. 


.2 


3 


3 


3- 


.3 


4 


9 


4. 


.4 


6 


6 


6. 


.5 


8 


2 


8. 


.6 


9 


9 


9. 


.7 


11 


5 


11. 


8 


13 


2 


12. 


.9 


14 


8 


14. 





15 


15 


.1 


1.5 


1.5 


.2 


3.1 


3.0 


.3 


4-6 


4.5 


.4 


6-2 


6-0 


.5 


7.7 


7.5 


.6 


93 


9.0 


.7 


10.8 


10-5 


.8 


12.4 


12.0 


.9 


13.9 


13.5 





14 


14 


.1 


1.4 


1-4 


o 


2.9 


28 


.3 


4-3 


4-2 


.4 


5.8 


5.6 


.5 


7.2 


7.0 


.6 


8.7 


8.4 


.7 


10.1 


9 8 


.8 


11-6 


11.2 


■ 9 


130 


12.6 



P. p. 



TABLE v.— LOGARITHMS OF NUMBERS. 




P. P. 



13 

1-4 



14 


1.4 


2 


8 


4 


2 


5 


6 


7 





8 


4 


9 


8 


11 


2 


12 


6 





15 


13 


.1 


13 


13 


.2 


2.7 


2 B 


.3 


4.0 


3.9 


.4 


54 


5.2 


.5 


6.7 


6.5 


• 6 


8.1 


7.8 


.7 


9.4 


91 


.8 


10.8 


10.4 


.9 


12.1 


11.7 



.1 

.2 
.3 
.4 
.5 
n 

7 

8 

.9 



1^ 


12 


1-2 


1.2 


2.5 


2.4 


3.7 


3.0 


5.0 


48 


6.S 


6.0 


7-5 


7.2 


8.7 


8.4 


10.0 


9.6 


11 S 


10. ff 



p.p. 



644 



■■ ' • '■* 



N. 

J50 

151 






TABLE v.— LOGARITHMS OF NUMBERS. 











1 

419 


3 

431 


3 

444 


4 

456 
580 


5 

489 
592 


6 

481 
605 


7 

493 
617 


8 
506 


9 

518 


P. P. 


54 407 
530 


13 

.11 1.2 




543 


555 


568 


629 


642 




152 


654 


666 


679 


691 


703 


716 


728 


740 


753 


765 


.2 


2 


.f) 




(53 


777 


790 


802 


814 


826 


839 


851 


863 


876 


888 


.3 


3 


.7 




154 


900 


912 


925 


937 


949 


961 


974 


986 


998^*010 


• 4 


f) 


n 




{55 


55 023 


035 


047 


059 


071 


084 


096 


108 


120 


]33 


• 5 


6 


?. 




{56 


145 


157 


189 


181 


194 


206 


218 


230 


242 


254 


• 6 


7 


fS 




(57 


267 


279 


291 


303 


315 


327 


340 


352 


364 


376 


• 7 


8 


7 




{58 


388 


400 


412 


424 


437 


449 


461 


473 


485 


497 


.8 


10 


n 




{59 
560 


509 


521 
642 


533 
654 


545 
666 


558 
678 


570 
690 


582 


594 


606 


618 


• 9 


11 


2 




630 


702 


714 


726 


738 


12 




















(61 


750 


762 


775 


787 


799 


811 


823 


835 


847 


859 


.1 


12 




(82 


871 


883 


895 


907 


919 


931 


943 


955 


966 


97fi 


• 2 


2 


4 




(63 


990 


*002 


*014 


*026 


*038 


*050 


*062 


*074 


*086 


*098 


.8 


3 


R 




{64 


58 110 


122 


134 


146 


158 


170 


181 


193 


20 v5 


217 


.4 


4 


8 




(65 


229 


241 


253 


265 


277 


288 


300 


312 


324 


336 


.5 


6 







(66 


348 


360 


372 


383 


395 


407 


419 


431 


443 


455 


.6 


7 


2 




(67 


466 


478 


490 


502 


514 


525 


537 


549 


561 


573 


.7 


8 


4 




(68 


585 


596 


608 


620 


632 


643 


655 


667 


679 


691 


• 8 


9 


6 




(69 
3?0 


702 


714 


726 


738 
855 


749 
867 


761 
879 


773 
890 


785 
902 


796 


808 
925 


.8 


10 


8 




820 


832 


843 


914 


-. ^r 




}71 


937 


949 


961 


972 


984 


996 


*007 


*019 


*031 


*042 


.1 


XX 

1-1 




J72 


57 054 


066 


077 


089 


101 


112 


124 


136 


147 


159 


.2 


2 


3 




}73 


171 


182 


194 


206 


217 


229 


240 


252 


264 


275 


.3 


3 


4 




574 


287 


299 


310 


322 


333 


345 


357 


368 


380 


391 


.4 


4 


6 




J75 


403 


414 


426 


438 


449 


461 


472 


484 


495 


507 


.5 


5 


7 




J76 


519 


530 


542 


553 


565 


576 


588 


599 


611 


622 


.6 


6 


9 




577 


634 


645 


657 


668 


680 


691 


703 


714 


726 


737 


.7 


8 







378 


749 


760 


772 


783 


795 


806 


818 


829 


841 


852 


.8 


9 


2 




J79 


864 


875 


887 


898 


909 


921 


932 


944 


955 


967 


• 9 10.3 




380 


978 


990 


*001 


*012 


*024 
138 


*C35 
149 


*047 
161 


*058 
172 


*069 
183 


*G81 
195 


•fl -1 




381 


58 092 


104 


115 


126 


.1 


X X 

1-x 




382 


206 


217 


229 


240 


252 


263 


274 


286 


297 


308 


.2 


2 


2 




583 


320 


331 


342 


354 


365 


376 


388 


399 


410 


422 


.3 


3 


3 




384 


433 


444 


455 


467 


478 


489 


501 


512 


523 


535 


.4 


4 


4 




385 


546 


557 


568 


580 


591 


602 


613 


625 


636 


647 


.5 


5 


5 




386 


658 


670 


681 


692 


703 


715 


726 


737 


748 


760 


• 6 


6 


6 




387 


771 


782 


793 


804 


816 


827 


838 


849 


861 


872 


.7 


7 


7 




388 


883 


894 


905 


916 


928 


939 


950 


961 


972 


984 


.8 


8 


8 




389 


995 


*006 
117 
229 


*017 
128 
240 


*028 
140 
251 


*039 
151 
262 


*050 


*062 


*073 


*084 


*095 


.9 9-9 

H7i 




390 


59 106 
217 


162 


173 


184 


195 


206 




391 


273 


284 


295 


306 


317 


.1 


To 




392 


328 


339 


351 


362 


■ 373 


384 


395 


4ce 


417 


428 


.2 


2.1 




393 


439 


450 


461 


472 


483 


494 


505 


516 


527 


538 


.3 


3.1 




394 


549 


560 


571 


582 


593 


604 


615 


626 


637 


648 


.4 


4.2 




395 


659 


6701 


681 


692 


703 


714 


725 


736 


747 


758 


.5 


5-S 


, 


396 


769 


780 


791 


802 


813 


824 


835 


846 


857 


868 


.6 


6.3 




397 


879 


890 


901 


91^ 


923 


933 


944 


955 


966 


977 


.7 


7.3 




398 


988 


999 


*010 


•^021 


*032 


*043 


*053 


*064 


*075 


*086 


.8 


8.4 




399 


80 097 


108 


119 


130 


141 
249 

4 


151 
260 

5 


162 
271 

6 


173 
282 

7 


184 


195 


.9 9.i 




400 


2C6 


217 
1 


227 
2 


238 
3 


293 
8 


303 
9 




N. 





P 


.P. 







645 



r' 









TABLE v.— LOGARITHMS OF NUMBERS. 




N. 





1 
217 


2 


3 


4 

249 


5 


6 


7 
282 


8 

293 
401 


9 

303 
412 


P. 


P. 


400 


80 206 


227 238 


260 


271 






401 


314 


325 


336 347 


357 


368 


379 390 


402 


422 


433 


444' 455 


466 


476 


487 


498 


509 


519 






403 


530 


541 


552 563 


573 


584 


595 


606 


616 


627 




11 


404 


638 


649 


659 670 


681 


692 


702 


713 


724 


735 


.1 


1.1 


405 


745 


756 


767i 777 


788 


799 


810 


820 


831 


842 


.2 


2.2 


406 


852 


863 


874 884 


895 


906 


9161 927 


938 


949 


.3 


3-3 


407 


959 


970 


981' 991 


*002 


*013 


*023-*034 


*044 


*055 


• 4 


4-4 


408 


61 066 


076 


087 098 


108 


119 


130 


140 


151 


161 


• 5 


5.5 


409 


172 


183 


193 


204 


215 


225 


236 


246 


257 


268 


• 6 
.7 

• 8 

• 9 


6 6 
77 
8-8 
9.9 


410 


278 


289 


299 


310 


320 


33l 


342 


352 


363 


373 


411 


384 


394 


405 


416 


426 


437 


447 458 


468 


479 


412 


489 


500 


511 


521 


532 


542 


553 563 


574 


584 






413 


595 


605 


616 


626 


637 


647 


658 668 


679 


689 






414 


700 


710 


721 


731 


742 


752 


763 773 


784 


794 






415 


805 


815 


825 


836 


846 


857 


867 878 


888 


899 




10 


416 


909 


920 


930, 940 


951 


961 


972 982 


993 


*003 


.1 


l.D 


417 


62 013 


024 


034^ 045 


055 


065 


076 


086 


097 


107 


.2 


2.1 


418 


117 


128 


1381 149 


159 


169 


180 


190 


20b 


211 


.3 


3-1 


419 


221 


232 


242 252 


263 


273 


283 


294 


304 


314 


.4 
.5 
.6 
.7 
.8 


4-2 
5-2 
6-3 
7.3 
8.^ 


430 


325 
428 


335 


345| 356 


366 


376 
480 


387 
490 


397 


407 


418 
521 


421 


438 


449 459 


469 


500 


510 


422 


531 


.541 


552 562 


572 


582 


593 


603 


613 


624 


.9 


9.4 


423 


634 


644 


6541 665 


675 


685 


695 


706 


716 


726 






424 


736 


747 


757 767 


777 


788 


798 


808 


818 


828 






425 


839 


849 


859 


869 


879 


890 


900 


910 


920 


931 






426 


941 


951 


961 


971 


981 


992 


*002 


*012 


*022 


*032 






427 


63 043 


053 


063 


073 


083 


093 


104 


114 


124 


134 




10 


428 


144 


154 


164 


175 


185 


195 


205 


215 


225 


235 


.1 


1.0 


429 


245 


256 


266 


276 


286 


296 


306 


316 


326 


336 


.2 
.3 
.4 
.5 
.6 


20 
30 
4.0 
50 
6.0 


430 


347 


357 


367 


377 


387 


397 


407 


417 


427 


437 


431 


447 


458 


468 


478 


488 


498 


508 


518 


528 


538 


432 


548 


558 


568 


578 


588 


598 


608 


618 


628 


639 


• 7 


7-0 


433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 


.8 


80 


434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 


■ 9 


9 


435 


849 


859 


869 


879 


889 


899 


909 


919 


928 


938 






436 


948 


958 


968 


978 


988 


998 


*008 


*01'8 


*028 


*038 






437 


64 048 


058 


068 


078 


088 


098 


107 


117 


127 


137 






438 


147 


157 


167 


177 


187 


197 


207 


217 


226 


236 






439 


246 


2.56 


266 


276 


286 


296 


306 


315 


325 


335 


.1 
.2 
.3 

.4 


0% 

3.8 


440 


345 


355 


365 


375 


384 


394 
493 


404 


414 
512 


424 


434 


441 


444 


453 


463 


473 


483 


503 


522 


532 


442 


542 


552 


5621 571 


581 


591 


601 


611 


621 


. 630 


.5 


4-7 


443 


640 


650 


660, 670 


679 


689 


699 


709 


718 


728 


.6 


5.7 


444 


738 


748 


758: 767 


777 


787 


797 


806 


816 


826 


.7 


6.6 


445 


836 


846 


8551 865 


875 


885 


894 


904 


914 


923 


8 


7.g 


446 


933 


943 


953) 962 


972 


982 


992 


*00T 


*011 


*021 


• 9 


8.5 


447 


65 031 


040 


050' 060 


069 


079 


089 


098 


108 


118 






448 


128 


137 


147 


157 


166 


176 


186 


195 


205 


215 






449 


224 
32l 


234 


244 


253 


263 


273| 282 


292| 302 


311 






450 


331 
1 


340 
2 


350 


36C 
4 


369l 379 
1 . 


389 

7 


398 
S 


408 
9 


N. 





3 


5 


6 


»•• 


P, 



046 









TABLE V 


'.—LOGARITHMS OF NUMBERS. 










450 

451 





1 

331 

427 


2 

340 
437 


3 

350 
446 


4 

360 
456 


r 

369 
406 


G 

379 
475 


4 

389 
485 


8 
398 





r 


. P. 




65 32l 

417 


4Cj: 








494 


504 




452 


514 


523 


533 


542 


bb2 


562 


571 


581 


590 


eoc 




10 




453 


610 


619 


629 


638 


648 


657 


667 


677 


686 


696 


.1 


1-0 




454 


705 


715 


724 


734 


744 


753 


76S 


772 


782 


7G1 


• 2 


2.0 




455 


801 


810 


820 


830 


839 


849 


858 


8C8 


877 


887 


.3 


3-0 




456 


896 


906 


915 


925 


934 


944 


95S 


9C3 


972 


982 


• 4 


40 




457 


991 


*001 


=■=010 


*02G 


*029 


''=039 


*048 


*058 


*067 


=1=077 


■ 5 


50 




458 


66 GG6 


096 


105 


115 


124 


134 


143 


153 


162 


172 


6 


6 




459 
460 

461 


181 
276 
370 


190 
285 
379 


200 
294 
389 


209 
304 
398 


219 
313 
408 


228 
323 
417 


238 


247 


257 


266 


7 
8 
9 


70 
80 
9-0 




332 


342 


351 


360 




426 


436 


445 


455 




462 


464 


473 


483 


492 


502 


511 


520 


530 


539 


548 








463 


558 


567 


577 


586 


595 


605 


614 


623 


633 


642 








464 


652 


661 


670 


680 


689 


698 


708 


717 


726 


736 








465 


745 


754 


764 


773 


782 


792 


801 


810 


820 


829 




9 




466 


838 


848 


857 


~86G 


876 


885 


894 


904 


913 


922 


1 


0.9 




467 


931 


941 


950 


959 


969 


978 


987 


996 


*ooe 


*015 


2 


1 


.9 




468 


67 024 


034 


043 


052 


061 


071 


080 


08 9 


09S 


108 


3 


2 


8 




469 


117 


126 


136 


145 


154 


163 


173 


182 


191 


200 


4 


3 


8 





























4 


• 7 




470 

471 


210 
302 


219 
311 


228 
320 


237 
329 


246 
339 


256 
348 


265 


274 


283 


293 


6 
7 
8 


5 
6 
7 


•7 

■ 6 

6 




357 


366 


376 


385 




472 


394 


403 


412 


422 


431 


440 


449 


458 


467 


477 


9 


8 


5 




473 


486 


495 


504 


513 


523 


532 


541 


550 


559 


568 








474 


578 


587 


596 


605 


614 


623 


633 


642 


651 


660 








475 


669 


678 


687 


697 


706 


715 


724 


733 


742 


751 








476 


760 


770 


779 


788 


797 


806 


815 


824 


883 


842 








477 


852 


861 


870 


879 


888 


807 


9C6 


915 


824 


933 








478 


943 


952 


961 


970 


979 


988 


997 


*CC6 


*015 


*024 




9 




479 

■ 

480 


68 033 


042 


051 


060 
151 


070 
160 


079 
169 


088 

178 


097 
187 


106 


115 


1 
2 
3 




1 
2 


9 
8 
7 




124 


133 


142 


196 


205 '. 




481 


214 


223 


232 


24l 


250 


259 


268 


277 


286 


295 


4 
5 


3 
4 


6 
5 




482 


304 


313 


322 


331 


340 


349 


sea 


367 


376 


385 


6 


5 


4 




483 


394 


403 


412 


421 


430 


439 


448 


457 


466 


475 


7 


6 


3 




484 


484 


493 


502 


511 


520 


529 


538 


547 


556 


565 


8 


7. 


2 




485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


654 


9 


8. 


1 




486 


663 


672 


681 


690 


699 


708 


717 


726 


735 


744 








487 


753 


762 


770 


779 


788 


797 


806 


815 


824 


833 








488 


842 


851 


860 


868 


877 


886 


895 


904 


913 


922 








489 


931 

69 019 

108 


940 
028 
117 


948 
037 
126 


957 
046 
134 


966 


975 
064 

]52 


984 
073 


993 


*002 


*010 


1 
2 


8 




490 


055 

143 


081 


090 


099 




491 


16] 


170 


179 


187 '. 


0- 
1. 


o 
7 




492 


196 


205 


21A 


223 


232 


240 


249 


258 


267 


276 


8 


2. 


5 




>:93 


284 


293 


302 


311 


320 


328 


337 


346 


355 


364 


4 


3. 


4 




494 


372 


381 


390 


399 


408 


416 


425 


434 


443 


451 


5 


4. 


2 




495 


460 


469 


478 


487 


495 


504 


513 


522 


530 


539 


6 


5. 


1 




496 


548 


557 


565 


574 


583 


592 


eoo 


609 


618 


627 


7 


5. 


9 




497 


635 


-644 


653 


662 


670 


679 


688 


697 


705 


714 


8 


6. 


8 




498 


723 


731 


740 


749 


758 


766 


775 


784 


792 


8C1 


g 


7- 


6 




499 


810 
897 




819 
905 

1 


827 
914 

2 


836 
923 

3 


845 
931 

4 


85? 

940 

5 


862 
949 

6 


871 
958 

7 


879 


888 








600 


966 
8 


975 




N. 


9 


P. 


P. 





647 







TABLE V 


— loctAUithms of numbers. 




N. 





1 


3 


3 


4 


5 


6 


7 


8 


9 


P. P. 


500 


69 897 


905 


914 


923 


93T 


940 


949 
*036 


958 

*044 


966 
*053 


975 
*06l 




501 


984 


992 


*001 


*010 


*018 


*027 




502 


70 070 


079 


087 


096 


105 113 


122 


131 


139 


148 


9 


503 


157 


165 


174 


182 


191 20C 


208 


217 


226 


234 


.1 


0.9 


504 


243 


251 


260 


269 


277 286 


294 


3C3 


312 


320 


.2 


1.8 


505 


329 


337 


346 


355 


363 372 


380 


389 


398 


406 


• 8 


2.7 


506 


415 


423 


432 


44] 


449 


458 


466 


475 


483 


492 


.4 


36 


507 


501 


509 


518 


526 


535 


543 


552 


560 


569 


578 


• 5 


45 


508 


586 


. 595 


603 


612 


620 


629 


637 


646 


654 


663 


• 6 


5-4 


509 


672 


680 


689 


697 


706 


714 


723 


731 


740 


748 


• 7 
.8 
.9 


6.3 
72 
8.1 


510 


757 


765 


774 


782 


791 


799 


808 


816 


825 


833 


511 


842 


850 


859 


867 


876 


884 


893 


90l 


910 


918 




512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*003 




513 


71 Oil 


020 


028 


037 


045 


054 


062 


071 


079 


088 




514 


096 


105 


113 


121 


130 


138 


147 


155 


164 


172 




515 


180 


189 


197 


206 


214 


223 


231 


239 


248 


256 


r 8 


616 


265 


273 


282 


290 


298 


307 


315 


324 


332 


340 


.1' 


o.g 


517 


349 


357 


366 


374 


382 


391 


399 


408 


416 


424 


• 2 


1.7 


518 


•433 


441 


449 


458 


466 


475 


483 


491 


500 


508 


.3 


2.5 


619 


516 


525 


533 


542 


550 


558 


567 


575 


583 


592 


.4 
.5 


3.4 
4.2 


530 


600 
684 


608 


617 


625 
709 


633 
717 


642 
725 


650 
734 


659 
742 


667 
750 


675 
758 


.6 
.7 
.8 


5.1 
5.9 
68 


521 


692 


700 


522 


767 


775 


783 


792 


800 


808 


817 


825 


833 


842 


.9 


7.6 


523 


850 


858 


867 


875 


883 


891 


900 


908 


916 


925 




524 


933 


941 


949 


958 


966 


974 


983 


991 


999 


*007 




525 


72 016 


024 


032 


040 


049 


057 


065 


074 


082 


090 




526 


098 


107 


115 


123 


131 


140 


148 


156 


164 


173 




527 


181 


189 


197 


206 


214 


222 


230 


238 


247 


255 




528 


263 


271 


280 


288 


296 


304 


312 


321 


329 


337 


, 8 


529 


345 


354 


362 


370 


378 


386 


395 


403 


411 


419 


.1 
.2 

• 8 

• 4 
-5 


0.8 
1.6 
2.4 
3.2 
4.0 


530 


427 


436 


444 


452 


460 


468 


476 


485 


493 


501 


531 


509 


517 


526 


534 


542 


550 


558 


566 


575 


583 


532 


591 


599 


607 


615 


624 


632 


640 


648 


656 


664 


.6 


48 


533 


672 


681 


689 


697 


705 


713 


721 


729 


738 


746 


• 7 


5-6 


534 


754 


762 


770 


778 


786 


795 


803 


811 


819 


827 


.8 


6.4 


535 


835 


843 


851 


859 


868 


876 


884 


892 


900 


908 


.9 


7.2 


536 


916 


924 


932 


941 


949 


957 


965 


973 


981 


989 




537 


997 


*005 


*013 


*021 


*030 


*038 


*046 


*054 


*062 


*070 




538 


73 078 


086 


094 


102 


110 


118 


126 


134 


143 


151 




539 


159 
239 


167 
247 


175 


183 


191 
27l 


199 

279 


207 


215 


223 
303 


231 
311 




540 


255 


263 


287 


295 


.1 
• 2 


0%. 

1.5 f 


541 


319 


328 


336 


344 


352 


360 


368 


376 


384 


392 


542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


.8 


2.2 1 


543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


.4 


3 


544 


560 


568 


576 


584 


592 


600 


608 


615 


623 


631 


.5 


3."? 


545 


639 


647 


655 


663 


671 


679 


687 


695 


703 


711 


.6 


4.5 


546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 


.7 


5.2 
6 


547 


798 


806 


814 


822 


830 


838 


846 


854 


862 


870 


.8 


548 


878 


886 


894 


902 


909 


917 


925 


933 


941 


949 


9 


6.7- 


549 


957 


965 


973 


981 


989 


997 


*004 


*012 


*020 


*028 




550 


74 036 


044 
1 


052 
2 


060 


068 
4 


075 
5 


083 
6 


091 

7 


099 
8 


107 
9 


N. 





3 


" i 

P. P. 1 
















648 










r 



• 1 






TABLE v.— LOGARITHMS OF NUMBERS. 






550 

551 





1 


2 


3 

060 


4 

068 


5 

075 


6 

083 


7 
091 


8 
099 


9 

107 


P 


. P. 


74 036 


044 


052 






115 


123 


131 


139 


146 


154 


162 


170 


178 


186 




552 


194 


202 


209 


217 


225 


233 


241 


249 


257 


264 






553 


272 


280 


288 


296 


304 


312 


319 


327 


335 


343 






554 


351 


359 


366 


374 


382 


390 


398 


406 


413 


421 






555 


429 


437 


445 


453 


460 


468 


476 


484 


492 


499 






556 


507 


515 


523 


531 


538 


546 


554 


562 


570 


577 






557 


585 


593 


601 


609 


616 


624 


632 


640 


648 


655 


.1 
.2 
• 3 


8 


558 


663 


671 


679 


687 


694 


702 


710 


718 


725 


733 





.0 


559 


741 


749 


756 


764 


772 


780 


788 


795 


803 


811 


1 
2 


■ 6 

• 4 


560 


819 


826 


834 


842 


850 


857 


865 


873 


881 


888 


-4 
.5 


3 

4 


■ 2 
• 


561 


896 


904 


912 


919 


927 


935 


942 


950 


958 


966 


• 6 
.7 

• 8 
.9 


4 
5 
6 
7 


8 
6 
4 
2 


562 
563 
564 


973 

75 051 

128 


981 
058 
135 


.989 
066 
143 


997 
074 
151 


*004 
081 
158 


*012 
089 

166 


*020 
097 
174 


*027 
105 
182 


*035 
112 
189 


*043 
120 
197 


565 


205 


212 


220 


228 


235 


243 


251 


258 


266 


274 






566 


281 


289 


297 


304 


312 


320 


327 


335 


343 


350 






567 


358 


386 


373 


381 


389 


396 


404 


412 


419 


427 




, 


568 


435 


442 


450 


458 


460 


473 


480 


488 


496 


503 






569 

570 

571 
572 
573 
574 


511 


519 


526 


534 


541 


549 

625 

70l 
777 
853 
929 


557 
633 


564 
641 


572 
648 


580 

656 


.1 
.2 
.3 
.4 
.5 

• 6 
■ 7 

• 8 
.9 




587 

663 
739 
815 
891 


595 


602 


610 


618 

694 
770 
846 
92l 




671 
747 
823 
899 


679 
755 
830 
906 


686 
762 
838 
914 


709 
785 
861 
936 


717 
792 
868 
944 


724 
800 
876 
951 


732 
808 
883 
959 


7 


1 
2 
3 
3 
4 
5 
6 
6. 


f 
7 
5 
2 

7 


575 
578 
577 


967 

76 042 

117 


974 
050 
125 


982 
057 
132 


989 
065 
140 


997 
072 
147 


*004 
080 
155 


*012 
087 
162 


*019 
095 
170 


*027 
102 
178 


*034 
110 
185 


578 
579 

580 

581 


193 
268 


200 
275 


208 
283 

358 

432 


215 
290 

365 

440 


223 
298 

372 

447 


230 
305 

380 

455 


238 
313 

387 


245 
320 


253 
328 


260 
335 


ii 


7 


343 


350 


395 


402 


410 




417 


425 


462 


470 


477 


485 




582 


492 


500 


507 


514 


522 


529 


537 


544 


552 


559 






583 


567 


574 


582 


589 


596 


604 


611 


619 


626 


634 






584 


641 


648 


656 


663 


671 


678 


686 


693 


700 


708 






585 


715 


723 


730 


738 


745 


752 


760 


767 


775 


782 






586 


790 


797 


804 


812 


819 


827 


834 


841 


849 


856 




7 
0.7 
1.4 
2-1 


587 


864 


871 


878 


886 


893 


901 


908 


915 


923 


930 


.1 

.2 

• 3 
.4 
.5 
.6 

• 7 
.8 
.9 


588 
589 

590 

591 


937 
77 Oil 


945 
019 


952 

026 


960 
033 

107 


967 
041 


974 
048 


982 
055 

129 


989 
063 

136 


997 
070 


*004 
078 


085 
158 


092 

166 


100 


114 


122 


144 


151 


28 
3.5 


173 


181 


188 


195 


203 


210 


217 


225 


4-2 
4.9 
5.6 
6.3 


592 
593 
594 


232 
305 
378 


239 
313 
386 


247 
320 
393 


254 
327 
400 


261 
335 
408 


269 
342 
415 


276 
349 
422 


283 
356 
430 


291 
364 
437 


298 
37l 
444 


595 


451 


459 


466 


473 


481 


488 


495 


503 


510 


517 






596 


524 


532 


539 


546 


554 


561 


568 


575 


583 


590 






597 


597 


604 


612 


619 


626 


634 


641 


648 


655 


663 






598 


670 


677 


684 


692 


699 


706 


713 


721 


728 


735 






599 


742 


750 


.757 


764 


771 


779 


786 
858 


793 


809 


808 






600 


815 


822 

1 


829 


837 
3 


844 


85l 
5 


866 

7 


873 

8 


880 




N. 





2 


4 


6 


9 


P. 


P. 



649 







TABLE V 


.—LOGARITHMS OF NUMBERS. 




i 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


• ^- 1 


600 


77 815 


822 


829 


837 


844 


85l 


858 


866 


873 


880 






601 


887 


894 


902 


909 


916 


923 


931 


938 


945 


952 






602 


959 


967 


974 


981 


988 


995 


*003 


*010 


*017 


*024 






603 


78 031 


039 


046 


053 


060 067 


075 


082 


089 


096 




604 


103 


111 


118 


125 


132 139 


147 


154 


161 


168 






605 


175 


182 


190 


197 


204 211 


218 


226 


233 


240 






606 


247 


254 


261 


269 


276 


283 


290 


297 


304 


311 




1.5 
2.2 


607 
608 
609 


319 
390 
461 


326 
397 
469 


333 
404 
476 


340 
412 
483 


347 
419 
490 


354 
426 
497 


362 
433 
504 


369 
440 
511 


376 
447 
518 


383 

454 
526 


.1 

.2 
• 3 


610 


533 


540 


547 


554 


56l 


568 


575 


583 


590 


597 


.4 
.5 


30 
3-7 


611 
612 
613 
614 


604 
675 
746 
817 


611 
682 
753 
824 


618 
689 
760 
831 


625 
696 
767 
838 


632 
703 
774 
845 


639 
710 
781 
852 


646 
717 
788 
859 


654 
725 
795 
866 


661 
732 
802 
873 


668 
739 
810 
880 


.6 
.7 
-8 
.9 


4-5 - 1 
5.2 
60 
6.7 


615 


887 


894 


901 


908 


915 


923 


930 


937 


944 


951 






616 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 






617 


79 028 


035 


042 


049 


056 


063 


070 


078 


085 


092 






618 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 






619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 




1 


630 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 






621 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 


• 1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


7 


622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


u 
1 
2 
2 
3 
4 
4 
5 
6 


1 

% 1 


623 
624 
625 


449 
518 
588 


456 
525 
595 


462 
532 
602 


469 
539 
609 


476 
546 
616 


483 
553 
622 


490 
560 

629 


497 
567 
636 


504 
574 
643 


5li 
581 
650 


626 
627 
628 
629 


657 
727 
796 
885 


664 
733 
803 
872 


671 
740 
810 
879 


678 
747 
816 
886 


685 
754 
823 
892 


692 
761 
830 
899 


699 
768 
837 
906 


706 
775 
844 
913 


713 
782 
851 
920 


720 
789 
858 
927 


630 


934 
80 003 


941 
010 


948 
016 


954 
023 


96l 
030 


968 
037 


975 


982 


989 
058 


996 
065 




■ 


631 


044 


051 


632 


071 


078 


085 


092 


099 


106 


113 


120 


126 


133 




i 


633 


140 


147 


154 


161 


168 


174 


181 


188 


195 


202 




1 


634 


209 


216 


222 


229 


236 


243 


250 


257 


263 


270 






vJ35 


277 


284 


291 


298 


304 


311 


318 


325 


332 


339 






636 


345 


352 


359 


366 


373 


380 


386 


393 


400 


407 




7? 


637 
638 
639 


414 
482 
550 


421 
489 
557 


427 
495 
563 


434 
502 
570 


441 
509 
577 


448 
516 
584 


455 
523 
591 


461 
529 
597 


468 
536 
604 


475 
543 
611 


.1 
.2 
.3 




1 
1 


6 
3 
9 ' 


640 


618 


625 


63l 


638 


645 


652 


658 


6b5 


672 


679 


.4 
.5 


2 
3 


i ' 


641 

642 
643 
644 


686 
753 
821 
885 


692 
760 
828 
895 


699 
767 
834 
902 


706 
774 
841 
909 


713 
780 
848 

915 


719 
787 
855 
922 


726 
794 
861 
929 


733 
801 
868 
936 


740 
807 
875 
942 


746 
814 
882 
949 


.6 
.7 
.8 
• 9 


3 
4 
5 
5 


i ! 

2 


645 


956 


962 


969 


976 


983 


989 


996 


*003 


*010 


*016 




I 


646 


81 023 


030 


036 


043 


050 


057 


063 


070 


077 


083 




' 


647 


095 


097 


104 


110 


117 


124 


130 


137 


144 


151 






648 


157 


164 


171 


177 


134 


191 


197 


204 


211 


218 






649 


224 


231 


238 


244 


251 


258 


264 


271 


278 


284 






650 


291 


298 

1 


304 
2 


3ll 
3 


318 
4 


324 
5 


33l 
6 


338 

7 


345 
8 


35l 
9 




■i 


N. 





P 


P. 

i 
















650 












i 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 

298 


2 

304 
37l 


3 

311 
378 


4 

318 
385 


5 

324 
39l 


6 

331 
398 


7 

338 

405 


8 


9 

351 
418 


50 


81 29l 


345 
411 


51 


358 


365 


52 


425 


431 


438 


444 


451 


458 


464 


471 


478 


484 


53 


491 


498 


504 


511 


518 


524 


531 


538 


544 


551 


54 


558 


564 


571 


577 


584 


591 


597 


604 


611 


617 


55 


624 


631 


637 


644 


650 


657 


664 


670 


677 


684 


56 


690 


697 


703 


710 


717 


723 


730 


736 


743 


750 


57 


756 


763 


770 


776 


783 


789 


796 


803 


809 


816 


58 


822 


829 


836 


842 


849 


855 


862 


869 


875 


882 


59 


888 


895 


901 


908 
974 
040 


915 
980 

046 


921 
987 
053 


928 
994 
059 


934 

*000 

066 


941 

*007 

072 


948 
*013 


6<) 


954 


961 


967 
033 


61 


82 020 


026 


079 


62 


086 


092 


099 


105 


112 


118 


125 


131 


138 


145 


63 


151 


158 


164 


171 


177 


184 


190 


197 


203 


210 


64 


217 


223 


230 


236 


''AS 


249 


256 


262 


269 


275 


65 


282 


288 


295 


302 


308 


315 


321 


328 


334 


341 


66 


347 


354 


360 


367 


373 


380 


386 


393 


399 


406 


67 


412 


419 


425 


432 


438 


445 


451 


458 


464 


471 


68 


477 


484 


490 


497 


503 


510 


516 


523 


529 


536 


69 


542 
607 
672 


549 
614 
678 


555 
620 
685 


562 


568 
633 
698 


575 


581 


588 
653 
717 


594 
659 


601 
666 
730 


70 


627 
691 


640 
704 


646 
711 


71 


724 


72 


737 


743 


750 


756 


763 


769 


775 


782 


788 


795 


73 


801 


808 


814 


821 


827 


834 


840 


846 


853 


859 


74 


866 


872 


879 


885 


892 


898 


904 


911 


917 


924 


75 


930 


937 


943 


949 


956 


962 


969 


975 


982 


988 


76 


994 


*001 


*007 


*014 


*020 


*027 


*033 


*G39 


*046 


*052 


77 


83 059 


065 


071 


078 


084 


091 


097 


103 


110 


116 


78 


123 


129 


136 


142 


148 


155 


161 


168 


174 


180 


79 


187 
251 
314 


193 
257 
321 


200 


206 


212 


219 


225 


231 


238 


244 


80 


263 
327 


270 
334 


276 
340 


283 


289 


295 


302 


308 


81 


346 


353 


359 


365 


372 


82 


378 


385 


391 


397 


404 


410 


416 


423 


429 


435 


83 


442 


448 


455 


461 


467 


474 


480 


486 


493 


499 


84 


505 


512 


518 


524 


531 


537 


543 


550 


556 


562 


85 


569 


575 


581 


588 


594 


600 


607 


613 


619 


626 


86 


632 


638 


645 


651 


657 


664 


670 


676 


683 


689 


87 


695 


702 


708 


714 


721 


727 


733 


740 


746 


752 


88 


759 


765 


771 


778 


784 


790 


796 


803 


809 


815 


J9 


822 
885 
948 


828 
891 
9-54 


834 
897 
960 


841 
904 
966 


847 


853 


859 


866 


872 


878 
94l 


.90 


910 
973 


916 
979 


922 


929 


935 
998 


91 


985 


992 


*004 


92 


84 010 


017 


023 


029 


035 


042 


048 


054 


061 


067 


93 


073 


079 


086 


092 


098 


104 


111 


117 


123 


129 


94 


136 


142 


148 


154 


161 


167 


173 


179 


186 


192 


95 


198 


204 


211 


217 


223 


229 


236 


242 


248 


254 


96 


261 


267 


273 


279 


286 


292 


298 


304 


311 


317 


97 


323 


329 


335 


342 


348 


354 


360 


367 


373 


379 


98 


385 


392 


398 


404 


410 


416 


423 


429 


435 


441 


99 


447 
510 


454 
516 


460 

522 


466 
528 


472 
534 

4 


479 
541 

5 


485 
547 

6 


491 
553 

7 


497 


503 


00 


559 
8 


565 
9 


N. 





1 


3 


3 



P. p. 





7 


1 


0.7 


2 


1-4 


3 


2.1 


4 


28 


5 


3.5 


6 


4.2 


7 


49 


8 


5.6 


9 


6.3 



1 





2 


1 


3 


1 


4 


2 


5 


3 


6 


3 


7 


4 


8 


5 


9 


5 



6_ 

6 
3 
9 
6 
2 
9 
5 
2 
8 



1 





2 


1 


3 


1 


4 


2 


5 


3 


6 


3 


7 


4 


8 


4 


9 


5 



6 

• 6 
.2 

8 
-4 

• 

• 6 
■ 2 

• 8 

• 4 



P. P. 



651 









TABLE V .— LOGARITflMS OF NUMBERS. 








N. 





1 

516 


2 

522 


3 

528 


4 
534 


5 

541 


6 

547. 
609 


7 

553 
615 


8 

559 
62l 


9 

565 
627 


P. P. 


700 


34 510 
572 






701 


578 


584 


590 


596 


603 






702 


633 


640 


646 


652 


658 


664 


671 


677 


683 


689 








703 


695 


701 


708 


714 


720 


726 


732 


730 


745 


751 








704 


757 


763 


769 


776 


782 


788 


794 


800 


806 


813 






' 


705 


819 


825 


831 


837 


843 


849 


856 


862 


868 


874 








706 


880 


886 


893 


899 


905 


911 


917 


923 


929 


936 







707 


•942 


948 


954 


930 


966 


972 


979 


985 


991 


997 


• 1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


6_ 


708 


85 003 


009 


015 


021 


028 


034 


040 


046 


052 


058 


" 


D 


709 


064 

128 

187 
248 
309 
370 


070 

132 

193 
254 
315 
376 


077 

138 

199 
280 
321 
382 


083 

144 

205 
266 
327 
388 


089 

150 

2ll 
272 
333 
394 


095 

156 

217 
278 
339 
400 


101 

162 

223 
284 
345 
406 


107 

168 

229 
290 
35l 
412 


113 


119 


1 
1 
2 
3 
3 
4 
5 
5 


3 

9 


710 


174 


181 


6 
2 
9 
5 
2 
8 




711 
712 
713 
714 


236 
297 
357 
418 


242 
303 
363 
424 


715 


430 


436 


443 


449 


455 


461 


467 


473 


479 


485 








716 


491 


497 


503 


509 


515 


521 


527 


533 


540 


546 








717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 








718 


612 


618 


624 


630 


636 


642 


648 


655 


661 


667 








719 


673 

733 

793 
853 


679 

739 

799 
859 


685 

745 

805 
865 


691 

751 

8ll 
872 


697 

757 

817 
878 


703 
763 


709 

769 

829 
890 


715 


721 


727 


.1 
.2 
.3 
.4 
.5 






730 


775 

835 
896 


781 

841 
902 


787 

847 
908 


721 
722 


823 
884 


6 

0-6 


723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


1 
1 

2 


8 

4 




724 
725 


974 
36 034 


980 
040 


988 
048 


992 
052 


998 
058 


*004 
063 


*010 
069 


*016 
075 


*022 
08l 


*028 
087 




72R 


093 


099 


105 


m 


117 


123 


129 


135 


141 


147 


3 







727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 


• 6 
.7 
.8 
.9 


3 

4 


6 
2 
8 
4 




728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 




729 


273 


278 


284 


290 


296 


302 


308 


314 


320 


326 


4 
5 




730 


332 
391 


338 

397 


344 
403 


350 
409 


356 
415 


362 
42l 


368 

427 


374 

433 


380 


386 


- 






731 


439 


445 


732 


451 


457 


463 


469 


475 


481 


486 


492 


498 


504 








733 


510 


516 


522 


528 


534 


540 


546 


552 


558 


563 








734 


569 


575 


581 


587 


593 


599 


605 


611 


617 


623 








735 


628 


634 


640 


646 


652 


658 


664 


670 


676 


682 








736 


688 


693 


699 


705 


711 


717 


723 


729 


735 


741 









737 
738 


746 
805 


752 
811 


758 
817 


764 
823 


770 
829 


776 
835 


782 
841 


788 

847 


794 
852 


800 
858 


.1 


5 

0-5 




739 


864 


870 


876 


882 


888 


894 


899 


905 


911 


917 


-2 
• 3 


1-1 

1-6 




740 


923 
982 


929 
987 


935 
993 


941 
999 


946 


952 


958 
*017 


964 
*023 


970 
*028 


976 

*034 


• 4 

• 5 
.6 
.7 
■ 8 
.9 


2.2 
2-7 
33 
3.8 
4.4 
4.9 




741 


*005 


*011 




742 


37 040 


046 


052 


058 


064 


069 


075 


081 


087 


093 




743 


099 


104 


110 


116 


122 


128 


134 


140 


145 


15] 




744 


157 


163 


169 


175 


180 


186 


192 


198 


204 


210 




745 


215 


221 


227 


233 


239 


245 


250 


256 


262 


268 








746 


274 


279 


285 


291 


297 


303 


309 


314 


320 


326 








747 


332 


338 


343 


349 


355 


361 


367 


372 


378 


384 








748 


390 


396 


402 


407 


413 


419 


425 


431 


436 


442 








749 


448 


454 


460 


465 


471 


477 


483 
541 


489 
546 


494 
552 


500 
558 








750 


506 


512 


517 


523 


529 


535 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


. P. 


















652 

















TABLE v.— LOGARITHMS OF NUMBERS. 



N. 



750 

751 
752 
753 
754 
755 
756 
757 
758 
759 

760 

761 
762 
763 
764 
765 
766 
767 
768 
769 

770 



771 

772 

773 

774 

775 

776 

777 |89 

778 

779 

780 i 



O 



87 506 



512 



564 570 

622 627 

679 685 

737: 743 

794 800 

852 858 

909, 915 

967 972 

88 024 030 



081 



087 



138 144 
195; 201 
252 258 



309 
366 
423 
479 
536 
595 



315 
372 
428 
485 
542 
598 



649 654 



705 
761 
818 
874 
930 
986 
042 
098 
153 

209 



781 
782 
783 
784 
785 
786 
787 
788 
789 

790 

791 
792 
793 
794 
795 
796 
797 
798 
799 

800 
N. 



265 
320 
376 
43l 
487 
542 
597 
652 
707 



76§ 



90 



817 
872 
927 
982 
036 
091 
146 
200 
254 

308 







711 
767 
823 
879 
936 
992 
047 
103 
159 



215 



270 
326 
38l 
437 
492 
548 
603 
658 
713 



768 



823 
^78 
933 
987 

042 
097 
151 
205 
260 

314 



517 



523 



575 581 
633 639 
6911 697 
748 754 
806: 812 
863 869 



921 
978 
035 



093 



927 
984 
041 



098 



150 
207 
264 
320 
377 
434 
491 
547 
604 



155 
21§ 
269 
326 
383 
440 
496 
553 
609 



660 666 



716 
773 
829 
885 
94l 



722 
778 
835 
891 
947 



997 *C03 
0531 059 
109' 114 
1651 170 



529 

587 
645 
702 
76C 
817 
875 
932 
990 
047 



104 



161 
218 
275 
332 
389 
445 
502 
558 
615 



535 

593 
650 
708 
766 
823 
881 
938 
995 
053 



6 



541 



110 

167 
224 
281 
337 
394 
451 
508 
564 
621 



67l 



220' 226 



728 
784 
840 
896 
952 
*008 
064 
120 
176 

23l 



2761 282 

332' 337 

387| 393 

442i 448 

498: 503 

553, 559 

6081 614 

663| 669 

71§ 724 



773! 779 



828 
883 
938 
993 
047 
102 
156 
211 
265 



834 
889 
943 
998 
053 
107 
162 
216 
271 



287 
343 
398 
454 
509 
564 
619 
674 
729 



784 



320 325 



839 
894 
949 
•=004 
058 
113 
167 
222 
276 

33Q 



677 

733 
790 
846 
902 
958 
*014 
070 
126 
181 



598 
656 
714 
771 
829 
886 
944 
*00l 
058 



546 



237 



293 
348 
404 
459 
514 
570 
625 
680 
735 



115 

172 
229 
286 
343 
400 
457 
513 
570 
626 

683 

739 
795 
851 
907 
964 
*019 
075 
131 
187 

243 



604 
662 
720 
777 
835 
892 
949 
*007 
064 



I2I 



298 
354 
409 
465 
520 
575 
630 
685 
740 



790 795 



845; 850 
900 905 
954 960 
*009,*015 
064! 069 
1181 124 
173' 178 
22?! 233 
282, 287 

336j 341 



653 



178 
235 
292 
349 
406 
462 
519 
575 
632 

688 



745 
801 
857 
913 
969 
*025 
081 
137 
193 



248 



304 
359 
415 
470 
525 
581 
636 
691 
746 



801 



552 

610 
668 
725 
783 
840 
898 
955 
*012 
070 



127 



184 
241 
298 
355 
411 
468 
525 
581 
638 



694 



750 
806 
863 
919 
975 
*031 
087 
142 
198 



9 



558 

616 
673 
73l 
789 
846 
904 
961 
*018 
075 



133 



190 
247 
303 
360 
417 
474 
530 
587 
643 



700 



254 

309 
365 
420 
476 
531 
586 
641 
696 
751 



806 



856 861 
911! 916 
9651 971 
*026 



*020 
075 
129 
184 
238 
292 

347 



080 
135 
189 
244 
298 

352 



756 
812 
868 
924 
980 
*036 
092 
148 
204 



259 



315 
370 
426 
481 
536 
592 
647 
702 
757 



812 



867 
922 
976 
*031 
086 
140 
195 
249 
303 

858 



P. P. 





6 


.1 


0.6 


• 2 


1 


2 


.3 


1 


8 


.4 


2 


4 


.5 


3 





.6 


3 


6 


.7 


4 


2 


• 8 


4 


8 


.9 


5 


4 





5 


.1 


0.5 


.2 


1 


1 


.3 


1 


6 


.4 


2 


2 


.5 


2 


7 


.6 


3 


3 


.7 


3 


8 


.8 


4 


4 


.9 


4 


d 



.1 


0- 


.2 


1- 


.3 


1. 


.4 


2- 


• 5 


2. 


• 6 


3- 


.7 


3- 


• 8 


4. 


.9 


4. 



-5 



.5 

■ 
.5 



■ 5 

■ 
.5 



P. P. 









TABLE v.— LOGARITHMS OF NUMBERS. 




N. 




90 309 
363 


1 
314 


2 


3 


4 

330 
385 


5 

336 


6 

34l 


7 


8 
352 


9 

358 

412 


P. P. 


800 


320 


325 
379 


347 
401 




801 


368 


374 


390 


396 


406 




802 


417 


423 


428 


433 


439 


444 


450 


455 


460 


466 




803 


47l 


477 


482 


488 


493 


498 


504 


509 


515 


520 




804 


525 


531 


536 


542 


547 


552 


558 


563 


569 


574 


♦ 


805 


579 


585 


590 


596 


601 


606 


612 


' 617 


622 


628 




806 


635 


639 


644 


649 


655 


660 


666 


671 


676 


682 




807 


687 


692 


698 


703 


709 


714 


719 


725 


730 


736 




808 


741 


746 


752 


757 


762 


768 


773 


778 


784 


789 




809 


795 
848 
902 


800 
854 


805 
859 


811 
864 


816 821 


827 


832 


838 


843 
896 
9{^0 




810 


870 875 


880 
934 


886 

939 


891 

945 




811 


907 


913 


918 


923 


929 


1 1 n ~ 


812 
813 
814 
815 
816 
817 
318 
819 


955 
91 009 
062 
116 
169 
222 
275 
328 


961 
014 
068 
121 
174 
227 
280 
333 


966 
019 
073 
126 
179 
233 
286 
339 


971 
025 
078 
131 
185 
238 
291 
344 


977 
030 
084 
137 
190 
243 
296 
349 


982 
036 
089 
142 
195 
249 
302 
355 

408 


987 
04] 
094 
147 
201 
254 
307 
360 


993 
046 
100 
153 
206 
259 
312 
365 

418 


998 
052 
105 
158 
211 
264 
318 
371 


*0U3 
057 

lie 

163 
217 
270 
323 
376 


•1 

.2 
.3 

• 4 
.5 
.6 
.7 
.8 

• 9 


1.1 
16 
2.2 
2-7 
3-3 
3.8 
4.4 
4.9 


830 


381 
434 


386 

439 


392 
445 


397 
450 


402 
455 


413 


423 


429 




821 


461 


466 


47l 


476 


482 




822 


487 


492 


497 


503 


508 


513 


519 


524 


529 


534 




823 


540 


545 


550 


556 


561 


566 


571 


577 


582 


587 




824 


592 


598 


603 


608 


614 


619 


624 


629 


635 


640 




825 


645 


650 


656 


661 


666 


671 


677 


682 


687 


692 




826 


698 


703 


708 


714 


719 


724 


729 


735 


740 


745 




827 


750 


756 


761 


766 


771 


777 


782 


787 


792 


798 




828 


803 


808 


813 


819 


824 


829 


834 


839 


845 


850 




829 


855 


860 


866 


871 


876 


881 


887 


892 

944 

996 
049 
101 
153 
205 
257 
309 
360 
412 


897 
949 


902 

955 

*007 
059 
111 
163 
215 
267 
319 
371 
423 




830 


908 


913 


918 


923 


928 


934 


939 




831 
832 
833 
834 
835 
836 
837 
838 
839 


960 
92 012 
064 
116 
168 
220 
272 
324 
376 


965 
017 
069 
122 
174 
226 
277 
329 
381 


970 
023 
075 
127 
179 
231 
283 
335 
386 


976 
028 
080 
132 
184 
236 
288 
340 
391 


981 
033 
085 
137 
189 
241 
293 
345 
397 


986 
038 
090 
142 
194 
246 
298 
350 
402 


991 

0'f:3 

096 
148 
200 
252 
303 
355 
407 


*002 
054 
106 
158 
210 
262 
314 
366 
417 


.1 
.2 
.3 
.4 
.5 

• 6 
.7 

• 8 
■ 9 


5 

0.5 
1.0 
1.5 
2.0 
2.5 
30 
3.5 
4.0 
4.5 












840 


428 
479 


433 
485 


438 


443 
495 


448 


454 


459 


464 


469 


474 




841 


490 


500 


505 


510 


515 


521 


526 


842 


531 


536 


541 


546 


552 


557 


562 


567 


572 


577 




843 


583 


588 


593 


598 


603 


608 


613 


619 


624 


629 




844 


634 


639 


644 


649 


655 


660 


665 


670 


675 


680 




845 


685 


691 


696 


701 


706 


711 


716 


721 


727 


732 




846 


737 


742 


747 


752 


757 


762 


768 


773 


778 


783 




847 


788 


79S 


798 


803 


809 


814 


819 


824 


829 


834 




848 


839 


844 


850 


855 


860 


865 


870 


875 


880 


885 




849 


891 
942 


896 


901 


906 


911 


916 
967 

5 


921 
972 


926 
977 

7 


931 


937 
988 

9 




850 


947 


952 


957 
3 


962 
4 


982 
8 


N. 





1 


2 


6 


P. P. 














6 


54 








• 









TABLE V 


.—LOGARITHMS OF NUMBERS. 








N. 





1 
947 


2 

952 
*003 


3 

957 
*008 


4 

962 


5 

967 
*018 


6 

972 
*023 


7 
977 


8 

982 


9 

988 
*039 


P 


P. 




850 


92 942 








851 


993 


998 


*013 


*028 


*034 




852 


93 044 


049 


054 


059 


064 


069 


074 


079 


084 


090 








853 


095 


100 


105 


110 


115 


120 


125 


130 


135 


140 








854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


191 








855 


196 


201 


207 


212 


217 


222 


227 


232 


237 


242 








856 


247 


252 


257 


282 


287 


272 


278 


283 


288 


293 




"e 




857 
858 
859 

860 

861 
862 


298 
348 
399 


303 
354 
404 


308 
359 

409 


313 
364 
414 


318 

369 
419 


323 
374 
424 


328 
379 

429 


333 
384 

434 


338 
389 

439 


343 
394 
445 


-1 
.2 
-3 
.4 
.5 
.6 
.7 
• 8 
.9 




1 
1 
2 
2 
3 
3 
4 


■ 5 

• 1 

■ 6 

■ 2 

• 7 

■ 3 
-8 

• 4 




450 


455 


460 


465 


•470 


475 


480 


485 


490 


495 

545 
596 




500 
550 


505 
556 


510 
561 


515 
566 


520 
571 


525 
576 


530 
581 


535 
586 


540 
591 




863 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 




864 


651 


656 


661 


666 


671 


676 


681 


686 


691 


696 


%' XJ 




865 


701 


706 


711 


716 


721 


726 


731 


736 


742 


747 








866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


79? 








867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 








868 


852 


857 


862 


887 


872 


877 


882 


887 


892 


897 








869 

870 

871 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 


-1 

• 2 

• 3 


5 

n R 




952 


957 


962 


967 


972 


977 


982 


987 


992 


997 

046 




94 002 


007 


012 


017 


022 


028 


03l 


036 


04l 




872 


051 


056 


061 


066 


071 


076 


081 


086 


091 


096 


1 
1 
2 
2 
3 
3 
4 
4 


• 
5 

5 

5 

5 




873 


lOl 


106 


111 


116 


121 


126 


131 


136 


141 


146 




874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 




875 
876 
877 


201 
250 
300 


206 
255 
305 


210 
260 
310 


215 
265 
315 


220 
270 
320 


225 
275 
324 


230 
280 
329 


235 
285 
334 


240 
290 
339 


245 
295 
344 


- 4 

.5 

■ 6 
• 7 

■ 8 
.9 




878 


■ 349 


354 


359 


364 


369 


374 


379 


384 


389 


394 




879 

!880 

881 


399 


404 


409. 


413 


418 


423 


428 


433 


438 
487 


443 
492 
542 




448 


453 


458 


463 


468 

517 


473 
522 


478 


483 






49? 


502 


507 


512 


527 


532 


537 




882 


547 


552 


556 


561 


566 


571 


576 


581 


586 


591 








883 


596 


601 


606 


611 


615 


620 


625 


630 


635 


640 








884 


645 


650 


655 


660 


665 


670 


674 


679 


684 


689 








885 


694 


699 


704 


709 


714 


719 


724 


728 


733 


738 








886 


743 


748 


753 


758 


763 


768 


773 


777 


782 


787 




J 




887 
888 
889 

890 

891 


792 
84l 
890 

939 

988 


797 
846 
895 


802 
851 
900 


807 
856 
905 


812 
861 
909 


817 
865 
914 

963 

*012 


821 
870 
919 

968 


826 
875 
924 

973 


831 
880 
929 


836 
885 
934 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


o; 

0. 
1- 
1- 
2- 
2- 
3- 
3. 
4 


i 

9 

3 
8 
2 
7 
1 
6 





944 


949 


953 


958 
*007 


978 


983 
03l 




992 


997 


*002 


*017 


*022 


*026 




1892 


95 036 


041 


046 


051 


056 


061 


065 


070 


075 


080 




!893 
894 


085 
134 


090 
138 


095 
143 


099 
148 


104 
153 


109 
158 


114 
163 


119 
167 


124 
172 


129 
177 




895 


182 


187 


192 


197 


201 


206 


211 


216 


221 


226 








896 


231 


235 


240 


245 


250 


255 


260 


264 


269 


274 








B97 


279 


284 


289 


294 


298 


303 


308 


313 


318 


323 








S98 


327 


332 


337 


342 


347 


352 


356 


361 


366 


371 








|599 
900 

N. 


376 


381 


385 


390 


395 
443 

4 


400 


405 


410 


414 
463 

8 


419 
467 

9 








424 


429 
1 


434 
2 


438 
3 


448 
5 


453 
6 


458 

7 







P. 


P. 




' 














655 





















TABLE V 


—LOGARITHMS OF NUMBERS. 








N. 





1 


2 

434 


3 

438 
487 


4 

443 


5 

448 


6 

453 


7 
458 


8 
463 


9 

467 
516 


P. P. 


900 


95 424 
472 


429 




901 


477 


482 


492 


496 


50l 


506 


511 




902 


520 


525 


530 


535 


540 


544 


549 


554 


559 


564 




903 


569 


573 


578 


583 


588 


593 


597 


602 


607 


612 






904 


617 


621 


626 


631 


636 


641 


645 


650 


655 


660 






905 


665 


669 


674 


679 


684 


689 


693 


698 


703 


708 






906 


713 


717 


722 


727 


732 


737 


741 


746 


751 


756 






907 


760 


765 


770 


775 


780 


784 


789 


794 


799 


804 






908 


808 


813 


818 


823 


827 


832 


837 


842 


847 


851 






909 


856 
904 


861 
909 


866 


870 


875 


880 


885 


890 


894 


899 


5 

line 




910 


913 


918 


923 
971 


928 

975 


933 


937 


94? 


947 




911 


952 


956 


96l 


966 


980 


985 


990 


994 




912 
913 
914 
915 
916 
917 
918 
919 


999 
96 047 
094 
142 
189 
237 
284 
33l 


*004 
052 
099 
147 
194 
241 
289 
336 


*009 
056 
104 
151 
199 
246 
293 
341 


*014 
061 
109 
156 
204 
251 
298 
345 


*018 
066 
113 
161 
208 
256 
303 
350 


*023 
071 
118 
166 
213 
260 
308 
355 


*028 
075 
123 
170 
218 
265 
312 
360 


*033 
080 
128 
175 
222 
270 
317 
364 


*037 
085 
132 
180 
227 
275 
322 
369 


*042 
090 
137 
185 
232 
279 
327 
374 


•1 
•2 

3 
.4 
.5 
.6 
• 7 
.8 
.9 


1 
1 

2 
2 
3 
3 
4 
4 


u 

5 

5 

5 

5 




























9^0 


379 
426 


383 


388 


393 


397 


402 
449 


407 


412 


416 


421 






921 


430 


435 


440 


445 


454 


459 


463 


468 




922 


473 


478 


482 


487 


492 


496 


501 


506 


511 


515 






923 


520 


525 


529 


534 


539 


543 


548 


553 


558 


562 






924 


567 


572 


576 


581 


586 


590 


595 


600 


605 


609 






925 


614 


619 


623 


628 


633 


637 


642 


647 


651 


656 






926 


661 


666 


670 


675 


680 


684 


689 


694 


698 


703 






927 


708 


712 


717 


722 


726 


731 


736 


741 


745 


750 






928 


755 


759 


764 


769 


773 


778 


783 


787 


792 


797 






929 


801 


806 


811 


815 


820 


825 


829 


834 
881 


839 


843 






930 


848 


853 


857 


862 


867 


871 

918 
965 


876 

923 
969 


885 


890 

937 
983 




931 

932 


895 
94J 


899 

946 


904 
951 


909 
955 


913 
960 


927 
974 


932 
979 


.1 
.2 
.3 
.4 

• 5 
.6 

• 7 
8 

• 9 


0^ 
0-9 
1.3 
18 
2.2 
2.7 
31 
36 
4.S 


933 
934 
935 
936 


988 

97 034 

081 

127 


993 
039 
086 

132 


997 
044 
090 
137 


*002 
048 
095 
141 


*007 
053 
099 
146 


*011 
058 

104 
151 


*016 
062 
109 
155 


*020 
067 
113 
160 


*025 
072 
118 
164 


*030 
076 
123 
169 




937 
938 
939 


174 
220 
266 

313 

359 


178 
225 
271 


183 

229 
276 


188 

234 
280 

326 


192 
239 
285 


197 
243 
289 

336 

382 


202 
248 
294 


206 
252 
299 


211 
257 
303 

349 


215 
262 
308 

354 




940 


317 
363 


322 


331 


340 
386 


345 




941 


368 


373 


377 


39l 


396 


400 


942 


405 


409 


414 


419 


423 


428 


432 


437 


442 


446 






943 


451 


456 


460 


465 


469 


474 


479 


483 


488 


492 






944 


497 


502 


506 


511 


515 


520 


525 


529 


534 


538 






945 


543 


548 


552 


557 


561 


566 


570 


575 


580 


584 






946 


589 


593 


598 


603 


607 


612 


616 


621 


626 


630 






947 


635 


639 


644 


649 


653 


658 


662 


667 


671 


676 






948 


681 


685 


690 


694 


699 


703 


708 


713 


717 


722 






949 


726 


731 


736 


740 


745 


749 


754 
800 

6 


758 
804 

7 


763 


768 


; 




950 


772 


777 


781 


786 


790 
4 


.795 
5 


809 
8 


813 
9 






1 


2 

1 




N. 





3 


1 
P. P. 
















656 














1 

'i 







TABLE V.- 


-LOGARITHMS OF NUMBERS. 






N. 





1 

777 


3 

78l 


3 

786 
831 


4 

790 


795 


6 

800 


7 
804 


8 
809 


9 


P. 


P. 




950 


97 772 
818 


813 
859 








951 


822 


827 


836 


841 


845 


850 


854 




952 


863 


868 


873 


877 


882 


886 


891 


895 


900 


904 








953 


909 


914 


918 


923 


927 


932 


936 


941 


945 


950 








954 


955 


959 


964 


968 


973 


977 


982 


986 


991 


996 








955 


98 000 


005 


009 


014 


018 


023 


027 


032 


036 


041 








956 


046 


050 


055 


059 


064 


068 


073 


077 


082 


086 




5 

o.s 

1.0 
1.5 
2-0 
2.5 
30 
3-5 
4-0 
4-5 




957 
958 
959 

960 

961 
962 
963 
964 


091 
136 
182 


095 
141 
186 


100 
145 
191 

236 

28l 
326 
37l 
416 


105 
150 
195 


109 
154 
200 


114 
159 
204 


118 
163 
209 


123 
168 
213 


127 
173 
218 


132 
177 
222 


.1 

.2 
.3 
.4 
• 5 
.6 
.7 
.8 
■ 9 




227 

272 
317 
362 
407 


231 

277 
322 
337 
412 


240 


245 


249 


254 


259 


263 


268 




286 
331 
376 

421 


295 
335 
380 

425 


295 
340 
385 
430 


299 
344 
389 
434 


304 
349 
394 
439 


308 
353 
398 
443 


313 
358 

403 
448 




965 


452 


457 


461 


466 


470 


475 


479 


484 


488 


493 








986 


497 


502 


506 


511 


515 


520 


524 


529 


533 


538 








967 


542 


547 


551 


556 


560 


565 


569 


574 


578 


583 








968 


587 


592 


596 


601 


605 


610 


614 


619 


623 


628 








969 

970 

971 
972 
973 
974 
975 
976 
977 
978 
979 

989 

981 


632 
677 


637 
68l 


641 
686 


646 
690 


650 
695 


655 
699 


659 


663 

708 

753 
798 
842 
887 
931 
976 
*020 
065 
109 


668 

713 


672 
717 


.1 
o2 
.3 
.4 
.5 
.6 
• 7 
.8 
.9 


0.4 

0.9 
1.3 
1-8 
2-2 
2 = 2 
3.1. 
3-6 
4^0 




704 

749 
793 
838 
882 
927 
971 
*016 
060 
105 




722 
766 
811 
856 
900 
945 
989 
99 034 
078 

122 

167 


726 
771 
815 
860 
905 
949 
994 
038 
082 

127 

17l 


731 
775 
820 
865 
909 
954 
998 
043 
087 

131 


735 
780 
824 
889 
914 
958 
*003 
0.7 
09l 

136 

180 


740 
784 
829 
873 
918 
933 
*007 
051 
098 

140 

184 


744 
789 
833 
878 
922 
967 
*011 
056 
100 


757 
802 
847 
89l 
936 
980 
*025 
069 
113 


762 
807 
851 
896 
940 
985 
*029 
074 
118 

162 




145 


149 


153 
198 


158 




176 


189 


193 


202 


206 




982 


211 


215 


220 


224 


229 


233 


237 


242 


246 


251 








983 


255 


260 


264 


288 


273 


277 


282 


286 


290 


295 








984 


299 


304 


308 


312 


317 


321 


326 


330 


335 


339 








985 


343 


348 


352 


357 


36i 


365 


370 


374 


379 


383 








986 


387 


392 


396 


401 


405 


409 


414 


418 


423 


427 




4 

0.4 
0.8 
1.2 
1.6 
2.0 
2-4 
28 
3.2 
3.6 




987 
988 


431 

475 


436 
480 


440 
484 


445 
489 


449 
493 


453 
497 


458 

502 


462 
506 


467 
511 


471 
515 


.1 
.2 
-3 

• 4 
-5 
.6 

• 7 
.8 

• 9 




989 
)90 

J91 


519 


524 


528 


533 


537 


541 


546 


550 


554 
598 


559 




563 
607 


568 


572 


576 


581 


585 


590 


594 


603 
647 




-61] 


616 


620 


625 


629 


633 


638 


642 




392 


651 


655 


660 


664 


6B8 


673 


877 


682 


888 


890 




593 


695 


699 


703 


708 


712 


717 


721 


725 


730 


734 




394 


738 


743 


747 


751 


756 


760 


765 


769 


773 


778 




)95 


782 


78fi 


791 


795 


8on 


804 


808 


813 


817 


821 








)96 


8'^6 


830 


834 


839 


843| 


847 


852 


856 


861 


885 








)97 


869 


874 


878 


882 


887 


891 


895 


900 


904 


908 








)98 


913 


917 


922 


92R 


930 


935 


93P 


943 


948 


952 








199 
-[>00 


956 
00 000 


961 


965 


969 


974 


978 


982 


987 


991 
034 


995 
039 








004 


008 


013 


017 


021 


026 


030 









1 


2 


3 


4 


5 


6 


7 


8 


9 


P.] 


P. 





657 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 



1000 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1010 

11 

12 
13 
14 
15 
16 
17 
18 
19 

1030 

21 
22 
23 
24 
25 
26 
27 
28 
29 

1030 

31 
32 
33 
34 
35 
36 
37 
38 
39 



O 



000 000 

434 
867 

001 301 
733 

002 166 
598 

003 029 
460 
891 

004 321 

751 

005 180 
609 

006 038 
466 
893 

007 321 
748 

008 174 

600 

009 025 
451 
875 

010 300 
724 

Oil 147 
570 
993 

012 415 

837 

013 258 
679 

014 100 
520 
940 

015 360 
779 

016 197 
615 



043 



1040 017 033 

41 450 

42 867 

43 018 284 



44 
45 
46 
47 
48 
49 

1050 



700 

019 116 
531 
946 

020 361 
775 

021 189 







477 
911 
344 
777 
209 
641 
072 
503 
934 



087 



364 

794 
223 
652 
081 
509 
936 
363 
790 
217 



642 



068 
493 
918 
342 
766 
189 
612 
*035 
457 



879 



301 
722 
142 
562 
982 
401 
820 
239 
657 



075 



521 
954 
387 
820 
252 
684 
115 
546 
977 



130 



407 

837 
266 
695 
123 
551 
979 
40C 
883 
259 

685 



111 
536 
960 
385 
808 
232 
655 
*077 
500 

921 



343 
764 
184 
604 
*024 
443 
862 
281 
699 

117 

534 
951 
367 
783 
199 
614 
*029 
444 
858 



564 
997 
431 
863 
295 
727 
159 
590 
*020 



173 



450 

880 
309 
738 

166 
594 
*022 
449 
875 
302 

728 



153 
578 

'=003 
427 
851 
274 
697 

*120 
542 

963 



217 



607 
*041 

474 

906 

339 

770 

202 

633 676 
*063 *106 



651 
*084 
517 
950 
382 
814 
245 



493 

923 
352 



536 



966 
,-_ 395 
781 824 
252 
680 
*107 
534 
961 



209 
637 
*064 
491 
918 
3441 387 



770 813 



196 
621 

*045 
469 
893 
316 
739 

*162 
584 



*006 



385 
806 
226 
646 
*086 
485 
904 
323 
741 



238 
663 

*088 
512 
935 
359 
782 

*20| 
626 

*048 



6 



260 



694 
*127 
560 
993 
425 
857 
288 
719 
*149 



158 



576 
992 
409 
825 
241 
656 
*071 
485 
899 

313 
3 



427 
848 
268 
688 
*108 
527 
946 
364 
782 



200 

617 
*034 
451 
867 
282 
697 
*112 
527 
941 

354 



469 
890 
310 
730 
'-150 
569 
988 
406 
824 



579 

*009 
438 
866 
295 
722 

*150 
577 

*CC3 
430 

855 

281 
7C6 

*130 
554 
978 
40l 
824 

*246 
668 

*090 



242 

659 
*076 
492 
908 
324 
739 
*154 
568 
982 

896 



5 



511 
932 
352 
772 

*192 
611 

*030 
448 
866 



284 



701 

*117 

534 

950 

365 

780 

*195 

610 

*024 

437 



304 



737 

*171 

604 

*036 

468 

900 

33l 

762 

*192 



622 



*05l 
481 
909 
33'7 
765 

*193 
620 

*046 
472 



898 



347 

781 

*214 

647 

*079 

511 

943 

374 

805 

*235 



665 



*09| 
523 
952 
380 
808 

*235 
662 

*089 
515 



940 




553 595 
974*016 



394 
814 

*234 
653 

*072 
490 
908 



325 

742 

*159 

575 

99l 

407 

822 

*237 

651 

*065 

478 

7 



436 
856 

*276 
695 

*113 
532 
950 



367 



784 
*201 

617 
*033 

448 

863 
*278 

692 
"=106 

520 



9 



P. P. 



708 



983 



216 



637 



409 



826 
*242 

659 
*074 

490 

905 
*320 

734 
*148 

561 



43_ 

4-3 



43 

43 



43 43 



4.2 


4-2 m 


8 


K 


8 


4 " 


12 


7 


12 


6 


17 


P 


16 


8 


21 


2 


21 





25 


5 


25 


2 


29 


7 


29 


4 


34 





33 


6 


38 


2 


37 


8 



41_ 

41 



41 

41 



32.8 
36-9 



P. P, 



658 



I 









TABLE V- 


-LOGARITHMS OF NUMBERS. 








N. 





1 


2 


3 


4 5 


6 


7 


8 


9 


P 


.P. 


1050 


021 


189 
602 


230 
644 


272 
685 


313 
726 


354 


396 


437 
850 


478 
892 


520 
933 


561 

974 


1 


41 

4.1 


51 


768 


809 


52 


022 


015 


057 


098 


:i.39 


18i 


222 


263 


304 


346 


387 


2 


8 


3 


53 




428 


469 


511 


552 


593 


634 


676 


717 


758 


799 


3 


12 


4 


54 




840 


882 


923 


964 


*005 


*046 


=^088 


*129 


*170 


*211 


4 


16 


6 


55 


023 


252 


293 


335 


376 


417 


458 


499 


540 


581 


62^ 


5 


20 


7 


56 




664 


705 


748 


787 


828 


889 


910 


951 


993 


*034 


6 


24 


9 


57 


024 


075 


116 


157 


198 


239 


280 


321 


362 


403 


444 


7 


29 





58 




485 


526 


568 


609 


650 


691 


732 


773 


814 


855 


8 


33 


2 


59 


025 


896 
306 


937 
347 


978 
388 


^019 


*060 


*101 


=*=142 


*183 


*224 


*265 


9 


37 


3 


1060 


429 


469 


510 


55l 


592 


633 


674 


41 

4.1 


6i 




715 


756 


797 


838 


879 


920 


961 


*002 


*042 


=*=083 


1 


62 


026 


124 


165 


206 


247 


288 


329 


370 


410 


451 


492 


2 


8 


2 


63 




533 


574 


615 


656 


696 737 


778 


819 


860 


901 


3 


12 


3 


64 




941 


982 


*023 


*064 


*105 *145 


*188 


=^227 


*268 


*309 


4 


16 


4 


65 


027 


349 


390 


431 


472 


512 553 


594 


635 


675 


716 


5 


20 


5 


66 




757 


798 


838 


879 


920 


961 


*001 


*042 


*083 


*123 


6 


24 


6 


67 


028 


164 


205 


246 


286 


327 


388 


408 


449 


490 


530 


7 


28 


7 


68 




571 


612 


652 


693 


734 


774 


815 


856 


896 


937 


8 


32 


8 


89 


029 


977 


*0I8 


*059 


*099 


^140 


•1=18] 


■•=22] 


*262 


*302 


*343 


9 


36 


9 


1070 


384 


424 


465 


505 


546 


.586 


62? 


668 


708 


749 


40. 
























71 




789 


830 


870 


911 


951 


992 


*032 


*073 


*114 


*154 


1 


4.0 


72 


030 


195 


235 


276 


316 


357 


397 


438 


478 


519 


559 


2 


8 


1 


73 




599 


640 


680 


721 


761 


802 


842 


883 


923 


964 


3 


12 


1 


74 


031 


004 


044 


085 


125 


166 


206 


247 


287 


327 


368 


4 


16 


2 


75 




408 


449 


489 


529 


570 


610 


651 


691 


731 


772 


5 


20 


2 


76 




812 


852 


893 


933 


973 


=^=014 


*054 


*094 


*135 


*175 


6 


24 


3 


77 032 


215 


256 


296 


336 


377 


417 


457 


498 


538 


578 


7 


28 


3 


78 




619 


659 


699 


739 


780 


820 


860 


900 


941 


981 


8 


32 


4 


79 


033 


021 


GBl 


102 


142 


182 


222 


263 


303 


343 


383 


9 

1 


36 


4 


10«80 


424 


484 


504 


544 


584 


625 
*026 


665 


705 


745 


785 




81 


825 


868 


906 


946 


986 


*066 


*107 


147 


187 


40 

40 


82 


034 


227 


287 


307 


347 


388 


428 


468 


508 


548 


588 


2 


8 





83 




628 


668 


708 


748 


789 829 


869 


909 


949 


989 


3 


12 





1 84 


035 


029 


069 


109 


149 


189 229 


269 


309 


349 


389 


4 


16 





85 




429 


470 


510 


550 


590 630 


670 


710 


750 


790 


5 


20 





86 




830 


870 


910 


950 


990 *029 


*069 


*109 


*149 


*189 


.6 


24 





87 


036 


229 


269 


309 


349 


389 


429 


469 


509 


549 


589 


.7 


28 





88 




629 


669 


708 


748 


788 


828 


868 


908 


948 


988 


.8 


32 





89 
1090 

1 91 


037 


028 


068 


107 


147 


187 


227 


287 


307 


347 


386 


• 9 


36 





426 


466 


506 


546 


586 


625 


665 


705 


745 


785 


39 

3.9 




825 


864 


904 


944 


984 *023 


*063 


*103 


143 


183 


.1 


' 92 

1 93 


038 


222 


-262 


302 


342 


381 421 


461 


501 


540 


580 


.2 


7 


9 




620 


660 


699 


739 


779 819 


858 


898 


938 


977 


.3 


11 


8 


1 94 


039 


017 


057 


096 


136 


176 216 


255 


295 


335 


374 


• 4 


15 


.8 


; 95 




414 


454 


493 


533 


572 612 


652 


691 


731 


771 


.5 


19 


•7 


96 




810 


850 


890 


929 


969 *008 


*048 


*088 


*127 


*167 


.6 


23 


•7 


97 


040 


206 


246 


286 


325 


365 404 


444 


483 


523 


563 


.7 


27 


6 


98 




602 


642 


681 


721 


760 800 


839 


879 


918 


958 


• 8 


31 


•6 


99 


041 


997 


*037 


*076 


*116 


*15"5 


=^195 


*234 


*274 


*313 


*353 


.9 


35 


• 5 


100 


•392 


432 


471 


511 


550 


590 


629 


669 


708 


748 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


.P. 



659 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin (/) = log (f)" + S. 
Log tan 4> = log (j," + T. 



log 4>" 
log 4>" 



log -sin (f> + (S'. 
log tan ^ + r'. 



tr 


r 


S 


T I.< 


>g. Sin. 


S' 


T' 


T.og. Tan. 








4.685 57 


57 


— eo 


5.314 42 


42 


— 00 


60 


1. 


57 


57 6 


-46 372 


42 


42 


6. 46 372 


120 


^ 


57 


57 


.76 475 


42 


42 


76 475 


180 


8 


57 


57 


.04 084 


42 


42 


94 084 


240 


4 
5 


57 
4.685 57 


57 7 


-08 578 


42 
5.314 42 


42 


7.C6 578 


300 


57 7 


-16 269 


42 


7. 16 269 


360 


6 


57 


57 


-24 187 


42 


42 


.24 188 


420 


7 


5Z 


57 


-30 882 


42 


42 


■ 30 882 


480 


8 


57 


57 


-36 681 


42 


42 


• 36 681 


540 


9 


57 


57 


.41 797 


42 


42 


■ 41 797 


600 


10 


4-685 57 


57 7 


.46 372 


5.314 42 


42 


7. 46 372 


660 


11 


57 


57 


-50 512 


42 


42 


■ 50 512 


720 


12 


57 


57 


-54 290 


42 


42 


• 54 291 


780 


13 


57 


57 


-57 767 


42 


42 


-57 767 


840 


14 


57 


57 


.60 985 


42 


42 


• 60 985 


900 


15 


4-685 57 


58 7 


63 981 


5-314 42 


42 


7 63 982 


960 


16 


57 


58 


.66 784 


42 


42 


-66 785 


jf020 


17 


57 


58 


.69 417 


4§ 


42 


.69 418 


1080 


18 


57 


58 


.71 899 


42 


42 


.71900 


1140 


19 


57 


58 


. 74 248 


42 
5.314 43 


42 
42 


• 74 248 


1200- 


20 


4.685 57 


58 7 


• 76 475 


7 76 476 


1260 


21 


57 


58 


-78 594 


43 


42 


-78 595 


1320 


22 


57 


58 


-80 614 


43 


42 


-80 615 


1380 


23 


57 


58 


-82 545 


43 


42 


.82 546 


1440 


24 


57 


58 


-84 393 


43 


42 


• 84 394 


1500 


25 


4-685 57 


58 7 


.86 166 


5.31443 


4l 


7-86 167 


1560 


26 


57 


58 


-87 869 


43 


41 


87 871 


1620 


27 


57 


58 


-89 508 


43 


41 


• 89 510 


1680 


28 


57 


58 


91 088 


43 


41 


-91 089 


1740 


29 


57 


58 


92 612 


43 


41 


92 613 


1800 


30 


4-685 57 


58 7 


94 084 


5.314 43 


41 


7. 94 086 


1860 


31 


57 


58 


95 508 


43 


41 


• 95 510 


1920 


32 


57 


58 


96 887 


43 


41 


• 96 889 


1980 


33 


57 


59 


98 223 


43 


41 


• 98 225 


2040 


34 


57 


59 


99 520 


43 


41. 


•99 522 


2100 


35 


4.685 56 


59 8- 


00 778 


5.314 43 


41 


8. GO 781 


2160 


36 


56 


59 


02 002 


43 


41 


-02 004 


2220 


37 


56 


59 


03 192 


43 


41 


-03 194 


2280 


38 


56 


59 


04 350 


43 


40 


-04 352 


2340 


39 


56 ' 


59 


05 478 


43 


40 
40 


• 05 481 


2400 


40 


4-685 56 


59 8- 


06 577 


5.314 43 


806 580 


2460 


41 


56 


59 


07 650 


43 


40 


-07 653 


2520 


42 


56 


59 


08 696 


43 


40 


-08 699 


2580 


43 


56 


60 


09 718 


43 


40 


.09 721 


2640 


44 


56 " 


60 


10 716 


43 


40 
40 


.10 720 


2700 


45 


4.685 56 


60 8- 


11 692 


5.314 44 


8-11 696 


2760 


46 


56 


60 


12 647 


44 


40 


-12 651 


2820 


47 


56 


60 


13 581 


44 


40 


-13 585 


2880 


48 


56 


60 


14 495 


44 


39 


-14 499 


2940 


49 


56 


60 


15 390 


44 


39 


-15 395 


3000 


50 


4.685 56 


60 8- 


16 268 


5.314 44 


39 


8-16 272 


3060 


51 


56 


60 


17 128 


44 


39 


-17 133 


3120 


52 


56 


61 


17 971 


44 


39 


-17 976 


3180 


53 


56 


61 


18 798 


44 


39 


.18 803 


3240 


54 
55 


55 


61 


19 610 


44 


39 


.19 615 


3300 


4.685 55 


61 8. 


20 407 


5-314 44 


39 


8.20 412 


3360 


56 


55 


61 


21 189 


44 


38 


.21 195 


3420 


57 


55 


61 


21 958 


44 


38 


.21 964 


3480 


58 


55 


61 


22 718 


44 


38 


.22 719 


SL540 


59 


55 


62 


23 455 


44 


38 


.23 462 



660 



IrABLE VI.— 


LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 


Log sill 4> = log gS" + 5. 




r 


log (j>" = log 


sin (A + S\ 


■ i.og taa <!> = log <p" + T. 




log<^' 


' = log tan <j> + r'. 


1 " ' 

1 


/ 


^S 


T I.og. Siu. 


S' 


T' 


liOg. Tan. 


1600 





4.685 55 


62 8 


.24 185 


5.314 44 


38 


8.24 192 


i660 


1 


55 


62 


.24 903 


45 


38 


.24 910 


;720 


2 


55 


62 


.25 609 


45 


38 


.25 616 


1780 


3 


55 


62 


.26 304 


45 


37 


.26 311 


;840 


4 
5 


55 


62 


.26 988 


45 


37 
37 


.26 995 


900 


4.685 55 


62 8 


.27 66l 


5-314 45 


8-27 669 


i960 


6 


55 


63 


.28 324 


45 


37 


.28 332 


.020 


7 


54 


63 


.28 977 


45 


37 


.28 985 


=080 


8 


54 


63 


.29 620 


45 


37 


.29 629 


=140 


9 


54 


63 


• 30 254 


45 


36 


• 30 263 


:200 


10 


4.685 54 


63 8 


30 879 


5.314 45 


36 


8-30 88S 


:260 


11 


54 


63 


31 495 


45 


36 


.31 504 


:320 


12 


54 


64 


32 102 


45 


36 


.32 112 


:380 


13 


54 


64 


32 701 


46 


36 


• 32 711 


440 


14 


54 


64 


33 292 


46 


36 
35 


.33 302 


:500 


15 


4. 685 54 


64 8 


33 875 


5.314 46 


8-33 885 


t560 


16 


54 


64 


34 450 


46 


35 


.34 461 


:620 


17 


54 


65 


35 018 


46 


35 


• 35 029 


t680 


18 


54 


65 


35 578 


46 


35 


.35 589 


t740 


19 


53 


65 


36 131 


46 


35 


-36 143 


1800 


30 


4. 685 53 


65 8 


36 677 


5-314 46 


34 


8-36 689 


I860 


21 


53 


65 


37 217 


46 


34 


.37 229 


1920 


22 


53 


65 


37 750 


46 


34 


.37 762 


t980 


23 


53 


66 


38 276 


46 


34 


.38 289 


J040 


24 


53 


66 


38 796 


47 


34 


-38 809 


ilOO 


25 


4-685 53 


66 8. 


39 310 


5.314 47 


33 


8-39 323 


)160 


26 


53 


66 


39 818 


47 


33 


.39 831 


)220 


27 


53 


67 


40 320 


47 


33 


.40 334 


5280 


28 


52 


67 


40 816 


47 


33 


.40 830 


)340 


29 


52 


67 


41 307 


47 


33 
32 


-41 321 


5400 


30 


4.685 52 


67 8. 


41 792 


5.31447 ' 


8-41 807 


5460 


31 


52 


67 


42 271 


47 


32 


.42 287 


5520 


32 


52 


68 


42 746 


47 


32 


.42 762 


5580 


33 


52 


68 


43 215 


48 


32 


.43 231 


5640 


34 


52 


68 


43 680 


48 


31 


.43 696 


5700 


35 


4.685 52 


68 8- 


44 139 


5.314 48 


31 


8-44 156 


5760 


36 


52 


69 


44 594 


48 


31 


.44 611 


5820 


37 


51 


69 


45 044 


48 


31 


• 45 061 


5880 


38 


51 


69 


45 489 


48 


30 


.45 507 


5940 


39 


51 


69 


45 930 


48 


30 


-45 948 


6000 


40 


4.685 5l 


69 8. 


46 366 


5.314 48 


30 


8-46 385 


6060 


41 


51 


70 


46 798 


49 


30 


.46 817 


6120 


42 


51 


70 


47 226 


49 


30 


.47 245 


6180 


43 


51 


70 


47 650 


49 


29 


.47 669 


6240 


44 


51 


70 


48 069 


49 


29 


.48 089 


6300 


45 


4.685 50 


71 8. 


48 485 


5.314 49 


29 


8-48 505 


6360 


46 . 


50 


71 


48 896 


49 


28 


.48 917 


6420 


47 


50 


71 


49 304 


49 


28 


.49 325 


6480 


48 


50 


72 


49 708 


49 


28 


.49 729 


6540 


49 


50 


72 


50 108 


50 


28 


-50 130 


6600 


50 


4.685 50 


72 8- 


50 504 


5-314 50 


27 


8-50 526 


6660 


51 


50 


72 


50 897 


50 


27 


.50 920 


6720 


52 


50 


73 


51 286 


50 


27 


.51 310 


6780 


53 


49 


73 


51 672 


50 


27 


.51 696 


684.0 


54 


49 


73 


52 055 


50 
5.314 50 


26 


.52 079 


6900 


55 


4.685 49 


73 8- 


52 434 


26 


8-52 458 


6960 


56 


49 


74 


52 810 


51 


26 


-52 835 


7020 


57 


49 


74 


53 183 


51 


25 


.53 208 


7080 


58 


49 


74 


53 552 


51 


25 


.53 578 
.53 944 


7140 


59 


49 


75 


53 918 


51 


25 



661 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES 



Log sin (f) = log (/>■■' -f S. 




2° 


log <j>' 


= log Rin (5 4- S', 


Log tan (j) = log ^" + T.. 




log 4>" 


= log tan <fi + T', j 


ff 


> 


S 


T 


J^og. Sin. 


S' 


T' I.og 


;. Tim. 


7200 





4.685 48 


75 


8-54 282 


5.314 51 


25 8 


54 30g 


7260 


1 


48 


75 


-54 642 


51 


24 


54 66S 


7320 


2 


48 


75 


-54 999 


5l 


24 


55 027 


7380 


3 


48 


76 


.55 354 


52 


24 


55 381 


7440 


4 
5 


48 


76 
76 


.55 705 


52 


23 


55 733 


7500 


4-685 48 


8-56 054 


5-314 52 


23 8 


56C83 


7560 


6 


48 


77 


-56 400 


52 


23 


56 429 


7620 


7 


47 


77 


-56 743 


52 


22 


56 772 


7680 


8 


47 


77 


-57 083 


52 


22 


57 113 


7740 


9 


47 


78 


.57 421 


52 
5-314 53 


22 


57 452 


7800 


10 


4.685 47 


78 


8-57 756 


22 8 


57 787 


7860 


11 


47 


78 


-58 089 


53 


21 


58 121 


7920 


12 


47 


79 


-58 419 


53 


21 


58 451 


7980 


13 


46 


79 


-58 747 


53 


21 


58 779 


8040 


14 


46 
4-685 46 


79 
80 


.59 072 


53 
5-314 53 


20 


59 105 


8100 


15 


8.59 395 


20 8 


59 428 


8160 


16 


46 


80 


-59 715 


54 


20 


59 749 


8220 


17 


46 


80 


• 60 033 


54 


19 


60 067 


8280 


18 


46 


81 


.60 349 


54 


19 


60 384 


8340 


19 


45 


81 


.60 662 


54 
5.314 54 


19 


60 698 


8400 


30 


4-685 45 


8l 


8.60 973 


18 8. 


61 009 


8460 


21 


45 


82 


-61 282 


54 


18 


61319 


8520 


22 


45 


82 


-61 589 


55 


18 


61 626 


8580 


23 


45 


82 


-61 893 


55 


17 


61 931 


8640 


24 


45 


83 
83 


.62 196 


55 


17 


62 234 


8700 


25 


4-685 44 


8.62 496 


5.314 55 


16 8. 


62 535 


8760 


26 


44 


83 


-62 795 


55 


16 


62 834 


8820 


27 


44 


84 


-63 091 


55 


16 


63 131 


8880 


28 


44 


84 


-63 385 


56 


]5 


63 425 


8940 


29 


44 


84 


.63 677 


56 


15 


63 718 


9000 


30 


4.685 43 


85 


8.63 968 


5-314 56 


15 8. 


64 009 


9060 


31 


43 


85 


-64 256 


56 


14 


64 298 


9120 


32 


43 


86 


-64 543 


56 


14 


64 585 


9180 


33 


43 


86 


-64 827 


57 


14 


64 870 


9240 


34 


43 


86 


.65 110 


57 
5-314 57 


13 


65 153 


9303 


35 


4-685 43 


87 


8.65 391 


13 8 


65 435 


9360 


36 


42 


87 


-65 670 


57 


12 


65 715 


9420 


37 


42 


87 


.65 947 


57 


12 


65 993 


9480 


38 


42 


88 


.66 223 


58 


12 


66 269 


9540 


39 


42 
4.685 42 


88 


.66 497 


58 


11 


66 54^ 


9600 


40 


89 


8.68 769 


5-314 58 


11 8. 


66 816 


9660 


41 


41 


89 


-67 039 


58 


10 


67 087 


9720 


42 


41 


89 


.67 308 


58 


10 


67 356 


9780 


43 


41 


90 


-67 575 


59 


10 


67 624 


9840 


44 
45 


41 


90 


.67 840 


59 


09 


P7 RfiO 


9900 


4-685 41 


91 


8.68 104 


5-314 59 


09 8 


68 154 


9960 


46 


40 


91 


-68 36C 


59 


08 


68 417 


10020 


47 


40 


9l 


-68 627 


59 


08 


68 678 


10080 


48 


40 


92 


-68 886 


60 


08 


68 938 


10140 


49 


40 


92 


.69 144 


60 
5-314 60 


07 


69 196 


10200 


50 


4-685 40 


93 


8.69 400 


07 8 


69 453 


10260 


51 


39 


93 


-69 654 


60 


06 


69 708 


10320 


52 


39 


93 


.69 907 


60 


08 


69 961 


10380 


53 


39 


94 


-70 159 


SI 


06 


.70 214 


10440 


54 
55 


39 
4-685 38 


94 


.70 409 


61 


05 


.70 464 


10500 


95 


8.70 657 


5-314 61 


05 8 


.70 714 


10560 


56 


38 


95 


.70 905 


61 


04 


.70 962 


10620 


57 


38 


96 


-71 150 


61 


04 


= 71 2U8 


10680 


58 


38 


96 


-71 395 


62 


03 


,71453 


10740 


59 


38 


97 


.71 638 


62 


03 


. 71 697 



662 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



170'' 



Log. Sin. 



— 00 _ 

46 372 
76 475 
94 084 
06 578 



16 269 
24 187 
30 882 
36 681 
41 797 



46 372 
50 512 
54 290 
57 767 
60 985 



63 981 
66 784 
69 417 
71 899 
74 248 



76 475 
78 594 
80 614 
82 545 
84 393 



86 166 

87 869 
89 508 

91 088 

92 612 



94 084 

95 508 

96 887 

98 223 

99 590 



00 778 

02 002 

03 192 

04 350 

05 478 



08 577 

07 650 

08 696 

09 718 

10 716 



11 692 

12 647 

13 581 

14 495 

15 390 



Ifi 268 
17 128 

17 971 

18 798 

19 RIO 



20 407 

21 189 

21 958 

22 713 

23 4 55 



8-24 185 
Log. Cos. 



D 



30103 
17609 
12494 

9691 
7918 
6695 
579? 
5115 

4575 
4139 
3778 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1772 
1703 
1639 
1579 
1524 

1472 
1424 
1379 
1336 
1296 

1258 
1223 
1190 
1158 
1128 

1099 
1072 
1046 
1022 
998 

976 
954 
934 
914 
895 

877 
860 
843 
827 
811 

797 
782 
768 
755 
742 

730 



Log. Tan. 



— 00 _ 

46 372 
76 475 
94 084 
06 578 



16 269 
24 188 
30 882 
36 681 
41 797 



46 372 
50 512 
54 291 
57 767 
60 985 



63 982 
66 785 
69 418 
71 900 
74 248 



76 476 
78 595 
80 615 
82 546 
84 394 



86 167 

87 871 
89 510 

91 089 

92 613 



94 086 

95 510 

96 88? 

98 225 

99 522 



00 781 

02 004 

03 194 

04 352 

05 481 



06 580 

07 653 

08 69? 

09 721 

10 720 



11 696 

12 651 

13 585 

14 499 

15 395 



16 272 

17 133 

17 976 

18 803 

19 615 



20 412 

21 195 

21 964 

22 719 

23 462 

24 192 



Log. Cot, 



Com. D. 



30103 
17609 
12494 

9691 
7918 
6694 
579? 
5115 

4575 
4139 
3779 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1773 
1703 
163£ 
1579 
1524 

1472 
1424 
1379 
1336 
1296 

125? 
1223 
1190 
1158 
1128 

109? 
1072 
1046 
1022 
999 

976 
954 
934 
914 
895 

877 
860 
843 
827 
812 

797 
783 
768 
755 
742 

730 



Com. D. 



Log. Cot. 



+ 00 

53 627 
23 524 
05 915 
93 421 



83 730 
75 812 
69 117 
63 318 
58 203 



53 627 
49 488 
45 709 
42 233 
39 014 



36 018 
33 215 
30 582 
28 09? 
25 751 



23 524 
21405 
19 384 
17 454 
15 605 



13 832 
12 129 
10 490 
08 910 
07 386 



05 914 
04 490 
03 111 
01 774 
00 478 



99 21? 
97 995 
96 805 
95 647 
94 519 



93 419 
92 347 
91 300 
90 278 
89 279 



88 303 
87 349 
86 415 
85 500 
84 605 



83 727 
82 867 
82 023 
81 196 
80 384 



79 587 
78 804 
78 036 
77 280 
76 538 

75 808 

Log. Tan. 



Log. Cos. 



00 000 
00 000 
00 000 
00 000 
00 000 



00 000 
00 000 
00 000 
00 000 
00 000 



00 000 
00 000 
99 99? 
99 99? 
99 999 



99 999 
99 999 
99 999 
99 99? 
99 999 



99 999 
99 999 
99 999 
99 999 
99 999 



99 999 
99 99? 
99 998 
99 998 
99 998 



99 998 
99 998 
99 998 
99 998 
99 998 



99 997 
99 997 
99 997 
99 997 
99 997 



99 997 
99 997 
99 997 
99 996 
99 996 



99 996 
99 996 
99 996 
99 996 
99 995 

99 995 
99 995 
99 995 
99 995 
99 994 



99 994 
99 994 
99 994 
99 994 
99 993 

99 993 

Log. Sin. 



663 



89" 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 

AND COTANGENTS. 



178° 



B5 
56 
57 
58 
69_ 

60 



log. S'tn. 



8.24 185 
8-24 903 
8-25 609 
8-26 804 
8-26 988 



8-27 661 
8-28 324 
8-28 977 
8-29 620 
8-80 254 



8-30 879 
8-31495 
8-82 102 
8-32 701 
8-33 292 



^-33 875 
8-34 450 
8-35 018 
8-35 578 
8-36 13l 

8-36 677 
8-37 217 
8.37 750 
8-38 276 
8-38 796 

8-39 310 
8-39 818 
8-40 320 
8.40 816 
8-41 307 



8-41 792 
8.42 271 
8-42 746 
8-43 215 
8 . 43 680 



8-44 139 
8-44 594 
8-45 044 
8-45 489 
8 ■ 45 930 

8-46 366 
8-46 798 
8-47 226 
8-47 650 
8-48 069 



8-48 485 
8-48 896 
8-49 304 
8-49 708 
8-50 108 



8-50 504 

8.50 897 

8.51 286 
8.51 672 
8-52 055 



52 434 

52 810 

53 183 
.53 552 
-53 918 



8-54 282 



718 
706 
694 
684 

673 
663 
653 
643 
634 

625 
616 
607 
599 
591 

583 
575 
567 
560 
553 

546 
539 
533 
526 
520 

514 
508 
502 
498 



485 

479 
474 
469 
464 

459 
454 
450 
445 
440 

436 
432 
428 
423 
419 
415 
411 
407 
404 
400 

396 
393 
389 
386 
382 

379 
375 
373 
369 
366 

383 



Log. Tan. 



24 192 

24 910 

25 616 

26 311 
26 995 



8.27 669 

8.28 332 

8.28 985 

8.29 629 
8-30 263 



8-30 888 
8.31 504 
8-32 112 
8-32 711 
8 • 33 302 



8 - 33 885 
8-34 461 
8-35 029 
8-35 589 
J_36i43_ 

8-38 689 
8-37 229 
8-37 762 
8-38 289 
8-38 809 



8-39 323 
8-39 831 
8-40 334 
8-40 830 
8-41 321 



8-41 807 
8-42 287 
8-42 762 
8-43 231 
8-43 696 



Log. Cos. 



D 



8.44 156 
8-44 611 
8-45 061 
8-45 507 
8-45 948 

8-46 385 
8-46 817 
8-47 245 
8-47 669 
8-48 089 



.48 505 
.48 917 
.49 325 
.49 729 
.50 130 



8-50 r?6 
8 . 50 1 20 
8.51 310 
8.51 696 
8-52 079 



8-52 458 
8-52 835 
8.53 208 
8.53 578 
8-53 944 



8-54308 



Com. D. 



Log,, Cot. 



718 
706 
695 
684 

673 
663 
653 
643 
634 

625 
616 
607 
599 
591 

58C 
575 
568 
560 
553 

546 
539 
533 
527 
520 

514 
508 
502 
496 
491 

485 
480 
475 
469 
464 

460 
455 
450 
445 
441 

437 
432 
428 
424 
419 

416 
412 
408 
404 
400 

396 
393 
390 
386 
383 

379 
376 
373 
370 
366 

364 



Com. 



Log. Cot, 



1-75 808 
1.75 090 
1.74 383 
1.73 688 
1-73 004 



72 331 
71 667 
71 014 
70 371 
69 736 



1.69 111 
1.68 495 
1.67 888 
1.67 288 
1.66 697 



1.66 114 
1.65 539 
1.64 971 
1 . 64 410 
1-63 857 



68 310 
62 771 
62 238 
61711 
61 191 



1.60 676 
1.60 168 
1 . 59 €66 
1 . 59 169 
1-58 678 



1-58 193 
1-57 713 
1-57 238 
1 - 56 768 
1-56 304 



1 . 55 844 
1.55 389 
1.54 938 
1 . 54 493 
1-54 052 

1.53 615 
1.53 183 
1.52 754 
1.52 330 
1.51 911 



1.51495 
1.51083 
1.50 675 
1.50 270 
1-49 870 



1-49 473 
1.49 080 
1.48 690 
1.48 304 
1-47 921 



47 541 
47 165 
46 792 
46 422 
46 055 



Log. Cos. 



1-45 691 



Log. Tan, 



99 993 
99 993 
99 993 
99 992 
99 992 



99 992 
99 992 
99 992 
99 99l 
99 991 



99 991 
99 990 
99 990 
99 990 
99 980 

99 989 
99 989 
99 989 
99 989 
99 988 



99 988 
99 988 
99 987 
99 987 
99 987 



99 986 
99 986 
99 986 
99 986 
99 985 



99 985 
99 985 
99 984 
99 984 
99 984 



99 983 
99 983 
99 982 
99 982 
99 982 

99 981 
99 981 
99 981 
99 980 
99 980 



99 979 
99 979 
99 979 
99 978 
99 978 



99 978 
99 977 
99 977 
99 976 
99 976 



99 975 
99 975 
99 975 
99 974 
99 974 



9-99 973 



Log, Sin. 



m" 



664 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



177" 



Log. Sin. 



8-54 282 
8-54 642 
8-54 999 
8-55 354 
8.55 705 



8-56 054 
8 . 56 400 
8-56 743 
8.57 083 
8.57 421 



57 756 

58 039 
58 419 

58 747 

59 072 



59 395 

59 715 

60 033 
60 349 
60 662 



8-60 973 
8.61 282 
8.61 589 

8.61 893 

8.62 196 



8.62 496 

8.62 795 

8.63 091 
8.63 385 
8-63 677 



8.63 968 

8.64 256 
8 . 64 543 

8. 64 827 

8.65 110 



8.65 
8.65 
8.65 
8.66 
8.66 



391 
670 
947 

223 
497 



8.66 
8.67 
8.67 
8.67 
8. 67 



769 
039 
308 
575 
840 



8.68 104 
8.68 366 
8.68 627 

8.68 886 

8.69 144 



8 . 69 400 
8. 69 654 

8.69 907 

8.70 159 
8 . 70 409 



8.70 657 

8.70 905 
8-71 150 

8. 71 395 
8.71 638 



8 71 880 



360 
357 
354 
351 

348 
346 
343 
340 
338 

335 
332 
330 
327 
325 

323 
320 
318 
316 
313 

311 
309 
306 
304 
302 

300 
298 
296 

294 
292 

290 
288 
286 
284 
282 

281 
279 
277 
275 
274 

272 
270 
268 
267 
265 

264 
262 
260 
259 
257 

256 
254 
253 
251 
250 

248 
247 
245 
244 
243 
24l 



Log. Cos. 



D 



Log, Tan. Com. D. 



8.54 308 

8.54 669 

8.55 027 
8.55 381 
8. 55 733 



56 083 
56 429 

56 772 
.57 113 

57 452 



8.57 787 

8.58 121 
8.58 451 

8.58 779 

8.59 105 



8. 



8 



59 428 

59 749 

60 067 
60 384 
60 698 



8-61 009 
8.61 319 

8.61 626 
8-61 931 

8. 62 234 



8. 62 535 

8.62 834 

8.63 131 
8.63 425 
8. 63 718 



8.64 009 
8. 64 298 
8. 64 585 

8. 64 870 

8. 65 153 



65 435 
65 715 

65 993 

66 269 
66 543 



8.66 816 
8-67 087 

8. 67 356 
8-67 624 
8. 67 890 



68 154 
68 417 
68 678 

68 938 

69 196 



8-69 453 
8. 69 708 
8. 69 961 
8-70 214 
8 . 70 464 



8-70 714 

8.70 962 
8-71 208 

8.71 453 
8.71 697 



8-71 939 
Log. Cot. 



360 
358 
354 
352 

349 
348 
343 
341 
338 

335 
333 
330 
328 
325 

323 
320 
318 
316 
314 

3ll 
309 
307 
305 
303 

300 
299 
297 
294 
293 

291 
288 
287 
285 
283 

281 
280 
278 
276 

274 

272 
271 
269 
267 
266 

264 
262 
261 
259 
258 

256 
255 
253 
252 
250 

249 
248 
246 
245 
243 
242 

Com, D. 



Log. Cot. I Log. Cos 



1.45 691 
1.45 331 
1.44 973 
1.44 618 
1.44 263 



1.43 917 
1.43 571 
1.43 227 
1.42-886 
1.42 548 



1.42 212 
1.41 879 
1.41 548 
1.41 220 
1-40 895 



1.40 571 
1.40 251 
1-39 932 
1.39 616 
1-39 302 



1 


38 990 


1 


38 681 


1 


38 374 


1 


38 068 


1 


37 765 



1.37 465 
1.37 166 
1.36 869 
1.36 574 
1. 36 281 

1.35 990 
1.35 702 
1.35 414 
1.35 129 
1.34 846 



1.34 565 
1.34 285 
1.34 007 
1.33 731 
1-33 456 

1-33 184 
1.32 913 
1.32 643 
1.32 376 
1.32 110 



1-31 845 
1-31 583 
1.31 321 
1.31 062 
1-30 803 



1-30 547 
1-30 292 
1-30 038 
1 - 29 786 
1-29 535 



1-29 286 
1-29 038 
1-28 791 
1.28 546 
1.28 303 



1-23 060 



999 973 
9-99 973 
9-99 972 
9-99 972 
999 971 



9-99 971 
9-99 971 
9-99 970 
9.99 970 
9.99 969 



9-99 969 
9.99 968 
9.99 968 
9.99 967 
9.99 967 



9.99 966 
9. 99 966 
9.99 965 
9-99 965 
9.99 964 



9.99 964 
9.99 963 
9.99 963 
9.99 962 
9-' 99 962 



9.99 961 
9.99 961 
9.99 960 
9.99 959 
9.99 959 



9.99 958 
9.99 958 
9.99 957 
9.99 957 
9-99 956 

9.99 956 
9.99 955 
9.99 954 
9.99 954 
9.99 953 
9.99 953 
9. 99 952 
9. 99 952 
9.99 951 
9.99 950 



9.99 950 
9.99 949 
9.99 948 
9.99 948 
9. 99 947 



9.99 947 
9. 99 946 
9. 99 945 
9.99 945 
9 . 99 944 



9 . 99 943 
9.99 943 
9 . 99 942 
9-99 942 
9.99 941 



Log. Tan. 1 Log. Sin 



9-99 940 



60 

59 
58 
57 
-56 
55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 
44} 
39 
38 
37 
36 

35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 

17 
16 

15 
14 
13 
12 
11 

10 

9 

8 

7 

6 

5 
4 
3 
2 
1 

O 



665 



sr' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



176^ 



Log. Sin. 



71 880 

72 120 
72 359 
72 597 
72 833 



73 06? 
73 302 
73 535 
73 766 
73 997 



74 226 
74 453 
74 680 

74 905 

75 129 



75 353 
75 574 

75 795 

76 015 
76 233 



76 451 
76 667 

76 883 

77 097 
77 310 



77 522 
77 733 

77 943 

78 152 
78 360 



d. 



78 567 
78 773 

78 978 

79 183 
79 386 



79 588 
79 789 

79 989 

80 18? 
80 387 



80 585 
80 782 

80 977 

81 172 
81 366 



81 560 
81 752 

81 943 

82 134 
82 324 



82 513 
82 701 

82 888 

83 075 
83 260 



83 445 
83 629 
83 813 

83 995 

84 177 



Log. Tan. 



8. 84 358 



240 
23? 
237 
236 

235 
233 
233 
231 
230 

22? 
227 
226 
225 
224 

223 
221 
221 
219 
218 

21? 
216 
215 
214 
213 
21§ 
211 
210 
209 
208 

207 
206 
205 
204 
203 

202 
20l 
200 
19? 
198 

197 
197 
195 
195 
194 

193 
192 
19l 
191 
189 

189 
188 
187 
186 
185 

185 
184 
183 
182 
182 

181 



Log. Cos. 



d. 



939 
180 
420 
659 
898 

131 
366 
59? 
831 
062 

292 
520 
748 
974 
199 

422 
645 
867 
087 
306 

524 
741 
958 
172 
386 

599 
8ll 
022 
232 
441 



648 
855 
061 
266 
470 



673 
875 
076 
276 
476 



674 
87l 
068 
264 
459 

653 
846 
038 
230 
420 

610 
79? 
987 
175 
361 



83 
83 
83 
84 
84 



547 
732 
916 
100 
282 



84 464 



Log. Cot. c.d. 



c.d, 

241 
240 
238 
237 

235 
235 
233 
232 
231 
229 
228 
227 
226 
225 

223 
223 
221 
220 
219 

218 
217 
216 
214 
214 

213 
212 
210 
210 
209 

207 
207 
206 
204 
204 

203 
202 
201 
200 
199 

198 
197 
197 
195 
195 

194 
193 
192 
191 
190 

190 
188 
188 
187 
186 

185 
185 
184 
183 
182 

182 



Log. Cot.jLog. Cos. 



28 060 9 

27 81989 
27 579 9 
27 341 9 
27 104 9 



26 868 
26 633 
26 400 
26 168 
25 937J9 

25 708 9 
25 479 9 
25 252 9 
25 026 9 
24_801 9 

24 577 9 
24 354 9 
24 133 9 
23 913 9 
23 693 9 

23 475 9 
23 258 9 
23 042 9 
22 827 9 
22 613 9 



22 400 9 
22 188 9 
21 978 9 
21 768 9 
21 559 9 



21 351 9 
21 144 9 
20 938 9 
20 734{9 
20 53019 



20 327|9 
20 125 9 
19 923 9 
19 723 9 
19 524 9 

19 326 9 
19 128 9 
18 93l 9 
18 736 9 
18 541 9 



18 347 9 
18 154 9 
17 961 9 



17 77 
17 57 



17 389 
17 201 
17 012 
16 825 
16 638 



16 453 
16 268 
16 083 
15 900 
15 717 



1-15 535 
Loff, Tan. 



99 940 
88 940 
99 939 
99 938 
99 938 



99 937 
99 936 
99 935 
99 935 
99 934 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 



99 933 50 
99 933 49 
99 932 48 
99 931 47 
99 931 J 6_ 

45 
44 
43 
42 
41 



99 930 
99 92? 
99 928 
99 928 
99 927 

99 926 
99 R25i 39 
99 925 "~ 
99 924 
99^3 

99 922 
99 922 
99 921 
99 &20 
99J919 



38 
37 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 



25 
24 
23 
22 
21 



99-'919 
99 918 
99 917 
99 936 
9^9_916 

99 915 
99 9141 
99 913 
99 912 
99 912 

99 91lbO 
99 910 19 
99 909 18 
99 908 17 
99 907 H 

15 
14 
13 
12 
11 
10 



99 907 
99 906 
99 905 
99 904 
99 903 



99 902 
99 9021 
99 901 
99 9001 
99 899 



99 898 
99 897 
99 896 
99 896 
99 895 



9. 99 894 



93° 



Log. Sin 
666 







P. P. 




330 330 310 300 


6 


33.0 


32.0 


31.0 


30.0 


7 


38.5 


37-3 


36.1 


35.0 


8 


44.0 


42 6 


41.3 


40.0 


9 


49-5 


48. 


46.5 


45-0 ■ 


10 


55.0 


53.3 


51-6 


50.0 


20 


110.0 


106.6 


103.3 


100.0 


30 


165.0 


160. 


155.0 


150.0 


40 


220.0 


213.3 


206.6 


200.0 


50 


275.0 


266.6 


258-3 


250.0 



6 
7 
8 
9 

10 
20 
30 
40 
50 



6 
7 
8 
9 

10 
20 
30 
40 
50 



290 

29.0 

38.8 

38-6 

43-5 

48.3 

96-6 

145.0 

193.3 

241.6 



350 

25.0 

29 

33 

37 

41 

83 
125 
166 
208 



380 

28-0 

32-6 

37-3 

42.0 

46.6 

93-3 

140.0 

186.6 

233-3 

240 

24.0 

28.0 

32-0 

36.0 

40.0 

80.0 

120.0 

160.0 

200-0 



370 

27.0 

31.5 

36. 

40.5 

45. 

9C.0 

135.0 

180.0 

225.0 

330 

23.0 

26-8 

30-6 

34.5 

38.3 

76.6 

115.0 

153.3 

191.6 



360 

260 
30. 



34 

39 

43 

86 

130 

173 

216 



330 

22.0 
25. 



29 

33 

36 

73 

110 

146 

183 





310 


300 


190 


« 21.0 


20-0 


19.0 


7 24.5 


23-3 


22.1 


8 


28. 


26-6 


25.3 


9 


31.5 


30-0 


28.5 


10 


35.0 


33.3 


31-6 


20 


70.0 


66.6 


63.3 


30 


105.0 


100.0 


95.0 


40 


140.0 


133.3 


126.6 


50 


175-0 


166.6 


158.3 



180 

18. 
21-0 
24-0 
27.0 
30.0 
60.0 
90^0 
120.0 
150.0 



7 
8 
9 
10 
20 
30 
40 
50 



9 

0.9 
1.1 
1.2 
1.4 
1.6 
3.1 
4.7 
6.3 
7.9 



9 8 

0.8 
0.? 
1.0 
1-2 
1.3 
2.6 
4.0 
5.3 
6.6 



7 
0.7; 
08 
0.9 
1.0 
1.1 
2.3 
3.5 
4.6 
5.8 



6 

0.6 
0.7 



5.0 



5 

05 
0.3 
0.6 
0.7 
0.8 
1-6 
2.5 
3-3 
4.1 





1 


4 


3 


3 


1 





6 


0.4 


0.4 


0.3 


0.2 


0.1 


0.0 


7 


0.5 


0.4 


0.3 


0-2 


0.1 


0.0 


8 


0.6 


0.5 


0.4 


0.2 


01 


0.0 


9 


0.7 


0.6 


0.4 


0.3 


0.1 


0.1 


10 


0-7 


0.6 


0.5 


0.3 


0.1 


0.1 


20 


1.5 


1-3 


1.0 


0.6 


0.3 


0.1 


30 


2.2 


2.0 


1.5 


1.0 


0.5 


02 


40 


3.0 


2-6 


2.0 


1.3 


0.6 


0.3 


50 


3.7 


3-3 


2.5 


1.6 


08 


0.4 



P. p. 



86' 



TABLE VII — LOGARITHMIC SINES, COSINES. TANGENTS. 

AND COTANGENTS. lyg* 



Log. Sin. 



.84 358 
84 538 
84 718 

84 897 

85 75 

85 252 
85 429 
85 605 

85 780 
.J5 954 

86 128 
86 301 
86 474 
86 645 
86 816 



36 987 
.i}7 156 
.87 325 
.87 494 
■ 87 66l 

.87 828 
.87 995 
.88 160 
.88 326 
88 490 



88 654 
88 817 

88 980 

89 142 
■ 8h 303 



85 
36 
37 
38 S8 
39_8 

40 8 



89 464 
89 624 
89 784 

89 943 
_?0 101 

90 25^ 
90 417 
90 573 
90 729 
90 885 




55 
56 
57 
58 
59 
60 



91040 
91 195 
91349 
91 502 
91 655 



180 
180 
178 
178 

177 
176 
176 
175 
174 

174 
173 
172 
171 
171 

170 

1 

169 

168 

167 

167 
166 
165 
165 
164 

163 
163 
162 
162 
16l 

161 
16C 
159 
159 
158 

158 
157 
156 
156 
156 

155 
154 
154 
153 
153 

152 
151 
151 
150 



Log. Tan, 



84 464 
84 645 

84 826 

85 005 
85 184 

85 363 
85 540 
85 717 
85 893 
88 068 



92 561 
92 710 

92 858 

93 007 
93 154 



.93 301 
.93 448 
.93 594 
.93 740 
93 885 



8 

8.94 029 

jLog. Cos. 



150 
149 
148 
148 
147 

147 
146 
146 
146 
145 

144 
T. 



86 243 
86 417 
86 590 
86 763 
86 935 



87 106 
87 277 
87 447 
87 616 
87 785 



87 953 

88 120 
88 287 
88 453 
88 618 



88 783 

88 947 

89 111 
89 274 
89 436 



89 598 
89 759 

89 920 

90 080 
90^240 

90 398 
90 557 
90 714 

90 872 

91 028 



91 184 
91 340 
91 495 
91 649 
91 8C3 



91 957 

92 109 
92 262 
92 413 
92 565 



92 715 

92 866 

93 015 
93 164 
93 313 



93 461 
93 609 
93 756 

93 903 

94 049 



94 195 



c.d. 

181 
180 
179 
179 

178 
177 
176 
176 
175 

175 
174 
173 
172 
172 

171 
170 
170 
169 
169 
168 
167 
167 
166 
165 

165 

164 
163 
163 
165 

162 
161 
161 
160 
159 

158 
158 
157 
157 
156 

156 
155 
155 
154 
154 

153 
152 
152 
151 
15l 

150 
150 
149 
149 
149 

148 
148 
147 
146 
146 

145 



Log. Cot. 



15 535 
15 354 
15 174 
14 994 
.14 815 



1.14 637 
1.14 459 
1.14 283 
1.14 107 
1.13 93l 



Log. Cos. 



1-13 756 
1.13 582 
1-13 409 
1.13 237 
1.13 065 



1.12 893 
1-12 723 
1.12 553 
1-12 384 
1-12 215 



1-12 047 
1.11 880 
1.11 713 
1.11 547 
1-11 38l 

1.11 216 
1-11 052 
1-10 889 
1-10 726 
1-10 563 



1.10 401 
1-10 240 
1-10 079 



1.09 91§ 9 
1.09 760 9 



1.09 601 
1.09 443 
1-09 285 
1.09 128 
] .08 971 



.08 815 
.08 660 9 

08 505 " 
.08 350 

08 196 



1.08 043 9 
1.07 890 
1.07 738 
1.07 586 
1.07 435 



1.07 284 
1.07 134 
1.06 984 
1.06 835 
1.06 686 



Log. 



Cot c.d. 



1.06 538 
1.06 390 
1.06 243 
1.06 097 
1.05 950 

1.05 805 



Log. Tan 



99 894 
99 893 
99 892 
99 891 
99 890 



99 889 55 
99 888 54 
99 888 53 
99 887 52 
99 886 51 



60 

59 
58 
57 
56 



99 885 
99 884 
99 883 
99 882 
99 881 



50 

49 
48 
47 
46 



99 880 
99 879 
99 878 
99 877 
99 876 



99 875 
99 874 
99 874 
99 873 
99 872 



99 871 
99 870 
99 869 
99 868 
99 867 



99 866 
99 865 
99 864 
99 863 
99 862 



99 861 
99 860 
99 PM 
99 858 

99 857 



99 856 
99 855 
99 853 
99 852 
99 85l 



99 850 
99 849 
99 848 
99 847 
99 846 



99 845 
99 844 
99 843 
99 842 
99 841 



99 840 
99 839 
99 837 
99 836 
99 835 

99 834 



84'* 



Log. Sin 
667 



40 

39 
38 
37 
36 



30 

29 
28 
27 
26 



30 

19 
18 
17 
16 



10 

9 
8 

7 
6 



O 



P. P. 





18] 


L 


180 


178 


6 


18.1 


18.0 


17.8 


7 


21 


1 


21.0 


20.7 


8 


24 


1 


24.0 


23.7 


9 


27 


1 


27.0 


26-7 


10 


30 


1 


30.0 


29.6 


20 


60 


3 


60.0 


59.3 


30 


90 


5 


90.0 


89-0 


40 


120 


6 


120.0 


118.6 


50 


150 


8 


150.0 


148.3 





174 


173 


170 


6 


17.4 


17.2 


170 


7 


20 


3 


20 





19.8 


8 


23 


2 


22 


9 


22.6 


9 


26 


1 


25 


8 


25-5 


10 


29 





28 


6 


28.3 


20 


58 





57 


3 


56.6 


30 


87 





86 





85.0 


40 


116 





114 


6 


113.3 


50 


145 





143 


3 


141.6 





166 


164 


163 


6 


16. 6 


16.4 


16.21 


7 


19.3 


19.1 


18 


9 


8 


22.1 


21.8 


21 


6 


9 


24.9 


24.6 


24 


3 


10 


27.6 


27-3 


27 





20 


55.3 


54-6 


54 





30 


83.0 


82.0 


81 





40 


110.6 


109.3 


108 





50 


138. 3 


136.6 


135 








158 


156 


154 


15 


6 


15.8 


15. e 


15-4 


15. 


7 


18 


4 


18 


2 


17 


9 


17. 


8 


21 





20 


8 


20 


5 


20 


9 


23 


7 


23 


4 


23 


1 


22. 


10 


26 


3 


26 





25 


6 


25. 


20 


52 


6 


52 





51 


3 


50. 


30 


79 





78 





77 





76- 


40 


105 


3 


104 





102 


6 


101. 


50 


131 


6 


130 





128 


3 


126. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 
7 
8 

9 
10 
20 
30 

40 
5C 



150 

15. 

17 

20 

22 

25 

50 

75 
100 
125 



149 

14. 9 
17 
19 
22 
24 
49 
74 
99 
124 



148 

14.8 
17 



146 

14.6 
17 
19 
21 
24 
48 
73 
97 
121 



145 

14.5 
16 



19 
21 
24 
48 
72 
96 
120 



19 
22 
24 
49 
74 
98 
123 

T 

0.1 
0.2 
0.2 
0.2 
0.2 
0.5 
0-7 
1.0 



176 

17-6 
20-5 
23.4 
26.4 
29. § 
58.6 
88.0 
117.3 
146-6 

168 

16.8 
19.6 
22.4 
25.2 
28. 
56. 
84. 
112.0 
140-0 

160 

16.0 
18. 6 
21.3 
24.0 
26-6 
53-3 
80.0 
106. 6 
133-3 



147 

14-7 

17.1 

19.6 

22.0 

24. 

49. 

73 

98 



5 

5 


122-5 



1 
0.1 
0-1 
01 
0-1 
0.1 
0.3 
0.5 
0.6 
I.2I0.8 



o 

0.0 
0.0 

0.5 

0-1 
0-1 
0-1 
0.2 

o.S 
0.4 



p. p. 



85'' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 



.^ 



Log. Sin. 



94 029 
94 174 
94 317 
94 430 
94 333 



94 745 

94 887 

95 028 
95 169 
95 310 



95 450 
95 589 
95 728 
95 837 
98 O05 



96 143 
96 283 
96 417 
96 553 
96 689 



96 825 

96 960 

97 094 
97 229 
97 363 



97 498 
97 629 
97 762 

97 894 

98 026 



O 

1 
2 
3 

_4 

5 
6 
7 
8 

10 

11 

12 

13 

^i- 

15 

16 

17 

18 

19_ 

30 

21 
22 
23 
2£ 

25 

28 

27 

28 

29. 

30 

31 

32 

33 

34. 

35 
36 
37 
38 

39, 

40 

41 
42 
43 
44^ 

45 
46 
47 
48 
49_ 

BO 

1)1 
12 
1)3 
fi4_ 

55 
56 
57 
58 
59. 
60 9-01 923 



98 157 
98 288 
98 419 
98 549 
98 679 



d.. [Log. Tan. 



c,d. 



144? 

i43r 



98 8JJ 

98 937 

99 086 
99 194 

99 322 



99 449 
99 577 
99 703 
99 830 
99 953 



00 081 
00 207 
00 332 
00 456 
00 580 



00 704 
00 828 

00 951 
01073 

01 196 



01 318 
01440 
01 56l 
01 682 
01 803 



143 
143 

142 
142 
141 
141 
14J 

140 
13J 
133 
133 
133 

133 
137 
137 
133 
1331 

13a 
135 
134 
134 
134 

133 
133 
132 
132 
132 

13l 
131 
133 
130 
133 

123 
129 
123 
128 
127 

127 
127 
126 
125 
123 

125 
125 
125 
124 
124 

124 
123 
123 
122 
122 

122 
122 
121 
121 
120 
125 



94 195 
94 340 
94 485 
94 629 
94 773 



94 917 

95 059 

95 202 
95 344 
95 485 



95 626 
95 767 

95 907 

96 047 
96 186 



96 325 
96 464 
96 602 
96 739 
96 876 



97 013 
97 149 
97 285 
97 421 
97 556 



97 690 
97 825 

97 958 

98 092 
98 225 



98 357 
98 490 
98 62l 
98 753 
98 884 



99 015 
99 145 
99 275 
99 404 
99 533 



99 662 
99 791 
99 91? 
00 046 
00 174 



00 300 
00 427 
00 553 
00 67? 
00 804 



Log. Cos. d. 



00 930 

01 054 
01 179 
01 303 
.01 427 



.01 550 
01 673 
-01 796 
-01 91§ 
•02 040 

-02 162 



Cot. 



145 

144 
144 
144 

143 

142 
142 
142 
14l 

141 
141 
140 
140 
133 

139 
138 
133 
137 
137 

137 
136 
138 
135 
135 

134 
134 
133 
133 
133 

132 
132 
13l 
131 
131 

131 
130 

130 
129 

129 

129 
128 
128 
127 
127 

126 

126 
128 
125 
125 

125 
124 
124 
124 
124 

123 
123 
123 
122 
122 

12l 



Log. Cot JLog. Cos.! I 



P.P. 



1.05 805 9-99 834 
1.05 659 9-99 833 
1.05 515 9.99 832 
1.05 370 9.99 831 
1.05 226 9.99 830 



1.U5 083 
1.04 940 



9.99 829 
9.99 827 



1.04 798 9.99 826 
1.04 656 9.99 825 
1.04 514 9.99 824 



60 
59 
58 
57 
56 



55 
54 
53 
52 
51 
1.04 373J9.99 823|50 
1.04 232 9.99 8221 49 



1.04 092 
1.03 952 
1.03 813 

1.03 674 
1.03 536 
1.03 398 
1.03 260 
1.03 123 



9.99 821 48 
9.99 819 47 
9.99 818| 46 

45 
44 
43 
42 
41 



9.99 817 
9.99 816 
9.99 815 
9.99 814 
9.99 813 





145 


144 


143 


14;^ 


6 


14-5 


14-4 


14-3 


14-2 


7 


16-9 


16-8 


16.7 


16-5 


8 


19-3 


19-2 


19-0 


18.9 


9 


21.7 


21-6 


21-4 


21-3 


10 


24.1 


24-0 


23-8 


23-6 


20 


48-3 


48-C 


47.6 


47-3 


30 


72-5 


72-0 


71-5 


71-0 


40 


96-6 


96-0 


95-3 


94-6 


50 


120-8 


120-0 


119-1 


118-3 



1.02 986J9-99 811 40 



1.02 850 
1.02 714 
1.02 579 
1.02 444 



1.02 309 
1.02 175 
1.02 041 
1.01 908 
1.01 775 



1.01 642 
1.01 510 
1.01 378 
1.01 247 
1.01 116 



1-00 935 
1-00 855 
1.00 725 
1.00 595 
1-00 436 



9-99 8101 39 
9-99 809 38 
9-99 808 37 
9-99 807 36 



9-99 805 
3-99 804 
9-99 803 
9.99 802 
9.99 801 



35 
34 
33 
32 
31 



140 

14-0 
16 



18 
21 
23 
46 
70 
93 
116 



139 

13-9 
16 



18 
20 
23 
46 
69 
92 
115 



138 
13-8 
16 



18 

20 
23 
46 
69 
92 
115 



137 

13-7 
16-0 
18 2 
20-5 
22-8 
45-6 
68-5 
91-3 
114.1 



141 

14-1 
16-4 
18. 8 
2(.I 
2b. 5 
47.0 
70-5 
94.0 
117.5 



13-8 
15-8 
lG-1 
20-4 
22-6 
45-3 
68-0 
90-6 
113-3 



9.99 
9.99 
9-99 
9-99 
9-9£ 

3-99 
9-99 
9-99 
9-99 
9-99 



799 30 
798 29 



6 

7 

8 

9 

10 

20 

30 

40 

50 



135 

13-5 
15.7 
18.0 
20.2 
22.5 
45.0 
67.5 
90.0 
112.5 



134 

13.4 
15 



17 
20 
22 
44 
67 
89 
111 



133 

13.3 
15-5 
17.T 
19-9 
22.1 
44.3 
66.5 
88.6 
110-8 



797 
796 

794 

793 
792 
791 
789 
788 



1-00 33/ 9.99 787 



1.00 209 
1.00 081 
0-99 953 
0-99 828 



0.99 699 
0.99 573 
0-99 446 
0-99 321 
0-99 195 



0.99 070 
0-98 945 
0.98 821 
0-98 697 



9-99 786 
9-99 784 
9-99 783 
9.99 782 



28 
27 

25 
24 
23 
22 
21 



9.99 781 
9.99 779 
9.99 778 
9.99 777 
99 776 



33 
19 
18 
17 
16 



9-99 774 
9-99 773 
9-99 772 
9-99 778 



0-98 573 9-99 769 



0.98 450 9.99 768 
0.98 327 9.99 766 
0.98 204 9.99 765 
0-98 081 9.99 764 
0.97 959 9.99 763 



15 
14 
13 
12 
11 



10 
9 
8 
7 
6 

5 
4 
3 
2 
1 





131 


1.30 


129 


6 


13.1 


13.0 


12.9 


7 


15.3 


15.1 


15.0 


8 


17.4 


17-3 


17-2 


9 


19.6 


19-5 


19-3 


10 


21.8 


21-6 


21.5 


20 


43.6 


43-3 


43.0 


30 


65.5 


65.0 


64.5 


40 


87.3 


86.6 


860 


50 


109.1 


108.3 


107.5 



132 

13-2 
15-4 
17.6 
19.8 
22.0 
44.0 
66-0 
880 
110-0 

128 

12-8 
14-9 
17-9 
19-2 
21.3 
42-6 
64-0 
85-3 
106-6 





127 


126 


125 


124 


6 


12-7 


12-6 


12-5 


12-4 


7 


14-8 


14-7 


14 


6 


14-4 


8 


16.9 


16-8 


16 


6 


16-5 


9 


19.0 


18-9 


18 


7 


18-6 


10 


21.1 


21.0 


20 


8 


20-6 


20 


42-3 


42-0 


41 


6 


41-3 


30 


63-5 


830 


62 


5 


62-0 


40 


84-6 


840 


83 


3 


82-6 


50 


105-8 


105-0 


104 


1 


103-3 



dS" 



0.97 838 | 9.99 761 
Log.Tan.jLog. Sin.! ' 

"668 



122 

12.2 
14.2 
16-2 
18.3 
20.3 
40.6 
61.0 
81.3 
101.6 



121 

12-1 
14-1 
16-1 
18-1 
20.1 
40.3 
60.5 
80.6 
100.8 



120 

12-0 
14-0 
16-0 
18-0 
20.0 
40.0 
60.0 
80.0 
100-0 



1 

0.1 
0-? 
0.2 
0.2 



P.P. 



84* 



6* 



TABLE VIL— LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



173^ 



|Log> Sin. d. 



9-01 
9-02 
9-02 
9-02 
9.02 



923 
043 
163 
282 
401 



120 
119 
119 
119 

119 
118 
118 
118 
117 
117 
116 
lib 



Log. Tan. 

9-02 162 
9-02 283 
9.02 404 
9-02 525 
9-02 645 



C.d. 



9-02 765 
9-02 885 
9-03 004 
9.03 123 
9.03 242 



9.03 361 

9.03 479 

^^g|9.03 597 



9-03 714 
19.03 83l 



Log. Cot.jLog. Cos. 



P»P. 



|.04 
9-04 
9.04 
9.04 

9.04_ 

9-04 
9 04 
905 
9-05 
3Q.5_ 

9-05 
9-05 
905 
905 
^■9A 
905 
906 
9-06 
9.06 
908 



116 

}j|l9.03 948 
II? 9-04 065 
{l| 9.04 181 
Itl 9.04 297 
^^* 9.04 413 
114^ 
113 

}}lf9.04 758 
■^■^^9.04 872 
9.04 987 



9.04 528 
9.04 643 



9.05 101 
9.05 214 
9.05 327 
9.05 440 
9.05 553 



93G 
046 
155 
264 
37^1 




9.07 
9.07 
9.07 
9.07 
9.07 



113 
112 
112 

112 
11] 
111 

111 
110 

lie 

liC 
110 

109 
1C9 
109 
109 
108 

108 
108 
107 
107 
107 

107 
106 
106 
106 
106 

}g| 9.07 857 
{H2 9.07 962 
^"^ 9.08 071 



9.05 666 
9.05 778 

9.05 890 
9. 08 001 
9J36_113 

9.06 224 
9.06 335 
9.06 445 
9.06 555 
n.r6 665 

9.06 775 
9.06 884 

9.06 994 

9.07 102 
9. 07 2ll 

9.07 319 
.07 428 
9.07 535 
9-07 643 
9.07 750 



9.08 
9.08 
9.08 
908 
9.08 

I 9-08 



Log. 



072 
176 
279 
383 

_48j6 

589 
Cos. 



104 
104 

104 
104 
103 



908 177 
9.08 283 



121 
121 
120 
120 
120 
119 
119 
119 
119 

118 
118 
118 
117 
117 

117 
116 
116 
116 
115 

115 
115 
Hi 
114 
114 

114 
113 
113 
113 
113 

.112 

112 
112 

m 
111 

111 
111 

llO 
liG 
ixO 

109 
109 
109 
108 
109 

108 

108 
107 
107 
107 

107 
107 
106 
106 
106 



0'.97 83fa 9.99 76if 
0.97 716 9-99 760 
0.97 595 9-99 759 
0.97 475 9.99 757 
0.97 354i9 .99 756 

9'.99 754 



GO 
L9 
58 
57 



0.97 234 
0.97 115 
0-96 995 
0.96 876 
0.96 757 



9.99 753 
9-99 752 
9.99 750 
9.99 749] 



8 
9 

llO 
r,l 20 



0.96 639 9.99 748 
0.96 521 9.99 746 
096 403 9.99 745 



0.96 285 
096 168 



9.99 744 
9-99 742 



0.96 051 
0.95 935 
|i).95 818 
0.95 702 
0.95 587 

0.95 471 
0.95 356 
0.95 242 
0.95 127 
0.95 013 

0.94 899 



9.08 389 

9.O8 494 

|i^g 9.08 600 

^"^'9.08 705 



103 
103 



d. 



9.99 741 
9.99 739 
9.99 738 
9.99 737 
9.99 735 

9.99 734 
9.99 732 
9.99 731 
9.99 730 
9.99 728 



53 
52 
51 

50 

49 
48 
47 
M. 
45 
44 
43 
42 
41 



131 

12. l! 
14.2 
16.2 
18.2 
20.2 
40.5 
6O.7 
81.0 
101-2 



121 

12.1 
14.1 
16.1 
18-1 
20.1 
40.3 
60. 5 
80.6 
100.8 



120 

12.0 
U 



16 
18 
20 
40 
60 
80 
100 



119 

11.9 
13.9 
15.8 
l7.g 
19-8 
39-6 
59.5 
79.3 
99 -i 



lis 

11.8 
13.7 

15.7 
17.7 
19.6 
39.3 
59.0 
78.6 
98.3 



9.99 727 
0-94 785 9.99 725 



0.94 672 
0.94 559 
0.94 446 



9.99 724 

9.99 723 

9.99 72l 

0.94 334 9.99 720 
0.94 222 9.99 718 
0.94 110 9.99 717 
0.93 998 9.99 715 
0.93 887 9.99 714 

0.93 77619.99 712 



0.93 66E 
0-93 554 
0.93 444 
0.93 334 



9-99 711 
9.99 710 
9-99 708 
9.99 707 



0.93 225 9.99 705 
0.93 1a5 9.99 704 
0.93 006 9.99 702 
0.92 897 9.99 70i 
0.92 788 9.99 699 



0-92 680 9.99 698 
0.92 572 9.99 696 
0.92 464 9.99 695 
0.92 357 9.99 693 
0.92 24919.99 692 



40 

39 
38 
37 
36 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



0.92 142 9.99 690 
0.92 035 9.99 689 
0.91 929 9.99 687 
0.91 822 9.99 686 
- 0-91 7I6J9.99 684 

^^|0.91 61lf9.99 683 

105 



105 



9.08 8101^^^0.91 190 



9-08 914 



104 



0.91 505 9.99 681 

0.91 400 9.99 679 

0.91 295 9-99 678 

9.99 676 



Log. Cot. c.d. 



0.91 085 



Leg. Tan. 



9^ 



9-99 675 
Log. Sin. I 

669 



20 

19 
18 
17 

M 
15 
14 
13 
12 

ii 

10 

9 

8 

7 

_fi 

5 
4 
3 
2 
1 

O 



117 117 



11.7ill 
13.7 13 
15.6:15 
17.6 17 
10 19.6 19 
20 39.1i39 
SO 58.7158 
40 78.3178 
50 97-9197 



116 115 

ii.e;ii.5 
13.5 13. 
15.4 IE. 

17.417. 
19.3 19- 
38 6 38. 
58-0 57.5 
77-3 76.6 
96.6 95.8 



6 

7 
8 

9 
10 
20 
30 
40 
50 



115_114 113 112 

-2 
-0 



11.4 


11.4 


11.3 


Il- 


13-3 


133 


13.2 


ls- 


15.2 


15.2 


15-0 


14- 


17.2 


17.1 


16-9116- 


19.1 


19.0 


18-8jl8. 


38.1 


38.0 


37.6 37. 


57.2 


57.0 


56-5 56. 


76.3 


760 


75.3 74- 


95.4 


95.0 


94.1 93. 



111 

11.1 
12.9 

14.& 
16.6 
I8.5 
37.0 
55.5 
74.0 
92-S 



6 
7 
8 

9 
10 
20 
30 
40 
50 



110 110 



11.0 


11 





10. 


12.9 


12 


8 


12. 


14.7 


14 


6 


14. 


16.6 


16 


5 


16. 


18.4 


18 


3 


18. 


36.8 


36 


6 


3b. 


5o.2 


50 





5*. 


73.6 


73 


3 


7^. 


92. i 


9x 


6 


9o. 



109 

9 
7 
5 
3 
I 
3 
5 
6 
8 



108 

10.8 
12.6 
14.4 
162 
180 
360 
5t.O 
72.0 
9u.O 



6 

7 

8 

9 

10 

20 

80 

40 

50 



107 


107 


106 105 


10.7 


10.7 


10.6 


10.5 


12.5 


12.5 


12.3 


12.2 


14.3 


14.2 


14-1 


14.0 


16.1 


16.0 


15.9 


15.7 


17-9 


17.8 


17.6 


17.5 


35. R 


35.6 


35.3 


350 


53.7 


53.5 


530 


52.5 


71.6 


71.3 


70.6 


70.0 


89-6 


89-1 


883 


87.5 



6 
7 

8 

9 

10 

20 
30 
40 
50 



loS 

10.3 
12.1 

13.8 
15-5 
17.2 
34.5 
51.7 
69 
86.2 



103 3 

10.3 0.2 
CO. 2 
7 0-2 

? 0.3 

lio-3 

2 0.6 
5 1.0 

d 1-3 

W|l.6 
P.P. 



1 

0.1 
0.2 
0.2 
0.2 
0.2 
0.5 
0.7 
1-0 
1.2 



104 

10.4 

12.1 
13.8 
15.6 
17-5 
34.6 
520 
69.§ 
86.6 

1 

0.1 
0.1 
O.I 
0.1 

04 
0.3 
0.5 

0-6 
0.8-, 



sal* 



TABLE VII.— LOGARITHMIC SINES. COSINES, 
AND COTANGENTS. 



TANGENTS, 



O 

1 

2 

3 

± 

5 
6 
7 
8 
^ 

10 

11 
12 
13 
ii 
15 
16 
17 
18 

30 

21 

22 
23 
24 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39. 

40 

41 

42 

43 

44 

45 

46 

47 

48 

ii 

50 

51 

52 

53 

il 

55 

56 

57 

58 

59_ 

eo 



Log. Sin.) d, 



9.08 589 
9.08 692 
9. 08 794' 
9. 08 897| 
9. 08 999 



9.09 101! 
9-09 202i 
9 09 303 
9-09 404 

909 505 



9-09 603 
9- 09 706 
9. 09 806 

9. 09 903 
9-J-Q_P06 

9.10 105 
10 205 

9.10 303 
9.10 402 
9-10 501 



9.09 
9.09 
'9.09 
9.09 
..J9.09 

iggg.io 



10 599 
10 697 
10 795 
10 892 
10 990 



11087 
11 184 
11 281 
11 377 
11 473 



11570 
11 665 
11 761 
11 856 
11 952 



12 047 
12 141 
12 236 
12 330 
12 425 



9 
9 
9 
9_ 

s'.12 518 
9.12 612 
9.12 706 
9. 12 799 
9.12 892 

9-12 985 
13 078 
13 170 
13 263 
13 355 



9-13 447 

9.13 538 

9-13 630 

13 721 

13 813 



13 goa- 
ls 994 

14 085 

14 175 

-14 265 



9^14jj5 
Log. Cos. 



433 
536 
639 

742 
844 



Log. Cot. Log. Cos. 



0.91 
0.90 
0.90 
0-90 
0.90 



0-90 



104 
104 
103 
103 

103 

103 

947 }n?|0-90 
1^^0.89 
|"f 0.89 

^"^iO.89 



085 
981 
877 



99 
9.99 
9.99 



773 9.99 
670 9.99 



675 
673 
672 
670 
669 



56619.99 
463 9.99 
S60 9-99 
258 9-99 
155 9.99 



667 
665 

664 
662 



048 
150 
252 
353 



9.11 
9-11 
9.11 
|9.11 
9.11 



Jg{l0.89 
}gg0.89 



0-89 
0.88 



053 9-99 
951 9-99 
849 9-99 
748 9.99 
647 9^9^ 

546 9.99 

445 9-99 
344 9-99 
244 9.99 
1449:99 



661 51 I 



659 50 
658 49 
656 48 
654 47 
b53 46 



19.12 
9-12 
9.12 
9-12 
9.12 



0.88 
0.88 
0.88 

0.88 
0.38 
0.88 
0.88 
0.88 



04489 -99 
944 9.99 



845 
745 
-64_ 
548 
449 - 
851 9 
253 
155 



9.99 
9.99 
9.99 



0-88 
0.87 
0.87 
0.87 
0.87 



651 
650 
643 

646 
645 _ 

6431 40 
641 39 
640 38 
638 37 
63711 38 



96 



qr,p • ■'■"J 

^^19.13 

92 

91 



91 
90 
90 
90 

90 




9-14 



9.14 



319 
412 
19.14 504 
596 
688 
780 



9.14 
9.14 



9.14 



d. [Log. 



0-87 
'0-87 
iO.87 
0.87 
!0.87 

0.87 
0.86 
0.86 
0.86 
0.86 



05719 -99 
959 9.99 
862I9.99 
735 
_668 

571 
475 
379 
283 
187 



0.88 
0.86 
0.86 
086 
0.86 



635 35 
63 134 
632 33 
630 32 
_628 31 

627 30 
625 29 
623 28 



9-99 
9.99 
9.99 
9.99 
9.99 

9.99 
9.99 
9.99 
9.99 
9.99 



610 
608 
606 
605 
603 



616 9-99 
521 9.99 
427 9.99 
333 9.99 
239 9.99 



601 15 



086 
C.86 
0.85 
0.85 

0.85 
085 
0.85 
0.85 
085 



146 9.99 
052 9.99 
959 9.99 
(9 -99 
77319.99 

680 
588 

495 
403 
311 



g _^5 

Cot. c.d.jLog, 



jl| 
Tan. 



9.99 
9.99 
9.99 
9.99 
9.99 



584 
582 
580 
579 
577 

575 

Sin, 



IT, 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



104 103 



103 

2 



10 


4 


10.3 10- 


12 


1 


12-0'11. 


13 


8 


13.7 


13. 


15 


6 


1'"^.4 


15 


17 


3 


17-1 


17 


34 


6 


34.3 


34. 


52 





51.5 


51- 


69 


3 


G8.6 


68. 


86 


6 


85.8 


85. 



10 i 

10.1 
11. B 
13.4 
15.1 
8 



9 
6 
3 

0.16 
33 
50 
67 
0,84 



6 

7 
8 
9 
10 
20 
3^ 
iO 
50 



100 100 



10.0 

11.7 

13.4 
15.1 
16-7 
33.5 
50.2 
67.0 
83.7 



99 

9.9 
11.5 
13.2 
14.8 
16. 5 
33-0 
_49.5 
66. 6166. 0165 
83. 3182.5181 



io.g 

11.6 
13-3 
15-0 
16.6 
33.3 
50.0 



98 

98 



•4 
■ li 

• 7 
•3 

• 6 


a 

•6 





97 


97 


96 


6 


9.7 


9.7 


9.6 


7 


11 


4 


11.3 


11.2 


8 


13 





12.9 


12.81 


9 


14 


6 


14.5 


14.4! 


10 


16 


2 


16.1 


16.0 


20 


32 


5 


32.3 


32 


30 


48 


7 


48.5 


48-0 


4C 


65 





64.6 


64. 


50 


81 


2 


80.8 


80.0' 



6 

7 

8 

9 

10 

20 

30 

40 

50 



94 

9.4 
11.0 
12.6 
14.2 
15.7 



31 
47 
63 
78 



94 

9-4 
10 

12 
14 
15 
31 
47 
62 
78 



93 

9.3 

10.8 
12.4 
13.9 
15.5 
31.0 
46-5 
62.0 
77.5 



95 

9.S 
11.1 
12.6 
14-2 
15.8 
31-6 
47.5 
63.3 
79.1 



93 

9.2 

10. z 
12.2 
13.8 
15.1 

30.6 
46.0 
61-3 
76.6 





91 


91 


90 


2 




6 


9.1 


9. 11 9.010.2:0. 


7 


10.7 


10 


6;10 


50.2 


0. 


8 


12.2 


12 


1112 


00.2 


0. 


9 


13. 7 


13 


6jl3 


50.3 


0. 


10 


15.2 


15 


1 15 





0.3 


0. 


20 


30.5 


30 


3I3O 





0.6 


0. 


30 


45.7 


45 


5 


45 





1.0 


0. 


40 


61.0 


60 


6 


60 





1.3 


1 


50 


76-2 


75 


8 


75 





1.6 


1. 



p, p. 



9^' 



670 



11 

1.2 

.2 

1.2 
• 2 

1.5 

7 


.2 



ss* 



8P 



TABLE VII —LOGARITHMIC SINES, COSINES. TANGENTS, 

AND COTANGENTS. I'S'l* 



(Loff. Sin. 



9!f 



9 
9 
9 
9_ 

5 9 

6 9 

7 9 

8 9 

J-\^ 

10 9 

11 9. 

12 9- 

13 9. 

14 9 



• 14 355 

.14 445 

14 535 

14 624 

14_7J,3 

14 802 
14 891 

14 980 

15 068 
15 157 



15 
16 
17 
18 
1L|9 

20 9 

21 |9 
22 
23 
24 9 

9 

26 29 

27 " 



28 

21 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39, 

40 9 

41 9 

42 19 

43 

45 



• 15 245 

• 15 333 

• 15 421 
■15 508 

15^595 

15 683 

15 770 
15 857 

15 943 

16 030 

16 116 
16 202 
16 288 
16 374 
16 460 

•16 545 

• 16 630 

• 18 716 

• 16 801 
ae 885 

• 16 970 

• 17 054 

• 17 139 

• 17 223 

• 17 307 



17 391 
9.17 474 
9.17 558 
9.17 641 
9.17 72A 



17 807 
17 890 

17 972 

18 055 
18 137 



47 

48 

41 

50 

51 

52 

5^;: 

5t 
56 
57 
58 
59 

60 



18 219 
18 301 
18 383 
18 465 
18 546 



18 628 
18 700 
18 790 
18 871 

18 952 

19 032 
19 113 
19 193 
19 273 
19 353 

19 433 



Log. Cos, 



90 
89 J 
89^ 
89 

89 
89 
88 
88 
88 

88 
88 
88 
87 
87 

87 
87 
87 
86 
86 

86 
86 
86 
86 
85 

85 
85 
85 
85 
84 

84 
84 
84 
84 
84 

84 
83 
83 
83 
83 

83 
83 
82 
82 
82 
82 
82 
82 
81 
81 

81 
81 9 
81 ^ 
80 

81 
80 
80 
80 
80 
80 

79 



Log. Tan. 

14 780 
14 872 

14 963 

15 054 
15 145 



15 236 
15 327 
15 417 
15 507 
15 598 



15 687 
15 777 
15 867 

15 956 

16 045 

16 134 
16 223 
16 312 
16 401 
16 489 



16 577 
16 665 
16 753 

16 841 
18 928 

17 015 
17 103 
17 190 
17 276 
17 363 



17 450 
17 536 
17 622 
17 708 
17 794 



178 

17 965 

18 051 
18 136 

18 221 

18 306 
18 390 
18 475 
18 559 
18 644 



18 728 
18 812 
18 896 

18 979 

19 063 



19 146 
19 229 
19 312 
19 395 
19 478 

19 560 
19 643 
19 725 
19 807 
19 889 



c.d, 



19 971 



Log. Cot. c.d 



91 
91 
91 
91 

91 
90 
90 
90 
90 

89 
90 
89 
89 
89 

89 
89 
89 
88 
88 

88 
88 
87 
88 
87 

87 

87 
87 
86 
87 

86 
86 
86 
86 
85 

83 
85 
85 
85 

85 

85 

84 
84 
84 
84 
84 
84 
84 
83 
83 

83 
83 
83 
83 
82 

'62 
82 
82 
82 

82 

82 



Log. Cot. 



—I 



85 219 
85 128 
85 037 9 
84 94b 
84 854 

84 763 
84 673 
84 582 
84 492 
84 402 



Log. 



^os.B 



99 575 GO 

99 573 59 

99 571 58 

9^99 570 57 

9^99 568 51 

55 
54 
53 
52 
51 



99 566 
99 564 
99 563 
99 561 
99 559 



84 
84 

84 
84 
83 

83 
83 
83 
83 
83 

83 422 
83 334 
83 247 
83 159 
83 071 



312 
222 
133|9 
043 9 
95419 

86B 
776 
687 

59e 

511 



82 984 
82 897 
82 810 
82 723 
82 63fc 

82 55C 

82 464 
82 377 
82 291 
82 206 

82 120 9 
82 034 9 
81 949 9 
81 864 9 

81 779 fl 



99 55V 60 
99 555 49 
99 55c "" 
99 552 
99 55C 



99 548 
99 546 
■99 544 
■99 542 
■99 541 



48 

47 
46 



45 
44 
43 
42 
41 



■ 99 539 40 
■99 537 39 

99 535 38 
■99 533 37 

■ 99 5311 36 



99 529 
93 528 
99 526 
99 524 
S9 522 



35 

34 
33 
32 
3L 

S0~ 
29 
28 
27 
26 

51l|25 
99 509 24 
99 507 23 
99 505 22 

99 5031 21 



■99 5201 
■99 518^ 
■99 516 
•99 514 
• 99 512, 



81 694 9-99 501 20 
81 609 9-99 499 19 
81 52589-99 4971 1 
81 440 
81 356 

272 
81 188 9 
81 104 
81 020 
80 937 




99 4911 15 
99 4891 14 
99 487 
99 485 
99 484 



80 85419 
80 770 9 
80 687 9 
80 604 9 
80 522 9 



99 482 
99 480 

99 478 
99 476 
99 474 



13 
12 
11 



10 
9 
8 
7 

_6 

5 
4 
3 
2 
__1 

0-80 02RS9^99 462 



80 439 9-99 472 
80 357 9 99 470 
80 274 9-99 468 
80 192 9^99 466 
80 110 9. 99 464 



Log. Tari.|Log. Sin. 



P. P. 



6 

7 
8 

9 

10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



91_ 

9.1 



91 



90 

9-0 



89 



8 

510 
11 



59 
0I74 



88 
8 ' 

10 
11 
13 
14 
29 
44 
59 
73 



88 87 86 





85 


85 


84 


83 


6 8-5 


8.5 


8-4 


8. 


7 


10.0 


9 


9 


9 


8 


9. 


8 


11-4 


11 


3 


11 


2 


11. 


9 


12.8 


12 


7 


12 


6 


12. 


10 


14.2 


14 


1 


14 





13- 


20 


28.5 


28 


3 


28 





27. 


30 


42 7 


42 


5 


42 





41. 


40 


57.0 


56 


6 


56 





55- 


50 


71-2 


70 


8 


70 





69- 



7 

8 

9 

10 

20 
80 
40 
50 



83 
82 



82 
8.2 
9 

10 

12 

13 

27 

41 

54 

68 



81 



3 

7 

.0 

• i 
8 
6 

•5 

• 3 
.1 



80 
80 





79 


2 


1 


6 


7^9!0^2 


0. 


7 


9 


3 


2 


0. 


8 


10 


60 


2 


0. 


9 


11 


90 


3 


0. 


10 


13 


20 


3 


0. 


20 


26 


5 





6 


0. 


30 


39 


7 


1 





0. 


40 


53 





1 


3 


1. 


50 


66 


2 


1 


6 


1. 



1 

1.2 
.2 
i.§ 
1.5 
1.7 

•Q 

.2 



P.P. 



66' 



671 



SV 



fFABLE VII.— LOGARITHMIC SINES, COSINES, TANGENliy, 
AND COTANGENTS. 



tm 



f^ 



Log. Sin. 



55 9 

56 9 

57 9 

58 9 

59 ) 



19 433 
19 513 
19 592 
19 672 
19 75] 



19 830 
19 909 

19 988 

20 066 
20 145 



20 223 
20 301 
20 379 
20 457 
20 535 

20 613 
20 690 
20 768 
20 845 
20 922 



20 999 

21 076 
21 152 
21 229 
21 305 



21 382 
21 458 
21 534 
21 609 
21 685 



21 761 
21 836 
21 911 

21 987 

22 062 



22 136 
22 211 
22 286 
22 380 
22 435 



22 509 
22 583 
22 657 
22 731 
22 805 



22 878 

22 952 

23 025 
23 098 
23 171 



23 244 
23 317 
23 390 
23 462 
23 535 



23 607 
23 679 
23 751 
23 823 
23 895 



IK 23 967 



log. Cos. 



d. 



80 
79 
79 
79 

79 
79 
79 
78 
78 

78 
78 
78 
78 
78 

77 
77 
77 
77 
77 

77 
77 
76 
76 
76 

76 
78 
76 
75 
76 

75 
75 
75 
75 

75 

74 
75 
74 
74 
74 

74 
74 
74 
73 
74 

73 
73 
73 
73 
73 

73 
72 
73 
72 
72 

72 
72 
72 
72 
72 

71 



Log. Tan. 



19 971 

20 053 
20 134 
20 216 
20 297 



20 378 
20 459 
20 540 
20 620 
20 701 



20 781 
20 862 

20 942 

21 022 
21 102 



21 181 
21 261 
21 340 
21 420 
21 499 



21 578 
21 657 
21 735 
21 814 
21 892 



21 971 

22 049 
22 127 
22 205 
22 283 



22 360 
22 438 
22 515 
22 593 
22 670 



22 747 
22 824 
22*900 

22 977 

23 054 

23 130 
23 206 
23 282 
23 358 
23 434 



23 510 
23 586 
23 661 
23 737 
23 812 



23 887 

23 962 

24 037 
24 112 
24 186 



24 261 
24 335 
24 409 
24 484 
24 558 



24 632 



c.d. 

8l 
8l 
8l 
81 

81 
81 
81 
80 
81 

80 
80 
80 
80 
80 

79 
79 
79 
79 
79 

79 
79 
78 
78 
78 

78 
78 
78 
78 
78 

77 
77 
77 
77 
77 

77 
77 
76 
77 
76 

76 
76 
76 
76 
73 

78 

75 
75 
75 
75 

75 
75 
75 
75 
74 

74 
74 
74 
74 
74 

74 



d. Log. Cot 



Log. Cot. Log. Cos. 



80 028 
79 947 
79 865 
79 784 
79 703 



79 622 
79 541 
79 460 
79 379 
79 298 



79 218 
79 138 
79 058 
78 978 
78 898 



78 818 
78 739 
78 659 
78 580 
78 501 

78 422 
78 343 
78 264 
78 186 
78 107 



78 029 
77 951 
77 873 
77 795 
77 717 



77 639 
77 562 
77 484 
77 407 
77 330 



77 253 
77 176 
77 099 
77 022 
76 946 



9 
9 
9 
9 
9 

76 870 9 
78 79319 
76 717 
76 641 
76 565 



76 489 
76 414 
78 338 
76 263 
76 188 



78 113 
76 038 
75 96319 
75 888 9 
75 813 9 



75 73919 
75 664 
75 590 
75 516 
75 442 



75 368 



99 462 60 
99 460 59 
99 458! 58 



99 456 
99 454 



99 452 
99 450 
99 448 
99 446 
99 444 



99 442 
99 440 
99 437 
99 435 
99 433 



99 431 
99 429 
99 427 
99 425 
99 423 



57 
56 



55 
54 
53 
52 
51 



.50 

49 
48 
47 

ii. 

45 
44 
43 
42 
41 



99 421 40 
99 419 39 
99 417 38 
99 415 37 
99 413 36 



99 411 
99 408 
99 406 
99 404 
99 402 



35 
34 
33 
32 
31 



99 400 
99 398 
99 396 
99 394 
99 392 



99 389 
99 387 
99 385 
99 383 
99 381 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 



99 379 
99 377 
99 374 
99 372 
99 370 



99 368 
99 366 
99 364 
99 361 
99 359 



99 357 
99 355 
99 353 
99 350 
99 348 



99 346 
99 344 
99 342 
99 339 
99 337 



20 

19 
18 
17 
16 

15 
14 
13 
12 
11 



99 335 



9»* 



c.d. Log. Tan. Log. Sin. 

e72 



10 

9 
8 
7 

' 5 
4 
3 
2 
1 

O 



P. P. 





81 




81 


80 


7£ 


8 


81 


8.1 


8.0 


7. 


7 


9 


5 


9 


4 


9 





9 


8 


10 


8 


10 


8 


10 


6 


10. 


9 


12 


2 


12 


1 


12 





11. 


10 


13 


6 


13 


5 


13 


3 


13. 


20 


27 


1 


27 





26 


6 


26. 


30 


40 


7,40 


5 


40 





39. 


40 


54 


3 54 





53 


3 


52. 


50 


67 


9 67 


5 


66 


6 


65- 



6 

7 

8 

9 

10 

20 

30 

40 

50 



78 78 



7 8 


78 


7. 


9 


1 


9 


1 


9- 


10 


4 


10 


4 


10. 


11 


8 


11 


7 


11. 


13 


1 


13 





12. 


26 


1 


26 





25. 


39 


o 


39 





38. 


52 


3 


52 





51. 


65 


4 


65 





64. 



77 
7 

2 
5 
8 
6 
5 

3 
1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



76 


•^6 75 


74 


7.6 


7.GI 7-5 


7. 


8 


9 


8 


8 8 


7 


8. 


10 


2 


10 


1 10 





9. 


11 


5 


11 


4 11 


2 


11. 


12 


7 


]2 


6 12 


5 


12. 


25 


5 


25 


3 25 


0j24. 


38 


2 


38 


37 


5 


37- 


51 





50 


6150 


0149. 


63 


7 


63 


3! 62 


5'61. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



73_ 

7-3 

8 

9 

11 
12 
24 
36 
40 
81 



73 

7 3 



73 
7.2 



3 

9 
10 
12 
24 
36.5|36 
48 6 48 
60.8160 





71 




71 , 2_ 


6 


7.1 


7.1I0.2I 


7 


8 


3 


8 


3 


3 


8 


9 


5 


9 


40 


3 


9 


10 


7 


10 


6 


4 


10 


11 


9 


11 


8 


A 


20 


23 


8 23 


60 


B 


30 


35 


7 


35 


5 1 


2 


40 


47 


6 


47 


3 1 


6 


50 


59 


8 


59 


12 


1 



2 

0.2 
0.2 
0.2 
0.3 
0-1 
0-6 
l.Q 
1.3 
1.6 



P P. 



8OP 



i(f 



TABLE VII— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 169« 



Log. Sin. 



O 

1 
2 
3 

5 
6 
7 
8 

1( 

n 

12 
13 
14 
15 
16 
17 
18 
19_ 

2G 

21 

22 

23 

24 

25 

28 

27 

28 

29_ 

3€ 

31 

32 

33 

34, 

35 

36 

37 

38 9 

39_ 9 

40 " 

11 

42 

43 

44 



45 
46 
47 
48 
49 

50 

51 

52 

53 

54 

55 
56 
57 
58 
59 
60 



23 967 

24 038 
24 110 
24 181 

24 252 



24 323 
24 394 
24 465 
24 536 
24 607 



24 677 
24 748 
24 818 
24 888 
24 958 



25 028 
25 098 
25 167 

25 237 
25 306 



25 376 
25 445 
25 514 
25 583 
25 652 



25 721 
25 790 
25 858 
25 927 
25 995 



26 063 
26 131 
26 199 
26 267 
26 335 



26 402 
26 470 
26 537 
26 605 
26 672 

26 739 
26 806 
26 873 

26 940 

27 007 

27 073 
27 140 
27 206 
27 272 
27 339 



27 405 
27 471 
27 536 
27 602 
27 688- 



27 733 
27 799 
27 864 
27 929 
27 995 



9 28 060 
Log. Cos. 



il 

7l 
71 
7! 
71 

71 
71 
71 
71 
70 

70 
70 
70 
70 
70 

69 
70 
69 
70 
69 

69 
69 
69 

69 9 
69 ^ 



Log. Tan. 



68 
69 
68 
63 
68 

68 
68 
68 
68 
67 

67 
68 

67 
67 
67 

67 
67 
67 
66 
67 

66 
66 
66 
66 
66 

66 

66 
65 
66 
65 

65 
65 
65 
65 
65 

65 



24 632 
24 705 
24 779 
24 853 
24 926 



25 000 
25 073 
25 146 
25 219 
25 292 



25 365 
25 437 
25 510 
25 582 
25 654 



25 727 
25 799 
25 871 

25 943 

26 014 

26 C8C 
26 158 
26 229 
26 30C 
26 371 



26 443 
28 514 
26 584 
26 655 
26 726 



26 796 
26 867 

26 937 

27 007 
27 078 



27 148 
27 218 
27 287 
27 357 
27 427 

27 496 
27 566 
27 635 
27 704 
27 773 



27 842 
27 911 

27 980 

28 049 
28 117 



28 186 
28 254 
28 322 
28 390 
28 459 



28 527 
28 594 
28 662 
28 730 
28 797 



c.d. 



28 865 



d. Log. Cot. 



73 
74 
73 
73 

73 
73 
73 
73 
73 

73 

72 
72 
72 
72 

72 
72 
72 
72 
71 

72 

7l, 

71 

71 

71 

71 

71 
70 
71 
70 

70 
70 
70 
7C 
70 

70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 
69 
68 
69 
68 

68 
68 
68 
68 
68 

68 
67 
68 
67 
67 
67 

Z6. 



Log. Cot.jLog. Cos.j 



75 368 9 

75 294 9 
75 220 9 
75 147 9 
75 078 9 



75 OOC 
74 927 
74 85489 
74 781 9 
74 70 819 

74 635 
74 562 
74 490 
74 417 
74 345 



9 
9 
9 
9 

74 273 9 
74 201 9 
74 129 9 
74 057 9 
73 985(9 



73 913 
73 842 
73 771 
73 699 
73^28 

73 557 
73 486 
73 415 
73 344 
73 274 



73 203 
73 133 
73 062 
72 992 
72 922 



72 852 
72 782 
''2 712 
72 642 
72 573 



72 503 9 
72 434 9 
72 365 9 
•72 295 9 
72 226 9 

72 157 9 
72 088 9 
72 O20I9 
71 951 9 
71 882 9 



71 814 9 
71 746 9 
71 677 9 
71 609 9 
71 54119 



71 473|9 
71 405 9 
71 337 9 
71 270 9 
71 20219 



99 335 60 
99 333 59 
99 330 58 
99 328 57 
89 3261^ 

55 
54 
53 
52 
51 



99 324 
99 321 
99 319 
99 317 

99 315 

312 50 
99 310 49 
99 308 48 
99 306 47 
99 303!j46 

45 
44 
43 

42 
41 

40 

39 
38 
37 
36 







P 


.P. 








74 73 73 


6 


7.4 


7-3 


7-3 


7 


8 


6 


8 


6 


8 


5 


8 


9 


8 


9 


8 


9 


7 


9 


11 


1 


11 





10 


9 


10 


12 


3 


12 


2 


12 


1 


20 


24 


6 


24 


5 


24 


3 


30 


37 





36 


7 


36 


5 


40 


49 


3 


49 





48 


6 


50 


61 


6 


61 


2 


60 


8 



99 301 
99 299 
99 296 
99 294 
99 292 

99 29C 
99 287 
99 285 
99 283 
99 280 



99 278 
99 276 
99 273 
99 271 
99 269 



35 
34 
33 
32 
li 
99 266|30 
99 264 29 
99 262 28 
99 259 27 
99 257 26 



99 255 
99 252 
99 250 
99 248 
99 245 

99 243 
99 240 
99 238 
99 236 
99 233 



99 231 
99 228 
99 226 
99 224 
99 221 



99 219 
99 216 
99 214 
99 212 
99 209 



99 207 
99 204 
99 202 
99 199 
99 197 



0-71 135 9-99 194 
Log. Tan.lLog. Sin. 



25 
24 
23 
22 
21_ 

20 

19 
18 
17 
16 



15 

14 
13 
12 
11 



10 

9 
8 
7 
6 



'S2 73 71 



7.2 
8 4 
96 
10 9 
10 12.1 
20 24 1 
3036 -2 
40 48 . 3 
50 60.4 



7-2 


7.1 


7. 


8 4 


8 3 


R. 


9-6 


9.5 


9. 


10.8 


10. 7 


10. 


12 


11.9 


11. 


24.0 


23-8 


23. 


36 


35.7 


35. 


48.0 


47.6 


47. 


60 


59 6 


59- 



7 
8 

9 
10 
20 
30 
40 
5Q 



6 
7 
8 

9 
10 
20 
80 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



70_ 

7.0 
82 
9.4 
10.6 
11.7 
23.5 
35.2 
47.0 
58-7 



68 

6 " 



70 

7.0 
8.1 

9.3 
10.5 
11.6 
23.3 
35.0 
46.6 
58.3 

68 
6-8 



69 

6.9 



67 

6 

7 

9 
10 
11 
22 
33 
45 
56 



71 

1 

3 

1 
6 

3 

6 
5 
3 
I 

69 

6.9 

8 

9 

10 
11 
23 
84 
46 
57 



67 

67 



66 


66 


65 


65 


6-6 


6-6 


6 5 


6- 


7 


7 


7 


7 


7-6 


7- 


8 


8 


8 


8 


8 


7 


8- 


10 





9 


9 


9 


8 


9. 


11 


1 


11 





10 


9 


10. 


22 


1 


22 


G 


21 


8 


21. 


33 


2 


33 





32 


7 


32. 


44 


3 


44 





43 


6 


43. 


55 


4 


55 





54 


6 


54. 



2 2 



6 


0.2 


0.2 


7 


0.3 


0.2 


8 


0.3 


0.2 


9 


0.4 


03 


10 


0.4 


o.§ 


20 


0.8 


0.6 


30 


12 


1.0 


40 


1.6 


1.3 


50 


2.1 


1.6 



P.P. 



100° 



673 



7^« 



11^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



168" 



O 

1 
2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 

35 j9 

36 9 

37 9 



Log, Sin, 



9-28 060 
9-28 125 
9.28 189 
9-28 254 
928 319 



9. 28 383 
9-28 448 
9. 28 512 
9. 28 576 
9-28 641 



9-28 705 
9. 28 769 
9- 28 832 
9. 28 896 
9-28 960 



9-29 023 
9 • 29 08Z 
9-29 150 
9-23 213 
9 29 277 
9-29 340 
9-29 403 
9.29 466 
9.29 528 
9.29 591 



29 654 
29 716 
29 779 
29 841 
29 903 



38 
39_ 

40 

41 

42 

43 

44 



■29 965 

■30 027 

■30 089 

30 151 

30 213 

30 275 
30 336 
30 398 
30 459 
30 fyO.O 



30 532 
30 643 
30 704 
30 765 
30 8^6 



45 


9 30 886 


46 


9 30 947 


47 


9 31008 


48 


9 31 068 


49 


9-31 129 


50 


9-31 189 


51 


9 31 249 


52 


9-31 309 


53 


9-31 370 


54 


9-31 429 


55 


9-31 489 


56 


9-31 549 


57 


9-31 609 


58 


9-31 669 


59 


9-31 728 


60 


9. 31 788 




Log. Cos. 



d, 

65 
64 
65 
6i 

64 
64 
65 
64 
64 

64 
64 
63 
64 
63 

63 
63 
63 
63 
63 

63 
63 
63* 
62 
63 

62 
62 
62 
62 
62 

62 
62 
62 
62 
61 

62 
61 
61 
6l 
61 

6l 
61 
61 
61 
61 

60 
61 
60 
65 
60 

60 
60 
60 
60 

59 

60 
60 
59 
60 
59 
59 



Log. Tan, 



• 28 865 

■ 28 932 

■ 29 000 

• 29 067 
■29 134 



.29 201 
■29 268 
■29 335 
■ 29 401 
• 29 468 



•29 535 
■29 801 

29 667 
•29 734 

29 800 

29 866 
•29 932 
■29 998 
■30 064 
• 3^0 ]jg 

•30 195 
•30 260 
•30 326 
•30 391 
•30 456 



c.d. 



•30 522 
•30 587 
■30 052 
•30 717 
■30 781 



9^ 
9^ 



30 846 
30 911 

30 975 

31 040 
31 104 



31 168 
31 232 
31 297 
31 331 
31 424 



31 488 
31 552 
31 616 
31 679 
31 743 



9 
9 
9 
9 
9_ 

1. 
9 

9^ 
1. 

9^ 
9^ 
9 



•31 80G 

•31 839 

•31 933 

31 996 

• 32 059 



32 122 
32 ■'85 
32 248 
32 310 
32 373 



d. 



32 433 
32 498 
32 580 
32 623 
32 685 



32 747 



Log. Cot. c. d. 



67 
67 
67 
67 

67 
66 
67 
66 
67 

66 
66 
66 
66 
66 

66 
66 
66 
66 
65 

65 
65 
65 
65 
65 

65 
65 
65 
65 
64 

65 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
84 
63 
65 
63 

63 
63 
63 
63 
63 

63 

63 
63 
62 
63 

62 
62 
62 
62 
62 

62 



Log. Cot. 



0-71 135 
0^71 OGV 
0.71 000 
0.70 933 
0.70 866 



070 798 
0-70 732 
0-70 665 
. 70 598 
0. 70 531 

0.70 465 
0.70 398 
0.70 332 
0.70 266 
0.70 200 



0.70 134 
0.70 068 
0.70 002 
0.69 936 
0.69 870 



Log. Cos. 



9-99 194 
J99 192 
9-99 189 
9-99 187 
9. 99 185 



99 182 
99 180 
99 177 
99 175 
99 172 



99 170 
99 167 
99 165 
99 162 
99 160 



9. 99 157 
9-99 155 
9-99 152 
9. 99 150 
9-99 14? 



0.69 
069 
069 
069 
69 



805 
739 
674 
608 
543 



0.69 
0.69 
0.69 
0.69 
0.69 



478 
413 
348 
283 
218 



9. 



99 145 40 

99 142 39 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



99 139 
99 137 
99 134 



•99 132 
•99 129 
■39 127 
■99 124 
99 122 



0.69 153 
0.69 089 
0.69 024 
0.68 9o0|9 
0-68 896 9 



0-68 831 
0-68 707 
0-68 703 
0.68 639 
0.68 57 5 

0-68 511 
0.68 447 
68 384 
0.68 320 
0.68 257 



99 119 
99 116 
99 114 
99 111 
99 109 



38 
37 
36 



35 

34 
33 
32 
31 



999 106 
9. 99 104 
9. 99 101 
9. 99 098 
999 096 



99 093 
99 091 
99 088 
99 085 
99 083 



68 193 9 
68 130 9 
68 067 9 
68 004 9 
67 941 9 



99 080 
99 077 
99 075 
99 072 
99 069 



067 878 
0.67 815 
0.67 752 
067 689 
0.67 626 



67 504 
67 501 
67 439 
67 377 
67 314 



0.67 252 
Log. Tan. 



99 067 
99 064 
99 062 
99 059 
99 056 



30 
29 
28 
27 
26 



25 
24 
23 
22 
21 



20 

19 
18 
17 
16 



15 
14 
13 
12 
11 



10 

9 
8 
7 
6 



•99 054 

• 99 051 

• 99 048 
99 046 

•99 043 



9.99 040 
Log, Sin, 



P. P. 



67 



7 
8 
9 
10 
20 
30 
40 
50 



6 


7 


6. 


7 


9 


7- 


9 





8. 


10 


1 


10 


11 


2 


11- 


22 


5 


22. 


33 


7 


33. 


45 





44. 


56 


2 


55. 



67 

7 
8 

9 

g 
1 

3 
5 
6 

s 





66 


66 


65 


6/ 


6 


6.6 


6-6 


6.5 


6. 


7 


7 


7 


7.7 


7 


6 


7- 


8 


8 


8 


88 


8 


7 


8- 


8 


10 





9.9 


9 


8 


9 


10 


11 


1 


11.0 


10 


3 


10. 


20 


22 


1 


22. 


21 


8 


21. 


30 


33 


2 


33. 


32 


7 


32. 


40 


44 


3 


44.0 


43 


6 


43- 


50 


55 


4 


55-0 


54 


6 


54. 



6 
7 
8 

9 

10 



64 

6.4 



64 



20|21 



30 
40 
50 



65 

63 



63 

63 





63 


63 


61 


6] 


6 


6.2 


62 


61 


6 


7 


7 


3 


7 


2 


7 


2 


7. 


8 


8 


3 


8 


2 


8 


2 


8. 


9 


9 


4 


9 


3 


9 


2 


9. 


10 


10 


4 


10 


3 


10 


2 


10. 


20 


20 


8 


20 


6 


20 


5 


20- 


30 


31 


2 


31 





30 


7 


30. 


40 


41 


6 


41 


3 


41 





40. 


50 


52 


1 


51 


6 


51 


2 


50. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



60_ 

60 



60 

60 



59 

5-9 





a 


1 


3 


6 


0.3(0-21 


7 





3 


3 


8 





40 


3 


9 





40 


4 


10 





510 


4. 


20 


1 








8 


30 


1 


5 


1 


2 


40 


2 





1 


6 


50 


2 


5 


2 


1 



039 
0I49 

3 

0.2 
0.2 
0.2 
0-3 
0.3 
0.6 
l.Q 
1.3 
1.6 



P. P. 



lor 



674 



78° 



TABLE VIT.— LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



167' 



Log. Sin. 



31 788 
31 847 
31 906 

31 966 

32 0^ 

32 084 

32 143 
32 202 
32 260 
32 319 
32 378 
32 43'6 
32 495 
32 553 
32 611 



32 670 
32 728 
32 786 
32 844 
32 902 



32 960 

33 017 
33 075 
33 133 
33 190 



33 248 
33 305 
33 362 
33 419 
33 476 



33 533 
33 590 
33 647 
33 704 
33 761 



33 817 
33 874 
33 930 

33 987 

34 043 



34 099 
34 156 
34 212 
34 268 
34 324 



34 379 
34 435 
34 491 
34 547 
34 602 



34 658 
34 713 
34 768 
34 824 
34 879 
34 934 

34 989 

35 044 
35 099 
35 154 



9 35 209 
Log. Cos. 



d. 



59 
59 
59 
59 

59 
59 
59 
5 '8 
59 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
57 
58 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

55 
56 
55 
56 
55 

55 
55 
55 
§5 
55 

55 
55 
55 
54 
55 

55 
T 



Log. Tan. 



32 747 
32 809 
32 871 
32 933 
32 995 



33 057 
33 118 
33 180 
33 242 
33 303 



33 364 
33 426 
33 487 
33 548 
33_6p9 

33 670 
33 731 
33 792 
33 852 
33 915 



33 974 

34 034 
34 095 
34 155 
34 215 



34 275 
34 336 
34 396 
34 456 
34 515 



C.d. 



34 575 
34 635 
34 695 
34 754 
34 814 



34 873 
34 933 

34 992 

35 05l 
35 110 



35 169 
35 228 
35 287 
35 346 
35 405 



35 464 
35 522 
35 581 
35 640 
35 698 



35 756 
35 815 
35 873 
35 931 
35 989 



36 047 
36 105 
36 163 
36 221 
36 278 



36 336 
Log. Cot. 



62 
62 
62 
62 

61 
61 
62 
61 
61 

61 
6l 
61 
61 
61 

60 
61 
61 
60 
61 

60 
60 
60 
60 
60 

60 
60 
60 
60 
59 
60 
60 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

58 
58 
59 
58 
58 
58 
58 
58 
58 
58 

58 
58 
57 
58 

57 

58 
C.d. 



Log. Cot. 



67 252 
67 190 
67 128 
67 066 
67 004 



66 943 
66 881 
66 819 
66 758 
66 696 



66 635 
66 574 
66 513 
66 452 
66 390 



66 330 
66 269 
66 208 
66 147 
66 086 



Log. Cos. 



66 026 
65 965 
65 905 
65 845 9 
65 784 9 



65 724 
65 664 
65 604 
65 544 
65 484 



65 424 
65 364 
65 306 
65 245 
65 186 



65 126 
65 067 
65 008 
64 948 
64 889 



64 830 
64 771 
64 712 
64 653 
64 594 



64 536 
64 477 
64 418 
64 360 
64 302 



64 24S 
64 185 
64 127 
64 068 
64 010 



63 952 
63 894 
63 837 
63 779 
63 72l 



0-63 663 
Log. Tan. 



99 040 
99 038 
99 035 
99 032 
99 029 

99 027 
99 024 
99 021 
99 019 
99 016 



60 

59 
58 
57 
56 



99 013 
99 010 
99 008 
99 005 
99 002 



98 999 
98 997 
98 994 
98 991 
98 988 



98 986 
98 983 
98 980 
98 977 
98 975 



98 972 
98 969 
98 966 
98 963 
98 961 



98 958 
98 955 
98 952 
98 949 
98 947 



98 944 
98 94l 
98 93i 
98 935 
98 933 



98 930 
98 927 
98 924 
98 921 
98 918 



98 915 
98 913 
98 910 
98 907 
98 904 



98 901 
98 898 
98 895 
98 892 
98 890 



98 887 
98 884 
98 881 
98 878 
98 875 

98 872 



Log. Si 

675 ~ 



n. 



50 

49 
48 
47 
46 



40 

39 

38 

37 

-36 

35 
34 
33 
32 
_31 
30 
29 
28 
27 
26 



25 
24 
23 
22 

m 

30 

19 
18 
17 
16 



15 
14 
13 
12 
11 



10 

9 
8 
7 
6 



P. P. 



63 61 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6-2 


6l 


6- 


7 


2 


7-2 


7. 


8 


2 


82 


8. 


9 


3 


92 


9 


10 


3 


10.2 


10. 


20 


6 


20-5 


20 


31 





30 7 


SO. 


41 


3 


41 


40. 


51 


6 


51. 2 


50. 



61 

1 
1 
1 
I 
I 
3 
5 
fi 
8 





65 


60 


59 


5S 


6 


60 


60 


5-9 


5- 


7 


7 





7 





6 


9 


6- 


8 


8 





8 





7 


Cl 


7. 


9 


9 


1 


9 





8 


(J 


8. 


10 


10 


1 


10 





9 


9 


9. 


20 


20 


1 


20 





19 


8 


19. 


30 


30 


2 


30 





29 


7 


29. 


40 


40 


3 


40 





39 


6 


89. 


50 


50 


4 


50 





49 


6 


49. 





5S 


58 


57 


5': 


6 


58 


58 


57 


5. 


7 


6 


8 


6 


7 


6 


7 


6- 


8 


7 


8 


7 


7 


7 


6 


7 


9 


8 


8 


8 


7 


8 


6 


8 


10 


9 


7 


9 


6 


9 


6 


9 


20 


19 


5 


19 


3 


19 


1 


19 


30 


29 


2 


29 





28 


7 


28. 


40 


39 





38 


6 


38 


3 


38. 


50 


48 


7 


48 


3 


47 


9 


47. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



56 


56 


55 


5^ 


5.6 


5.6 


55 


5 


6 


6 


6 


5 


6 


5 


6 


7 


5 


7 


4 


7 


4 


7. 


8 


5 


8 


4 


8 


3 


8 


9 


4 


9 


3 


9 


2 


9. 


18 


8 


18 


6 


18 


5 


18 


28 


2 


28 





27 


7 


27 


37 


6 


37 


3 


87 





36 


47 


1 


46 


6 


46 


2 


45 





54 


, 


3 


3 


6 


5.4 


0-3 


0. 


7 


6 


8 





3 


0. 


8 


7 


2 





4 


c 


9 


8 


2 





4 


0- 


10 


9 


1 





5 





20 


18 


1 


1 


G 


0. 


30 


27 


2 


1 


5 


1. 


40 


36 


3 


2 





1. 


50 


45 


4 


2 


5 


2. 



.2 
3 

.3 
4 
4 
8 

•2 
6 

.1 



h. 1'. 



77" 



13° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND C0TAN(;ENTS. 



ie( 



Log. Sin. 





1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 

15 
16 
17 
18 
19_ 

20 

21 
22 
23 
24 

25 
28 
27 
28 
29 

30 

31 

32 

33 

34_ 

35 

33 

37 

38 

3J. 

40 

41 

42 

43 

44 9 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
5i 
60 



35 209 
35 263 
35 318 
35 372 
35 427 



35 481 
35 536 
35 590 
35 644 
35 698 



d. Log. Tan. c. d. 



35 752 
35 806 
35 860 
35 914 
35 968 



36 021 
36 075 
36 128 
36 182 
36 235 

36 239 
36 342 
36 395 
33 443 
36 501 



36 55^ 
36 60^ 
36 660 
36 713 
36 766 



36 818 
36 871 
36 923 

36 976 

37 028 



37 081 
37 133 
37 185 
37 23? 
37 2R9 



37 3'lT 
37 30j 
37 445 
37 497 
37 548 



37 600 
37 652 
37 703 
37 755 
37 806 



37 857 
37 909 

37 960 

38 Oil 
38 062 



9. 38 113 
9.38 164 
9-38 215 
9-38 266 
9. 38 317 



9.38 367 



Log. Cos. 



54 
54 
54 
54 

54 
54 
54 
54 
54 

54 

54 
54 
53 
54 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
52 
53 

52 
52 
52 
52 
52 

52 

52 
52 
52 
52 

52 
52 
5l 
52 
51 

52 
51 
5l 
51 
51 

51 
51 
51 
5l 
51 

51 
51 
50 

51 
51 

50 



I9.36 336 
9 36 394 
9-36 451 
9.36 509 
9.36 585 



9.36 623 
9. 36 681 
9. 36 738 
9-36 795 
9.36 852 



9. 38 909 
9-38 966 
9. 37 023 
9-37 080 
9^.37 136 

9 
9 



S7 193 
37 250 
9-37 306 
937 363 
9-37 419 



37 475 
37 532 
937 588 
9-37 644 
37 700 



9-37 756 
9.37 812 
9.37 868 
9.37 924 
9.37 979 



9.38 035 
9.38 091 
9.38 146 
9-38 202 
938 257 



9-38 313 
938 368 
9-38 423 
938 478 
9-38 533 

9-38 589 
9. 38 644 
9. 38 698 
9.38 753 
9. 38 808 

9. 38 863 

9. 38 918 
9-38 972 

9. 39 027 
9. 39 081 



d. 



9.39 136 
9.39 190 
9.39 244 
9. 39 299 
9-h^9 353 

9 . 39 407 
9.39 481 
9.39 515 
9.39 569 
9. 39 623 



9.39 677 
Log. Cot. 



! 57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
56 
57 
56 

57 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
55 
56 
56 
55 

56 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
54 
55 
55 

54 
55 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 



Log. Cot.jLog. Cos, 



83 663 9 
- 63 60613 
0-63 548 9 



0.63 491 
063 433 



063 376 
0-63 319 
0-63 262 
0-63 204 
0-63 147 



83 090 
0-63 033 
0-62 977 
0. 62 920 
062 863 



98 872 
98 8G9 
98 866 
98 863 
98 860 



0.62 806 
062 750 
0-62 693 
0.62 637 
0-62 580 



-62 524 

-62 468 

62 4] 2 

62 356 

62 299 



62 243 
62 188 
-62 132 
62 076 
62 020 



0-81 
0-61 
0.61 
0.61 
0-61 



.98 858 
-98 855 
98 852 
-98 849 
-98 846 



98 843 
98 840 

■ 98 837 
98 834 

-98 831 



60 
59 
58 
57 
56 



-98 828 
-98 825 
-98 822 
-98 819 
-98 816 

-98 813 
-98 810 
-98 807 
-98 804 
-98 801 



98 798 
-98 795 
-98 792 
•98 789 

98 786 



964 9 
909 
853 9 
798 9 
742,3 



0-61 
061 
0-61 
0-61 
0-61 



687 

63J 
576 
521 
46R 



-98 783 

98 780 

-98 777 

-98 774 

98 771 



50 

49 
48 
47 
-46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



• 98 768 

-98 765 

98 762 

98 759 

.9fi 755 



35 
34 
33 

32 

11 
30 

29 
28 
27 
26 



0-61 411 9 
0-61 35P 9 
0-61 301 " 
061 246 
0-61 191 





9 
9 

0-61 137 9 
0-61 08? 9 
0^61 027 9 
0-60 973 9 
0.80 918 9 



98 75? 
-98 7'^9 
98 7^6 
98 748 
987^0 

98 737 
98 734 
98 731 
98 728 
98 725 



0.60 864 9 
0.60 809 9 
0.60 755 9 
060 701 9 
060 647 9^ 

9 
9 



0.60 592 
0.60 538 
0.80 484 
0-60 430 
0-80 376 



n.fiO 323 



c.tl. Log. Tan. 



9,8 721 
98 718 
98 715 
98 712 
98 709 



25 
24 
23 
22 
?T 

20 
19 
18 
17 
16 



98 706 
98 703 
98 700 
98 696 
98 693 



9. 98 P90 
Log, Sin. 



15 
14 
13 
12 

n_ 

10 

9 
8 
7 
6 



P. P. 





57 


57 


56 


56 


6 


5.7| 


5.7 


5.6; 5.6 


7 


6 


7 


6 


6 


6 


6 


6 


5 


8 


7 


6 


7 


6 


7 


5 


7 


4 


9 


8 


6 


8 


5 


8 


5 


8 


4 


10 


9 


6 


9 


5 


9 


4 9 


3 


20 


19 


1 


19 





18 


8 18 


6 


30 


28 


7 


28 


5 


28 


2 28 





40 


38 


3 


38 





37 


6 37 


3 


50 


47 


9 


47 


5 


47 


146 


6 





55 


55 


54 


6 


5-5 


5-5 


5-41 


7 


6-5 


6 


4 


6 


o 

O 


8 


7-4 


7 


3 


7 


2 


9 


8 3 


8 


2 


8 


2 


10 


9-2 


9 


1 


9 


] 


20 


18-5 


18 


S 


18 


1 


30 


27-7 


27 


5 


27 


2 


40 


37-0 


36 


6 


36 


3 


50 


46.2 


45 


8 


45 


4 



54 

5-4 





53 


6 


5.3| 


7 


6 


2 


(f 


7 


1 


9 


8 





ir 


8 


9 


2C 


'7 


R 


3C 


-^6 


7 


4C 


35 


6 


50 


44 


6 



53 

5-3 



53 
5-§ 



53 
5-2 





51 


5 


1 


5 


D - 


6 


5-] 


5-1 


50 


7 


6 





5 


9 


5 


& 


8 


6 


8 


8 


P 


6 


7 


9 


7 


7 


7 


6 


7 


6 


10 


8 


6 


8 


5 


8 


d. 


20 


17 


1 


17 


C 


16 


8 


30 


25 


7 


25 


5 


25 


2 


40 


34 


3 


34 





33 


6 


50 


42 


9 


42 


5 


42 


1 



1 

8 
9 

10 
20 
30 
40 
50 



3 

0.3 
n a 

0-4 
0-5 
0-R 
1. 1 
1-7 
2-3 
2-9 



3 

0.3 
0.3 
0-4 
0-4 
0-5 
1-0 
1-5 
20 
2-5 



2 _ 

0.2 
03 
0.3 
0.4 
0.4 
0.8 
1.2 
1.6 
2.1 



P.P. 



103° 



676 



76° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



165' 



O 

1 
•2 

3 
.4 

5 
6 
7 
8 

10 

11 
12 
13 

M 

15 
16 
17 
18 

li 
3C 

21 
22 
23 
24 

25 

26 

27 

28 

29, 

30 

31 

32 

38 

Si 

35 

36 

37 

38 

39 



40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 



Log. Sin. 



38 367 
38 418 
38 468 
38 519 
38 569 



38 620 
38 670 
38 720 
38 771 
38 821 



38 871 
38 921 

38 971 

39 021 
39 071 



39 120 
39 170 
39 220 
39 269 
39 319 



39 368 
39 418 
39 467 
39 516 
39 566 



39 615 
39 664 
39 713 
39 762 
39 811 



39 860 
39 909 

39 957 

40 006 
40 055 



40 103 
40 152 
40 200 
40 249 
40 297 



40 345 
40 394 
40 442 
40 490 
40 538 



40 586 
40 634 
40 682 
40 730 
40 777 



40 
40 
40 
40 

41 



825 
873 
920 
968 
015 



063 
110 
158 
205 
252 



299 



9-41 

Log. Cos. d, 



d. 



50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
48 
48 
49 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
47 

48 

47 

47 

147 

47 

47 
47 
47 
47 
47 
47 



Log. Tan. 



677 
731 
784 
838 
892 

945 
999 
052 
106 
159 



212 
265 
318 
372 
425 



478 
531 
583 
636 
689 



742 
794 
847 
899 
952 



004 
057 
109 
16l 
213 



266 
318 
370 

422 
474 



525 
577 
629 
681 
732 



784 
836 
887 
938 
990 



42 
42 
42 
42 
42 



041 
092 
144 
195 
246 



297 
348 
399 
450 
501 



42 
42 
42 
42 
42 

42 



552 
602 
653 
704 
754 

805 



C. d. 



54 
53 
54 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
52 
53 

52 

53 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

5l 
52 
52 
51 
5l 

5l 
52 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 
51 
50 

51 
50 
51 
50 
50 

50 



Log. Cot. c. d 



60 323 
60 269 
60 215 
60 16l 
60 108 



Log. Cot. Log. Cos. 



60 054 
60 001 
59 947 
59 894 
59 841 



59 787 
59 734 
59 68l 
59 628 
59 575 



59 522 
59 469 
59 416 
59 363 
59 311 

59 258 
59 20E 
59 15c 
59 lot 
59 G4f 



58 99L 
58 94£ 
58 89] 
58 838 
58 786 



58 734 
58 682 
58 63G 
58 578 
58 526 



58 474 
58 422 
58 37C 
58 31£ 
58 267 



58 216 
58 164 
58 112 
58 061 
58 010 



57 958 
57 9C7 
57 856 
57 805 
57 753 



57 702 
57 651 
57 600 
57 549 
57 499 



57 448 
57 397 
57 346 
57 296 
57 245 



0.57 195 



Log, Tan. 



98 690 
98 687 
98 684 
98 681 
98 678 



98 674 
98 671 
98 668 
98 665 
98 662 



98 658 
98 655 
98 652 
98 649 
S8 646 



98 642 
98 639 
98 636 
98 633 
98 630 



98 626 
98 623 
98 620 
98 617 
98 613 



98 610 
98 607 
98 604 
98 600 
98 59? 



98 594 
98 591 
98 587 
98 584 
98 581 



98 578 
98 574 
98 571 
98 568 
98 564 



98 561 
98 558 
98 554 
98 55l 
98 548 



98 544 
98 54l 
98 538 
98 534 
98 531 



98 528 
98 524 
98 52l 
98 518 
98 514 



98 511 
98 508 
98 504 
98 501 
98 498 



9. 98 494 



Log, Sin. 



d. 

3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
§ 
3 
3 

5 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 



d. 



60 
59 
58 
57 
_56 

55 

54 

53 

52 

-51 

50 

49 

48 

47 

j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 

32 
31 

30 

29 
28 

27 
26 

25 
24 
23 
22 

2L 

20 
19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 

2 
1 





P. P. 



6 

7 

8 

9 

10 

20 

3C 

4C 

50 



54 

54 
65 
7 



53 

5-3 



53 

5.3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



7 
8 
9 
10 
20 
30 
40 
50 



53 

5.2 



53 

5.2 



51 

5.1 



61 

51 





50 


50 


49 


49 


6 


5-0 


50 


4.9 


4. 


7 


5 


9 


5 


8 


5 


8 


5. 


8 


6 


7 


6 


6 


6 


6 


6. 


9 


7 


6 


7 


5 


7 


4 


7. 


10 


8 


4 


8 


3 


8 


2 


8. 


20 


16 


8 


16 


6 


16 


5 


16. 


30 


25 


2 


25 





24 


7 


24. 


40 


33 


6 


33 


3 


33 





32. 


50 


42 


1 


41 


6 


41 


2 


40. 



48_ 
48 



48 
48 

5 
6 
7 
8 

16 
24 
82 
40 



47 

4.7 



47 

4 

5 

6 

7 

7 
15 
23 
31 
39 



• 7 
•S 
.2 



• S 

•e 

.5 

;! 



6 
7 
8 
g 

10 
20 
30 
40 
50 






3 


0. 


c 


4 


0. 





4 


0. 





5 


0. 





6 


0. 


1 


1 


1. 


1 


7 


1. 


2 


3 


2. 


2 


9 


2. 



• 3 
.3 
■ 4 
.4 
.5 


.5 


.5 



r^. K. 



104' 



677 



75" 



15- 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



Log. Sin. 





1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 

ii 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29^ 

30 9 



9-41 2b)9 
9.41 346 
9-41 394 
9.41 441 
9.41 488 



41 534 

41 581 

9-41 628 

9-41 675 

9-41 72] 



31 
32 
33 
34 
85 
36 
37 
38 
39 



9-41 768 
9-41 815 
9-41 86l 
9.41 908 
9-41 954 



9. 42 000 
9-42 047 
9-42 093 
9-42 139 
9.42 185 



9-42 232 
9 . 42 278 
9-42 324 
9-42 369 
9. 42 415 



42 461 
42 507 
42 553 
42 598 
42 644 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

60 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



42 690 
42 735 
42 781 
42 826 
42 87l 



42 917 

42 962 

43 007 
43 052 
4^ m*' 



9-43 143 
9-43 188 
9-43 233 
9-43 278 
9-43 322 



9-43 367 
943 412 
9-43 457 
9-43 501 
9-43 546 



9.43 591 
9.43 635 
9.43 680 
9.43 724 
9. 43 768 



43 813 
43 857 
9. 43 90T 
9.43 945 

9. 43 989 

9.44 034 
log. Cos. 



d. 



47 
47 
47 
47 

46 
47 
47 
46 
46 

47 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
45 
46 

46 
46 
45 
45 
46 

45 
45 
45 
45 
45 

45 

45 
45 
45 
45 

45 

45 
45 
45 
44 

45 

44 
45 
44 
44 

45 

44 
44 
44 
44 

44 
44 
44 
44 
44 

44 



Log. Tan. 



42 805 
42 856 
42 906 

42 956 

43 007 



43 057 
43 107 
43 157 
43 208 
43 258 



43 308 
43 358 
43 408 
43 458 
43 508 



43 557 
43 607 
43 657 
43 706 
43 756 



43 806 
43 855 
43 905 

43 954 

44 003 



44 053 
44 102 
44 151 
44 200 
44 249 



44 299 
44 348 
44 397 
44 446 
44 484 



44 543 
44 592 
44 641 
44 690 
44 738 



44 787 
44 835 
44 884 
44 932 
44 981 



45 029 
45 077 
45 126 
45 174 
45 229. 

45 270 
45 318 
45 367 
45 415 
45 463 



45 510 
45 558 
45 rf06 
45 654 
45 702 

45 749 



Log. Cot. 



c.d. 

51 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
48 

49 
49 
48 

48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

47 
48 
48 
47 
48 

47 
Gid. 



Log. Cot- 



57 195 
57 144 
57 094 
57 043 
56 993 



56 942 
56 892 
56 842 
56 792 
56 742 



56 692 
56 642 
56 592 
56 542 
56 492 



58 442 
56 392 
56 343 
56 293 
56 243 



56 194 
56 144 
56 095 
56 045 
55 996 



55 947 
55 898 
55 848 
55 799 
55 750 



Log. Cos. 



9 
9 

55 701 9 
55 652 9 
55 603 9 
55 554 9 
5- *^. 505 9 

05 456 9 
55 407 9 



55 359 
55 310 
55 26l 



55 213 
55 164 
55 116 
55 067 
55 019 



54 970 
54 922 
54 874 
54 825 
54 777 



54 729 
54 68l 
54 633 
54 585 
54 537 



54 489 
54 441 
54 393 
54 346 
54 298 

54 250 



Log. Tan, 



98 494 
98 491 
98 487 
98 484 
98 481 



98 477 
98 474 
98 470 
98 467 

98 464 



98 460 
98 457 
98 453 
98 450 
98 446 



98 443 
98 439 
98 436 
98 433 
98 429 



98 426 
98 422 
98 419 
98 415 
98 412 



98 408 
98 405 
98 401 
98 398 
98 394 



98 391 
98 387 
98 384 
98 380 
98 377 



98 373 
98 370 
98 366 
98 363 
98 359 



98 356 
98 352 
98 348 
98345 
98 34l 



98 338 
98 334 
98 331 

98 327 
98 324 



98 320 
98 316 
98 313 
98 309 
98 306 



98 302 
98 298 
98 295 
98 29] 
98 288 

98 284 



Log. Sin. 



3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 

O 
kJ 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
4 
3 
3 

3 
3 
§ 
3 
3 

4 
3 
3 
3 
3 

4 
3 
3 
3 



d. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41_ 

40 

39 
38 
37 
36 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 



30 

19 
18 

17 
16 



15 
14 
13 
12 
11 



10 

9 
8 

7 



P.P. 





50 


50 


6 


5.0 


50 


7 


5 


9 


5 


8 


8 


6 


7 


6 


6 


9 


7 


6 


7 


5 


10 


8 


4 


8 


3 


20 


16 


8 


16 


6 


30 


25 


2 


25 





40 


33 


6 


33 


3 


50 


42 


1 


41 


6 





49 


49 


48 


6 


4.9 


4-9 


4.8 


7 


5 


8 


5 


7 


5 


6 


8 


6 


6 


6 


5 


6 


4 


9 


7 


4 


7 


3 


7 


3 


10 


8 


2 


8 


1 


8 


1 


20 


16 


5 


16 


3 


16 


1 


30 


24 


7 


24 


5 


24 


2 


40 


33 





32 


6 


32 


3 


50 


41 


2 


40 


8 


40 


4 



48 
4 8 



5.8 

6-4 

7.2 

80 

16.0 

24-0 

32.0 

40-0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



47 

4.7 



47 
4.7 



46 

4.6 

5.4 

6.2 

7.0 

7.7 

15-5 

23.2 

310 

38-7 



46 



4 


6 


5 


3 


6 


1 


6 


9 


7 


6 


15 


3 


23 





30 


6 


38 


3 





45 


45 


44 


44 


6 


4.5 


4-5 


4-4 


4. 


7 


5 


3 


5 


2 


5 


2 


5. 


8 


6 





6 





5 


9 


5. 


9 


6 


8 


6 


7 


6 


7 


6. 


10 


7 


6 


7 


5 


7 


4 


7. 


20 


15 


1 


15 





14 


8 


14. 


30 


22 


7 


22 


5 


22 


2 


22. 


40 


30 


3 


30 





29 


6 


29. 


50 


37 


9 


37 


5 


37 


1 


[36. 





4 


3 


3 


6 


0.4 


0.3 


0.3 


7 





4 


0.4 





3 


8 





5 


0.4 





4 


9 





6 


0.5 





4 


10 





6 


0.6 





5 


20 


1 


3 


1.1 


1 





30 


2 





1.7 


1 


5 


40 


2 


6 


2.3 


2 


50 


3 


3 


2.9 


2 


5 



P.P. 



106° 



678 



TABLE vn.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



163' 



Log. Sin. 



44 034 
44 078 
44 122 
44-166 
44 209 



44 253 
44 297 
44 341 
44 384 
44 428 



44 472 
44 515 
44 559 
44 602 
44 646 



44 689 
44 732 
44 776 
44 819 
44 862 



44 905 
44 948 

44 991 

45 034 
45 077 



45 120 
45 163 
45 206 
45 249 
45 291 



45 334 
45 377 
45 419 
45 462 
45 504 



45 547 
45 589 
45 631 
45 674 
45 716 



d. 



45 758 
45 800 
45 842 
45 885 
45 927 



45 969 

46 Oil 
46 052 
46 094 
46 13*) 



46 178 
46 220 
46 261 
46 303 
46 345 

46 386 
46 428 
46 469 
46 511 
46 552 



9-46 593 



Log. Cos. 



44 
44 
44 
43 

44 
44 
43 
43 
44 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
42 
43 
43 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
41 
42 
42 

41 
42 
41 
41 
42 

4l 
4l 
41 
41 
41 

41 
T 



Log, Tan. 



45 749 
45 787 
45 845 
45 892 
45 940 



45 987 
48 035 

46 082 
46 129 
46 177 

46 224 
46 271 
46 318 
46 366 
46 413 



46 460 
46 507 

46 554. 
46 601 
46 647 

46 694 
46 74l 
46 788 
48 834 
46 881 

46 928 

46 974 

47 021 
47 067 
47 114 



47 lec 

47 207 
•47 253 
• 47 2S9 
■ 47 345 



c.d. 



47 392 
.47 438 

47 484 
.47 530 
.47 576 



47 622 
47 668 
47 714 
■47 760 
47 806 



47 851 
47 897 
47 943 

47 989 

48 034 



48 080 
48 125 
48 171 
48 216 
48 262 



48 307 
48 353 
48 398 
48 443 
48 488 



9. 48 534 



48 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
46 

47 
47 
46 
46 
47 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
45 
46 
46 

45 
46 
45 
46 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 



0-54 250 9 • 
0.54 202 9. 
0.54 155 9. 
0-54 lO-^ 9. 
0^5j4j060 9^ 

0-54 012 9. 
0-53 965!9 
0-53 917 
0-53 870 
0-53 823 



Log. Cot. Log. Cos. 



98 284 
98 280 
98 277 
98 273 
98 269 



9- 



98 266 
98 262 
98 258 
98 255 
98 251 



0-53 776 
0-53 728 
0-53 681 
0-53 634 
0-53 587 

. 53 540 
0-53 493 
0.53 446 9 
53 399 9 
0-53 352 9 

0-53 305 9" 
0.53 258 9 
0.53 212 9 
0-53 165 9 
0-53 118 9 

0.53 072 9 



98 247 
98 244 
98 240 
98 236 
98 233 

98 229 
98 225 
98 222 
98 218 

98 214 

98 211 
98 207 
98 203 
98 200 
98 196 



53 025 9 
52 979 9 
0-52 932 9 
0-52 886 9 



52 839 
52 793 
52 747 
52 700 
52 654 



98 192 
98 188 
98 185 
98 181 
98 177 



98 173 
98 i7C 
98 166 
98 162 
98 158 



0-52 608 9-98 155 
9-98 151 
9-98 147 
9-98 143 
fi-98 140 



. 52 56'2 
0-52 516 
0-52 469 

^.Fi2 423 

0-52 377 
0-52 331 
0-52 286 
. 52 240 
0-52 194 

0.52 148 
0.52 102 
0.52 057 
0.52 Oil 
051 965 



5] 920 
51 874 
51 829 
51 783 
51 738 



Log. Cot, 



c.d. 



0-51 692 
0-51 647 
0-51 6O: 
0-5] 556 
0-51 511 
0.51 466 



98 136 
98 132 
98 128 
98 124 
98 121 



98 117 
98 113 
98 109 
98 105 
_98_102 

98 098 
98 094 
98 090 
98 086 
98 082 



98 079 
98 075 
98 071 
98 067 
98_063 
98 059 



Log. Tan.jLog, Sin. 
679 



d. 

3 
3 
3 
4 

3 
3 
4 
3 
3 
4 
§ 
3 
4 
3 

3 
4 
3 
3 
4 

3 
4 
3 
3 
4 

3 
4 
3 
4 

3 

4 

3 
4 

§ 
4 

3 
4 
3 
4 
3 

4 
3 
4 
4 
3 

4 
3 
4 
4 
3 

4 
4 
3 
4 
4 

3 
4 
4 
3 
4 

4 

T 



1 60 

59 
58 

57 
56 

55 
54 
53 
52 



50 

49 
48 
47 

45 
44 
43 
42 
4] 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



SO 

29 
28 
27 
26 

25 
24 
23 
22 
21 



30 

19 
18 
17 
16 



15 
'14 
13 

12 

21 

10 

s 

8 

7 



P. P. 





48 


47 


47 


6 


4.8 


4-7 


4. 


7 


5 


6 


5 


5 


5. 


8 


6 


4 


6 


3 


6- 


9 


7 


2 


7 


1 


7- 


10 


8 





7 


9 


7 


20 


16 





15 


8 


15- 


30 


24 





23 


7 


23- 


40 


32 





31 


6 


31. 


50 


40 





39 


6 


39. 





46 


46 


45 


45 


6 


4.6 


4.6 


4.5 


4. 


7 


5 4 


5 


3 


5 


3 


5. 


8 


62 


6 


1 


6 





6. 


9 


7 





6 


Q 


6 


8 


6. 


10 


7 


7 


7 


6 


7 


e 


7- 


20 


15 


5 


15 


3 


15 


1 


15. 


30 


23 


2 


23 





22 


7 


22 


40 


31 


r 


30 


6 


30 


3 


30 


50 


38 


7 


38 


3 


37 


9 


37. 





44 


43 


4; 


6 


4.4 


4.3 


4- 


7 


5 


1 


5 


1 


5- 


8 


5 


8 


5 


8 


5 


9 


6 


e 


6 


5 


6- 


10 


7 


3 


7 


2 


7. 


20 


14 


6 


14 


5 


14 


80 


22 





21 


7 


21. 


40 


29 


3 


29 





28. 


50 


36 


6 


36 


2 


35. 





42 


43 


41 


43 


6 


4.2 


4.2 


4.1 


4. 


7 


4 


9 


4 


9 


4 


8 


4. 


8 


5 


6 


5 


6 


5 


5 


5 


9 


6 


4 


6 


3 


6 


2 


6. 


10 


7 


1 


7 





6 


9 


6. 


20 


14 


1 


14 





13 


8 


13- 


30 


21 


2 


21 





20 


7 


20. 


40 


28 


3 


28 





27 


6 


27. 


50 


35 


4 


35 





34 


6 


34. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



4 3 



P. P. 



73? 



17^ 



TABLE VII —LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



165 





1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
M 

15 
16 
17 
18 
19_ 

30 

21 
22 
23 
2i 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 

50 

51 

52 

53 

^ 

55 

56 

57 

58 

59_ 

§0 



Log. Sin. 



46 593 
46 635 
46 676 
46 717 
46 758 



46 799 
46 840 
46 881 
46 922 
46 963 



47 004 
47 045 
47 086 
47 127 
47 168 



47 208 
47 249 
47 29Q 
47 330 
47 371 



47 411 
47 452 
47 492 
47 532 
47 573 



47 613 

■ 47 653 

■ 47 694 
47 734 
47 774 



47 814 
47 854 
47 894 
47 934 
47 974 



48 014 
48 054 
48 093 
48 133 
48 173 



48 213 
48 252 
48 292 
48 331 
48 371 



48 410 
48 450 
48 489 
48 529 
48 568 



9-48 607 
9.48 646 
9.48 686 
9.48 725 
9.48 764 



48 803 
48 842 
9.48 881 
9.48 920 
9.48 959 



9.48 998 



Lo«f. Cos. 



d. 



4l 
41 
41 
41 

41 
41 
41 
41 
41 

41 
41 

41 
40 
41 

40 
40 
41 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 

40 
39 
4C 
39 

40 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 
39 
39 
39 
39 
39 

38 



log, Tan. 



48 
48 
48 
48 
48 



534 
579 
624 
669 
71-1 



759 
804 
849 
894 
939 



984 
028 
073 
118 
162 



207 
252 
296 
341 
_385 

430 
474 
518 
563 
607 



651 
695 
740 
784 
828 



872 
916 
960 
004 
048 




529 
572 
616 
659 
702 



746 
789 
832 
876 
919 



962 
005 
048 
091 
134 



51 177 



Log. Cot. 



c.d, 



45 
45 
45 
45 

45 
45 
44 
45 
45 

45 
44 
45 
44 
44 

45 

44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
4-4 

43 
44 

44 
44 
43 
44 
43 

44 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 



Log. Cot.JLog. Cos. 



c.d, 



51 486 9 
51 42119 
51 376 
51 330 
51 285 



51 24Q 
51 195 
51 151 
51 106 
51 061 



51 016 
50 971 
50 926 
50 882 
50 837 



9 
9 
9 
j9 

- 

50 792 9 
50 748 9 
50 703 9 
50 659 9 
50 614 9 

50 570 9 
50 525 9 
50 481 9 
50 437 9 
50 392f9 



50 348|9 
50 304 9 
50 260 9 
50 216 9 
50 172 9 



50 128 
50 083 
50 039 9 
49 996 
49 952 



49 908J9 
49 864 9 
49 820 9 
49 776 9 
49 733 9 

49 689 9 
49 645 9 
49 602 9 
49 558 9 
49 514 9 



49 471 9 
49 427|9 
49 384 
49 340 
49 297 



9 

9 

49 254 9 
49 210 9 
49 167 9 
49 124 
49 081 



9 

9 

49 038 9 
48 994 9 
48 951 
48 908 
48 865 



48 822 
Log. Tan, 



98 059 
98 G5C 
98 052 
98 048 
98 044 

98 040 
98 036 
98 032 
98 028 
98 024 



98 021 
98 017 
98 013 
98 009 
98 005 



98 001 
97 997 
97 993 
97 989 
97 985 

97 981 
97 977 
97 973 
97 969 
97 966 



97 962 
97 958 
97 954 
97 950 
97 946 



97 942 
97 938 
97 934 
97 930 
97 926 



97 922 
97 918 
97 914 
97 910 
97 908 

97 902 
97 898 
97 894 
97 890 
97 886 



97 881 
97 877 
97 873 
97 869 
97 865 



97 861 
97 857 
97 853 
97 849 
97 845 



97 841 
97 837 
97 833 
97 829 
97 824 



97 820 



Log. Sin. 



3 
4 
4 
4 

3 
4 
4 
4 
4 

3 
4 
4 
d 

4 

4 
3 

4 
4 
4 

4 
4 
4 
4 
3 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 
4 
4 

4 
4 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 

ii 
40 

39 
38 
37 
36 



35 
34 
33 
32 

-11 
30 

29 
28 

27 
28 



25 
24 
23 
22 
21 



20 
19 
18 
17 
16 

15 
14 
13 
12 
11 



10 

9 
8 
7 
6 

5 
4 
3 
2 
1 



d. 



P. P. 





45 


45 


44 


44 


6 


4-5 


4-5 


4.4 


4 41 


7 


5 


3 


5 


2 


5.2 


5 


i 


8 


6 





S 





59 


5 


8 


9 


6 


8 


6 


7 


3 7 


6 


e 


10 


7 


6 


7 


5 


7-4 


7 


3 


20 


15 


1 


15 





148 


14 


6 


30 


.22 


7 


22 


5 


22.2 


22 


a 


40 


30 


3 


30 





29.6 


29 


3 


50 


37 


9 


37 


5 


37. 1 


36 


6l 



6 
7 
8 

9 
10 
20 
30 
40 
50 



43 



4.3 


4. 


5 


1 


5. 


5 


8 


5. 


6 


5 


6. 


•7 


2 


7. 


14 


5 


14. 


21 


7 


21. 


29 





28. 


36 


2 


35. 



43 

3 


7 
i 
1 
3 
5 
6 
8 

















41 


41 40 40 


6' 41 


4.1 40 


4-0 


7s 4 


8 


4 


8 4 


7 


4 6 


8 5 


5 


5 


4 5 


4 


53 


9 6 


2 


6 


1 


6 


1 


6 


10 6 


9 


6 


8 


6 


7| 6.g 


20 13 


8 


13 


6 13 


513.3 


30 20 


7 20 


5 20 


2 20 


40 27 


6 27 


3 27 


26 6 


50 34 


6 


34 


1>33 


7 33 3 





39 


3" 


9 


38 


6 


3.9 


3 


9 


3.8 


7 


4 


6 


4 


5 


4 


5 


8 


5 


2 


5 


2 


5 


1 


9 


5 


9 


5 


8 


5 


8 


10 


6 


6 


6 


5 


6 


4 


20 


13 


1 
J. 


13 





12 


8 


30 


19 


7 


19 


5 


19 


2 


40 


26 


3 


26 





25 


6 


50 


32 


9 


32 


5 


32 


1 



e 

7 
8 
9 
10 
20 
30 
40 
50 



7,3 



4 

040 

40 

.5 

•0 
.R 

3 
.0 
.6 
■ S 



P. P. 



1^7" 



6S0 



73° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



161' 



Log. Sin. 



48 998 

49 037 
49 076 
49 114 
49 153 



49 192 
49 231 
49 269 
49 808 
49 346 



49 385 
49 423 
49 462 
49 500 
49 539 



49 577 
49 615 
49 653 
49 692 
49 730 



49 768 
49 806 
49 844 
49 882 
49 920 



49 958 

49 996 

50 034 
50 072 
50 110 



50 147 
50 185 
50 223 
50 260 
50 298 



50 336 
50 373 
50 411 
50 448 
50 486 



50 523 
50 561 
50 598 
50 635 
50 672 



50 710 
50 747 
50 784 
50 821 
50 858 



50 895 
50 932 

50 96a 

51 006 
51 04 3 

51 080 
51 117 
51 154 
51 190 
51 227 



9-51.264 



Logi Cos 



d. 



Log. TaiT. 



39 
39 
38 
39 

38 
39 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
37 
38 
38 

37 
38 
37 
37 
38 

37 f 

37 

37 

37 

37 

37 
37 1 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

37 
36 
37 
36 
37 

36 



51 177 
51 220 
51 263 
51 306 
51 349 



51 392 
51 435 
51 477 
51 520 
51 563 



51 605 
51 648 
51 691 
51 733 
51 776 



51 818 
51 861 
51 903 
51 946 
51 988 



52 030 
52 073 
52 115 
52 157 
52 199 



52 241 
52 284 
52 326 
52 368 
52 410 



52 452 
52 494 
52 536 
52 578 
52 619 



52 661 
52 703 
52 745 
52 787 
52 828 

52 870 
52 912 
52 953 

52 995 

53 036 



53 078 
53 119 
53 161 
53 202 
53 244 

53 285 
53 326 
53 368 
53 409 
53 450 



53 491 
53 533 
53 574 
53 615 
53 656 



.53 697 
Log. Cot. 



43 
43 
43 
43 
42 
43 
42 
43 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
41 

42 
42 
4l 
42 
41 

4l 
42 
4l 
41 
41 

4l 

^1 
41 
4l 
41 

4l 
41 
4l 
41 
4l 

41 
4l 
41 
41 
41 

41 
cTd". 



Log. Cot. Log. Cos. 



48 822 
48 779 
48 736 
48 693 
48 65G 

48 608 
48 565 
48 522 
48 479 
48 437 



48 394 
48 351 
48 309 
48 266 
48 224 



48 181 
48 139 
48 096 
48 054 
48 012 



47 969 
47 927 
47 885 
47 842 
47 800 



47 758 
47 716 
47 674 
47 632 
47_19C 

47 548 
47 506 
47 464 
47 422 
47 380 



47 338 
47 296 
47 255 
47 213 
47 173 



47 13G 
47 088 
47 046 
47 005 
46 9631 



46 922 9 
46 880 9 
46 839 



46 797 
46 756 



46 714 
46 673 
46 632 
46 591 
46 549 

46 508 9 



46 467 
46 426 
46 385 
46 344 



0-46 303 
Log. Tan. 



97 820 
97 816 
97 812 
97 808 
97_8G4 

97 8"C0 
97 796 
97 792 
97 787 
97 78 3 

97 77§ 
97 775 
97 771 
97 767 
97 763 



97 758 
97 754 
97 750 
97 746 
97 742 

97 737 
97 733 
97 729 
97 725 
97 721 



97 716 
97 712 
97 708 
97 704 
97 700 



97 6S5 
97 691 
97 687 
97 683 
97 678 



97 674 
97 670 
97 666 
97 661 
97 657 

97 653 
97 649 
97 644 
97 640 
97 636 



97 632 
97 627 
97 623 
97 619 
97 614 

97 610 
97 606 
97 601 
97 597 
97 593 



97 588 
97 584 
97 580 
97 575 
97 57l 



9-97 567 
Log. Sin. 



d. 

4 
4 
4 
4 

4 
4 
4 

4 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 
4 
4 
4 
4 

4 
4 
4 
4 

4 

4 
4 
4 
4 
4 

f 19 
4 



60 

59 
58 
57 
16 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
j41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 



.30 

29 
28 
27 
26 



25 
24 
23 
22 
21 



18 
17 

16 



15 
14 
13 
12 
11 



10 

9 
8 
7 
6 

5 
4 
3 
2 

1 





P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



43 

4-3 



42 



43 

4-2 



41 



6 


4 


I 


4- 


7 


4. 


8 


4- 


8 


5 


5 


5- 


9 


6 


2 


6- 


10 


6 


9 


6- 


20 


13 


8 


13- 


80 


20 


7 


20- 


40 


27 


6 


27- 


50 


34 


6 


34- 



41 

1 
8 
4 
I 
8 
6 
5 
3 
1 



B 
7 
8 

9 
10 
20 
30 
40 
50 



39 

39 



38_ 
38 



38 

38 





37 


37 


3< 


6 


3-7 


3-7 


3- 


7 


4 


4 


4 


3 


4- 


8 


5 





4 


9 


4- 


9 


5 


6 


5 


5 


5- 


10 


6 


2 


6 


1 


6- 


20 


12 


5 


12 


3 


12- 


80 


18 


7 


18 


5 


18- 


40 


25 





24 


B 


24- 


50 


31 


2 


30 


8 


30- 





4 




4 


6 


0-4 


4 


7 





5 





4 


8 





6 





5 


d 





7 





6 


10 





7 





6 


20 


1 


5 


1 


3 


80 


2 


2 


2 





40 


3 





2 


Q 


50 


3 


7 


3 


3 



P. p. 



108° 



.681 



71' 



19° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



160 



Log. Sin. 





1 

2 

3 

_^ 

5 
6 
7 
8 
_9_ 

10 

11 

12 

13 

11 

15 

16 

17 

18 

19_ 

20 

21 
22 
23 
24 

i5 

16 

27 

28 

?! 

30 

31 

32 

33 

34 

35 
36 
57 
38 
39_ 

40 

■41 
42 
-43 
4£ 

45 

46 

47 

48 

41 

50 

51 

52 

53 

51 

55 

56 

57 

58 

59 



51 264 
51 301 
51 337 
51 374 
51 410 



51 447 
51 483 
51 520 
51 556 
51 503 



51 629 
51 665 
51 702 
51 738 
51 774 



51 810 
51 847 
51 883 
51 919 
51 955 

51 991 

52 027 
52 083 
52 099 
52 135 

52 170 
52 206 
52 242 
52 278 
52 314 

52 349 
52 385 
52 421 
52 456 
52 492 



52 527 
52 563 
52 598 
52 634 
52^89 

52 704 
52 740 
52 775 
52 810 
52_846 

52 881 
52 916 
52 951 

52 986 

53 021 



53 056 
53 091 
53 126 
53 161 
53 196 



53 231 
53 266 
53 301 
53 335 
53 370 



60 9-53 4riri 
jLog. Cos. 

109^ 



d. 



Log. Tan. 



37 

36 
36 
36 

36 
36 
36 
36 
36 

36 
33 
36 
36 
36 

36 
36 
36 
36 
36 

36 
38 
38 
36 
36 

35 
36 
36 
35 
36 

35 
35 
36 |2 
35 \l 

35 L- 

35 ^ 

"2 In 
ctF. \y ■ 
■^2 Iq 

6^ Ig. 



53 697 
53 738 
53 779 
53 820 
53 861 



53 902 
53 943 

53 983 

54 024 
54 065 



54 106 
54 147 
54 187 
54 228 
54 269 



54 309 
54 350 
54 390 
54 431 
54 471 



54 512 
54 552 
54 593 
54 633 
54 673 



54 714 
54 754 
54 794 
54 834 
54 874 



54 915 
54 955 

54 995 

55 035 
55 075 



35 

35 
35 
35 
35 
35 
35 
35 
35 
35 

35 

35 
35 
35 
35 

34 
35 
35 
34 
35 

34 



55 115 
55 155 
55 195 
55 235 
55 275 

55 315 
55 355 
55 394 
55 434 
55174 

55 514 
55 553 
55 593 
55 633 
55 672 



55 712 
55 751 
55 791 
55 831 
55 870 



55 909 
55 949 

55 988 

56 028 
56 067 

56 106 



c.d. 



41 
41 
41 
41 

41 
41 
40 
41 
41 

40 
41 
40 
40 
41 

40 
40 
40 
40 
4(5 

40 
-10 
40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
39 
40 
40 
40 

40 
40 
39 
40 
39 

40 
39 
40 
39 
39 

39 
39 
40 
39 
39 

39 

39 
39 
39 
39 

39 



Log. Cot. 



46 303 
46 262 
46 221 
46 180 
46 139 



46 098 
46 057 
46 016 
45 975 
45 934 



45 894 
45 853 
45 812 
45 772 
45 731 



45 690 
45 650 
45 609 
45 569 
45 528 



45 488 
45 447 
45 407 
45 367 9 
45 326 9 



Log. Cos, 



.45 286 
.45 246 
45 205 
45 165 
■ 45 125 

45 085 
45 045 
45 005 
44 965 
44 92 5 

44 884 
44 845 
44 805 
44 76519 

44 725 9 

44 685 9 
44 645 9 
44 605 9 
44 565 9 
44 526 9 



44 486 

44 446 
44 406 
44 367 
44 327 



44 288 
44 248 
44 208 
44 169 
44 129 



44 090 
44 051 
44 Oil 
43 97289 
4 3 93P 9 

43 893 9 



97 567 
97 562 
97 558 
97 554 
97 549 



97 545 
97 541 
97 536 
97 532 
97 527 



97 523 
97 51? 
97 514 
97 510 
97 505 



97 501 
97 497 
97 492 
97 488 
97 483 



97 479 
97 475 
97 470 
97 466 
97 461 



97 457 
97 452 
97 448 
97 443 
97 439 



97 434 
97 430 
97 425 
97 421 
97 416 



97 412 
97 407 
97 403 
97 398 
97 394 

97 389 
97 385 
97 380 
97 376 
97 371 



97 367 
97 362 
97 358 
97 353 
97 349 



97 344 
97 340 
97 335 
97 330 
97 326 



97 321. 
97 317 
97 312 
97 308 
97 303 



97 298 



d. jLog. Cot. c.d. Log. Tan.lLog. Sin. 

682 



d. 



d. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 

20 
19 
18 
17 
16 

15 

14 

13 

12 

11_ 

10 

9 

8 

7 

6 



5 
4 
3 
2 
_1^ 





P. P. 





41 


40 


40 


6 


4.1 


40 


40 


7 


4 


8 


4-7 


4 


6 


8 


5 


4 


5.4 


5 


3 


9 


6 


1 


6.1 


6 





10 


6 


8 


6:7 


6 


6 


20 


13 


6 


13.5 


13 


3 


30 


20 


5 


20.2 


20 





40 


27 


3 


27.0 


26 


6 


50 


34 


1 


33.7 


33 


3 



6 
7 
8 

9 
10 
20 
30 
40 
50 



39 

3 

4 



39 

3.9 



6 
7 
8 

9 
10 
20 
30 
40 
50 



37 

3.7 



36_ 

3.6 



36 

3.6 





35 


35 


34 


6 


3-5 


3.5 


3.4 


7 


4 


1 


4 


1 


4 





8 


4 


7 


4 


6 


4 


6 


9 


5 


3 


5 


2 


5 


2 


10 


5 


9 


5 


8 


5 


7 


20 


11 


8 


11 


6 11 


5 


30 


17 


7 


17 


5 17 


2 


40 


23 


6 


23 


3 23 





50 


29 


6 


29 


1 


28 


7 



6 

7 
8 
9 
10 
20 
30 
40 
50 






5 





4 


0. 





6 





5 


0. 





6 





6 


0. 





7 





7 


0. 





8 





7 


0. 


1 


6 


1 


5 


1. 


2 


5 


2 


2 


2. 


3 


3 


3 


n 


2. 


4 


1 


3 


7 


3- 



4 

4 
4 
5 
6 
6 
3 

6 
3 



P.P. 



70*" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



159° 



Log. Sin. 



9 



53 405 
53 440 
53 474 
53 509 
53 544 

53 578 
53 613 
53 647 
53 682 
53 716 



53 750 
53 785 

53 8ia 

53 854 
53 888 

53 922 
53 956 

53 990 

54 025 
54 059 



54 093 
54 127 
54 161 
54 195 
54 229 



54 263 
54 297 
54 331 
54 365 
54 398 



54 432 
54 466 
54 500 
54 534 
54 567 



54 601 
54 634 
54 668 
54 702 
54 735 



54 769 
54 802 
54 836 
54 869 
54 902 



54 936 

54 969 

55 002 
55 036 
55 069 



55 102 
55 135 
55 168 
55 202J 
55 235 



55 268 
55 301 
55 334 
55 367 
55 400 



9 55 433 
Log. Cos. 



d. 



35 
34 
34 
35 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
3i 
34 

34 
34 
34 
34 
34 

33 
34 
34 
34 
33 
34 
34 
33 
34 
33 

33 
33 
34 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

83 
33 
33 
33 
33 

33 



Log. Tan, 



56 106 
56 146 
56 185 
56 224 
56 263 



56 303 
56 342 
56 381 
56 420 
56 459 



56 498 
56 53? 
56 576 
56 615 
56 654 



56 693 
56 732 
56 771 
56 810 
56 848 

56 887 
56 926 

56 965 

57 003 
57 042 



57 081 
57 119 
57 158 
57 196 
57 235 



57 274 
57 312 
57 350 
57 389 
57 427 



c.d. 



57 466 
57 504 
57 542 
57 581 
57 619 

57 657 
57 696 
57 734 
57 772 
57 810 



57 848 
57 886 
57 925 

57 963 

58 001 



58 039 
58 077 
58 115 
58 153 
58 190 



58 228 
58 266 
58 304 
58 342 
58 380 

58 417 



Log. Cot. c. d 



39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
38 

39 
39 
39 
39 
38 

39 
38 
39 
38 
38 

39 
38 
38 
38 
38 

39 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
37 

38 
38 
38 
37 
38 

37 



Log. Cot. 



43 893 
43 854 
43 815 
43 775 



Log. Cos. 



43 736 9 



43 697 
43 658 
43 619 
43 580 
43 540 



43 501 
43 462 
43 423 
43 384 
43 346 



43 307 
43 268 
43 229 
43 ISC 
43 151 



43 112 
43 074 
43 035 
42 996 
42 958 



42 919 
42 880 
42 842 
42 803 
42 765 



42 726 
42 687 
42 649 
42 611 
42 572 



42 53_ 
42 495 
42 457 
42 419 
42 380 

42 342 
42 304 
42 266 
42 227 
42 189 



42 151 
42 113 
42 075 
42 037 
41 999 



41 961 
41 923 
41 885 
41 847 
41 809 



0-41 582 
Log. Tan. 



41 771 
41 733 
41 695 
41 658 9 
41 620 9 



97 298 
97 294 
97 289 
97 285 
97 280 



97 275 
97 271 
97 266 
97 261 
97 257 



97 202 
97 248 
97 243 
97 238 
97 234 



97 229 
97 224 
97 220 
97 215 
97 210 

97 206 
97 201 
97 196 
97 19l 
97 187 



97 182 
97 177 
97 173 
97 168 
97 163 



97 159 
97 154 
97 149 
97 144 
97 140 



97 135 
97 130 
97 125 
97 121 
97 116 



97 111 
97 106 
97 102 
97 097 
97 092 



97 087 
97 082 
97 078 
97 073 
97 068 



97 063 
97 058 
97 054 
97 049 
97 044 



97 0S9 
97 034 
97 029 
97 025 
97 020 



9-97 015 
Log. Sin. 



d. 



60 

59 
58 
57 
56_ 
55 
54 
53 
52 
51 

50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 

20 

19 
18 

17 
16 



15 
14 
13 
12 
11 

10 

9 
8 

7 
6 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



39 

3.9 

4 

5 

5 

6 
13 
19 
26 
32 



39 

39 

4 

5 

5 

6 
13 
19 
26 
32 





38 


38 


37 


6 


38 


3-8 


37 


7 


4 


5 


4 


4 


4 


4 


8 


5 


1 


5 





5 





9 


5 


8 


5 


7 


5 


8 


10 


6 


4 


6 


3 


6 


2 


20 


12 


8 


12 


6 


12 


5 


30 


19 


2 


19 





18 


7 


40 


25 


6 


25 


3 


25 





50 


32 


1 


31 


6 


31 


2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



35 

3-5 



34_ 

3-4 



34 

3-4 





33 


33 


6 


3-3 


3.3 


7 


3 


9 


3 


8 


8 


4 


4 


4 


4 


9 


5 





4 


9 


10 


5 


6 


5 


5 


20 


11 


1 


11 





30 


16 


7 


16 


5 


40 


22 


3 


22 





50 


27 


9 


27 


5 



6 

7 

8 

9 

10 

20 

30 

40 

50 






5 


0. 





6 


0. 





6 


0. 





7 


0. 





8 


0. 


1 


6 


1- 


2 


5 


2- 


3 


3 


3. 


4 


1 


3- 



.3 

• 5 

6 

7 

7 

•5 

■ 2 

Q 
•7 



P. P. 



683 



^9" 



31' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



158* 





1 

2 

3 

_4 

5 
6 
7 
8 
^ 

10 

11 
12 
13 
U 

15 
16 
17 
18 
19_ 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
5i 



Log. Sin. I d. (Log. Tan, 



9. 55 433^ 
9-55 438 
9. 55 498' 
9-55 531i 
9-55 564' 



33 

32 
33 
33 

32 
33 



9-58 417 
9.58 455 
9. 58 493 
19-58 531 
58 5-=!^ 



9. 55 597 
9-55 630, o5 
9.55 662 ''"' 
9-55 6951 
9-55 728; 



9-55 760 
9-55 793 
9-55 826 
9-55 858 
9-55 891. 



9-55 92o 
9-55 955 
9-55 988 
9-56 02(3 
9-56 ns<j 



9-56 08a 
9-56 118 
9-56 150 
9-56 182 
9-5*? 214 



9-55 247 
9-56 279 
9-56 311 
9-56 343 
9-56 3 75| 

9-56 401 
9-56 439 
9-56 471 
9-56 503 
9-56 535 



56 587 
56 599 
56 63l 
56 663 
56 69; "^ 

56 727 
58 758 
56 790 
9-56 822 
9 - 56 854 



56 885 
56 917 
56 949 

56 980 

57 012 



9-57 043 
9-57 075 
9-57 106 
9-57 138 
9-57 169 



9-57 201 
9-57 232 
9. 57 263 
9-57 295 
9-57 326 



9-57 357 



|Log.' 



Cos. 



33 

32 

32 
32 
33 
32 

32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
3l 
32 

32 
31 
32 
3l 
32 

3 1 
3l 
32 
31 
31 

3l 
31 
31 
31 
3l 

3l 
31 
31 
31 
31 

31 
T 



9-53 806 
9-58 644 
9. 58 68l 
958 719 
9- 58 756 



9-58 794 

958 831 
9-58 869 
9-58 906 
9 - 58 944 

9 - 58 981 
9-59 019 
9-59 056 
59 093 

959 1^"' 



59 168 
59 205 
59 242 
59 280 
59 317 



9 
9 
9 
9 
9 

9-59 354 
9-59 391 
59 428 
9-59 465 
9-59 51^^ 



9-59 540 
9-59 577 
9 • 59 814 
9.59 851 
9.59 688 



9 . 59 72-1 
59 781 
9.59 798 
9. 59 835 
9. 59 872 

9-59 909 
9-59 946 
9- 59 982 
9-60 019 
9-60 056 



9-60 093 
9-80 129, 
9-60 166 
9-60 203 
9-60 239 



9-60 276 
9. 80 312 
9-60 349 
9-60 386 
9-60 422 



9-60 459 
9-60 495 
9-60 531 
9-60 568 
9-60 60l 



9-60 641 



Log. Cot, 



c, d. Log. Cot 



33 
37 
38 
37 

37 
38 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 

37 

37 
37 
37 
37 
37 

36 
37 
37 
37 
38 

37 

36 
37 
36 

37 
36 
37 

36 
36 

36 
36 
37 
36 
36 

36 
36 
36 
36 
36 

36 



041 582 
0-41 544 
041 507 
0-41 469 
041 431 



0-41 394 
0-41 356 
0-41 318 
0-41 281 

-41 24G 



Log. Cos. 



9- 97 015 
1-97 010 
9- 97 005 
3-97 oog 
3. 96 995 



0-41 20C 
0-41 168 
0-41 131 
0-41 093 
0-41 056 



96 991 
96 986 
96 981 
96 976 

96 971 



c.d. 



0-41 018 
0.40 981 
0.40 944 
0.40 906 
0-40 R^q 

0-40 832 
0-40 794 
0.40 757 
0-40 720 
0-40 683 



96 966 
98 961 
96 956 
98 952 
96 947 



3- 96 942 
996 937 
9-96 932 
3-96 927 
').m 99,9, 



98 917 
96 912 
96 907 
96 902 
96 897 



0-40 646 
0-40 608 
0-40 57l 
0-40 534 
0-40 497 I 

9 
1 

9. 
!■ 

9' 
9 
9 
9 



0-40 460 
0-40 423 
0-40 386 
040 349 
0-40 313 



0-40 275 
0-40 238 
0-40 201 
0-40 164 
0-40 19.P 



96 892 
96 887 
96 882 
96 877 
96 873 



96 868 
96 863 
96 858 
96 853 
96 848 



0-40 
040 
040 
0-39 
039 



091 
054 
017 
980 
944 



0-39 
039 
039 
0-39 
039 



907 
870 
833 
797 
760 



0-39 724 
0-39 687 
0-39 65C 
0-39 81£ 
0-39 577 

0-39 541 
0-39 50^ 
039 468 
039 432 
0-39 395 

0.39 359 



'Log, Tan, 



96 843 
98 838 
96 833 
98 828 
96 823 



96 818 
96 813 
96 808 
. 98 802 
96 797 



96 792 
96 787 
96 782 
96 777 
96 772 



96 76Z 
96 762 
96 757 
96 752 
86 747 



96 742 
96 737 
96 732 
96 727 
96 72l 

9.96 716 
Log. Sin. 



d. 



5 
5 
5 

4 
5 
5 
5 
4 

5 
5 
5 

4 
5 

5 
5 
5 
5 

4 

5 
5 
5 
5 
5 

5 
5 
5 

5 
4 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 

17 



60 

59 
58 
57 
J8 

55 
54 
53 
52 
-51 
50 
49 
48 
47 
46 



45 
44 
43 
42 

JlL 

40 

39 
38 
37 
3A 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 



15 
14 
13 
12 
11 



25 
24 
23 
22 
21 

20 

19 
18 
17 
16 



10 

9 
8 
7 
6 



P. P. 





38 


37 


3 


6 


38 


3.7 


3. 


7 


4 


4 


4 


4 


4. 


8 


5 





5 





4. 


9 


5 


7 


5 


e 


5. 


10 


6 


3 


6 


2 


6- 


20 


12 


6 


12 


5 


12- 


3C 


19 





18 


7 


18- 


40 


25 


3 


25 


C 


24. 


50 


31 


6 


31 


2 


30- 





36 


36 


6 


3-6 


3 6 


7 


4 


2 


4 


2 


8 


4 


8 


4 


8 


9 


5 


5 


5 


4 


10 


6 


1 


8 





20 


12 


1 


12 





30 


18 


2 


18 





40 


34 


3 


24 





50 


30 


4 


SO 








33 


33 


3 


6 


3-3 


3-2 


3- 


8 


3 


8 


3 


8 


3'- 


7 


4 


4 


4 


3 


4 


9 


4 


9 


4 


9 


4. 


10 


5 


5 


5 


4 


5 


20 


11 





10 


8 


10 


30 


16 


5 


16 


2 


16 


40 


22 





21 


6 


21 


50 


27 


5 


27 


1 


26 



6 

7 

8 

9 

10 

20 

30 

40 

50 



31 



3-1 


3- 


3 


7 


3- 


4 


2 


4.. 


4 


7 


4. 


5 


2 


5. 


10 


5 


10. 


15 


7 


15. 


21 





20. 


26 


2 


25- 



31 

1 
6 
1 
6 
1 
3 

5 
6 
8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



5 

0.5, 



5 

0-5 



0.4 



P. P. 



111= 



684 



TABLE Vn.--LOGARITHMIC SINES, COSINES, TANGENTS. 
23° . AND COTANGENTS. I570 



Log. Sin. 



57 357 
57 389 
57 42G 
57 451 
57 482 



31 
32 
33 
34 



57 513 
57 544 
57 576 
57 607 
57 638 



57 669 
57 700 
57 731 
57 762 
57 792 





1 
2 
3 

_1 

5 
6 
7 
8 
J^ 

10 

11 
12 
13 
li 
15 
16 
17 
18 

11 

2(] 

21 

22 

23 

24 

25 

26 

27 

28 

29_ 

30 9. 58 284 
58 314 
58 345 
58 375 
58 406 



57 823 
57 854 
57 885 
57 916 
57 947 



.57 977 
58 00£ 
58 039 
58 070 
58 100 



9.58 131 
9. 58 162 
9-58 192 
9-58 223 
9.58 253 



35 
36 
37 
38 
39_ 

40 

41 |9 

42 9 

43 9 

44 9 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59_ 
60 



58 436 
58 466 
58 497 
58 527 
58 557 

58' 587 
58 618 
58 648 
58 678 
58 708 



9. 58 738 
9-58 769 
9-58 799 
9. 58 829 
9. 58 859 



9. 58 889 

9. 58 919 
9. 58 949 
9-58 97& 
9- 59 009 

9.59 038 
9.59 068 
9.59 098 

59 128 
9-59 158 

9-59 188 

Log. Cos. 



d. 



31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
30 

31 
31 
31 
30 
31 

30 
31 
30 
31 
30 

30 
31 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

25 
30 
30 
29 
30 

30 



Log. Tan. 



9. 60 
9. 60 
9-60 
9-60 
9-60 

9. 60 
9-60 
9. 60 
9-60 
9-60 



641 
677 
713 
750 
786 



822 
859 
895 
931 
967 



9. 61 
9. 61 
9-61 
9. 61 
9. 61 



003 
039 
076 
112 
148 



9. 61 
9 61 
9. 61 
9. 61 
q.61 

9.61 
9-61 
9-61 
9-61 
9.61 



184 

220 
256 
292 
328 

364 
400 
436 
472 
507 



c.d, 



9. 61 
7. 61 
9. 61 
9-61 
9. 61 



543 
579 
615 
651 
686 



9. 61 
9.61 
9-61 
9-61 
9.61 



722 
758 
794 
829 
865 



9. 61 
9. 61 
9-61 
9. 62 
9.R2 



901 
936 
972 
007 
043 



9. 62 
9.62 
9.62 
9-62 
9.62 



078 
114 
149 
185 
220 



9. 62 
9-62 
9.62 
9-62 
9.62 



9.62 
9-62 
9-62 
9. 62 
9.62 



9-62 
9.62 
9. 62 
9.62 
9. 62 

9-62 



256 
291 
327 
362 
JS97 

433 
468 
503 
539 
574 

609 
644 
679 
715 
750 

785 



d. Log. Cot. 



36 
36 
36 
36 

P 
36 

36 

36 

36 

36 
86 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
35 

36 
36 
35 
36 
35 

36 
35 
36 
35 
35 

36 
35 
35 
35 
35 

85 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 



Log. Cot. 



039 359 
0.39 322 
039 286 
0-39 250 
0.39 213 



039 177 
039 141 
039 105 
039 06P 
0-39 032 



038 996 
038 960 
0-38 924 
038 888 
038 852 



Log. Cos. 



96 716 
•96 711 
•96 706 

• 96 701 
96J696 

■ 96 691 

• 96 686 

• 96 681 

• 96 675 
96 670 



0-38 816 
038 780 
038 744 
038 708 
0.38 67? 



0-38 636 
0^38 6CC 
038 564 
038 528 
038 492 



■ 96 665 
• 96 660 
•96 655 
•96 650 
■96 644 

96 639 
96 634 
96 629 
96 624 
P6 619 



0.38 456 
038 420 
038 385 
38 349 
038 313 



96 613 
96 608 
9. 96 603 
996 598 
9 96 59'3 



038 277 
038 242 
038 206 
0.38 170 
038 135 



0. 38 099 
038 063 
038 028 
037 992 
0.37 957 



37 921 
37 886 
0.37 850 
0.37 815 
037 779 



9 96 587 
9. 96 582 
9-96 577 
9.96 572 
9-96 567 



996 561 
9-96 556 
996 551 
9 96 546 
9 96 540 



9. 96 535 
9. 96 530 
996 525 
996 519 
996 514 



0.37 744 
0.37 708 
0.37 673 
0.37 637 
037 602 



0.37 567 
0.37 53l 
0.37 496 
0-37 461 
0-37 426 



C.d. 



0-37 390 
0-37 355 
0-37 320 
0-37 285 
0.37 250 



9-96 509 
9^S6 503 
9. 96 498 
9-96 493 
9. 96 488 



9-96 482 
9. 96 477 
9-96 472 
9-96 466 
9. 96 461 



9-96 456 
9. 96 450 
9-96 445 
9-96 440 
9. 96 434 



0-37 215 



Log. Tan, 



9. 96 429 
9-96 424 
9. 96 418 
9. 96 413 
9-96 408 

9-96 402 

Log. Sin. 



d. 

5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
T 



60 

59 
58 
57 

55 
54 
53 
52 
-51 
50 
49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 

30 

29 
28 
27 
16 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 



15 
14 
13 
12 
11 

10 

9 
8 

7 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



36 



3-6 


3 


4-2 


4- 


48 


4. 


5-5 


5- 


6 1 


6- 


12-1 


12- 


18-2 


18- 


24-3 


24- 


30-4 


30- 



36 

6 

2 
8 
4 










35 


35 


6 


3-5 


3-5 


7 


4 


-1 


4-1 


8 


4 


7 


4-5 


9 


5 


3 


5-2 


10 


5 


9 


5-8 


20 11 


8 


11-fJ 


3017 


7 


17.5 


40 23 


6 


23.3 


50 29. 


eJ 


29 i 





31 


3- 


6 


SI 


3 


7 


3 


.7 


3. 


8 


4 


2 


4. 


9 


4 


7 


4. 


10 


5 


2 


5. 


20 


10 


5 


10. 


30 


15 


7 


15. 


40 


21 





20. 


50 


26. 


2 


25. 



• I 

.6 

• 1 
■ S 
-I 
.3 
-5 
.6 
•S 





30 


30 


3< 


6 


3.0 


3-0 


2. 


7 


3 


.5 


3 


.5 


3. 


8 


4 





4 


-0 


3 


9 


4 


6 


4 


5 


4. 


10 


5 


1 


5 





4. 


20 


10 


1 


10 





9. 


30 


15 


2 


15 





14- 


40 


20 


3 


20 





19. 


50 


25. 


4 


25. 





24. 



8 

7 

8 

9 

10 

20 

30 

40 

50 



5_ 5 

0-510-5 
-6 0-6 
-7i0.6 
■ 8,0.7 
-9|0-8 
-ill. 6 
-7 2-5 
-63-3 
6)4.1 



P. P. 



iia' 



685 



er 



33* 



TABLE Vll.— LOGAHITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. . i^g"! 



Log. Sin, 



59 ll>8 
59 217 
59 247 
59 277 
59 306 



59 336 
59 366 
59 395 
59 425 
59 454 



59 484 
59 513 
59 543 
59 572 
59 602 



59 631 

59 8S1 
59 690 
59 719 
59 749 

59 778 
59 807 
59 837 
59 866 
59 895 



59 324 
59 953 

59 982 

60 012 
60 041 



60 070 
60 009 
60 128 
60 157 
60 186 



60 215 
60 244 
60 273 
60 301 
60 330 



60 359 
60 388 
60 417 
60 445 
60 474 



60 503 
60 532 
60 560 
60 589 
60 618 



60 646 
GO 675 
60 703 
60 732 
60 760 



60 789 
60 817 
60 84R 
60 874 
60 903 



9-60 93l 



Log. Cos. 



d. 



29 
30 
29 
29 

30 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
28 
29 

29 
28 
29 
28 
29 

28 
29 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 



Log. Tan. 



62 785 
62 820 
62 855 
62 890 
62 925 



62 960 

62 995 

63 030 
63 065 
63 100 



63 135 
63 170 
63 205 
63 240 
63 275 



63 310 
63 344 
63 379 
63 414 
63 449 



63 484 
63 518 
63 553 
63 588 
63 622 



63 657 
63 692 
63 726 
63 761 
63 795 



63 830 
63 864 
63 899 
63 933 
63 968 



64 002 
64 037 
64 071 
64 106 
64 145 



64 174 
64 209 
64 243 
64 277 
64 312 



64 346 
64 380 
64 415 
64 449 
64 483 



64 517 
64 551 
64 585 
64 620 
64 654 



64 688 
64 722 
64 756 
64 790 
64 824 



64 858 



d. Log. Cot. c. d 



c.d. 



35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
34 
35 
35 
34 

35 
34 
34 
35 
34 

34 
35 
34 
34 
34 

34 

34 
34 
34 
34 

3? 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 



Log. Cot. 



37 215 
37 179 
37 .144 
37 109 
37 074 



37 039 
37 004 
36 969 
36 934 
36 899 



36 864 
36 829 
36 794 
36 760 
36 725 



36 69C 
36 655 
36 620 
36 585 
36 551 



36 516 
36 481 
36 447 
36 412 
36 377 



36 343 
36 308 
36 273 
36 239 
36 204 



36 170 
36 135 
36 101 
36 066 
36 032 9 



Log. Cos. 



35 997 
35 963 
35 82£ 
35 894 
35 859 



35 825 
35 791 
35 756 
35 722 
35 688 



35 653 
35 619 
35 585 
35 551 
35 517 



35 482 
35 44E 
35 414 
35 380 
35 346 



35 312 
35 278 
35 244 
35 209 
35 175 



35 141 



Log. Tan. 



96 402 
96 397 
96 392 
96 386 
96 381 



96 375 
96 370 
96 365 
96 359 
96 354 



96 349 
96 343 
96 338 
96 332 
96 327 



96 321 
96 316 
96 311 
96 305 
96 300 



96 294 
96 289 
96 283 
96 278 
96 272 



96 267 
96 261 
96 256 
96 251 
96 245 



96 240 
96 234 
96 229 
96 223 
96 218 



96 212 
96 206 
96 201 
96 195 
96 190 



96 184 
96 179 
96 173 
96 168 
96 162 



96 157 
96 151 
96 146 
96 140 
96 134 



96 129 
96 123 
96 118 
96 112 
96 106 



96 101 
96 095 
96 090 
96 084 
96 078 



9. 96 073 



log. Sin. 



d. 



d. 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
_46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
M 
35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 

17 

ii 

15 
14 
13 
12 
11 



10 

9 
8 

7 
_6 

5 
4 
3 
2 





P.P. 





35 




35 


6 


3.5 


35 


7 


4 


1 


4 


1 


8 


4 


7 


4 


6 


9 


5 


3 


5 


2 


10 


5 


9 


5 


8 


20 


11 


8 


11 


6 


30 


17 


7 


17 


5 


40 


23 


6 


23 


3 


50 


29 


6 


29 


1 





34 


34 


6 


3-4 


3-4 


7 


4 





3 


9 


8 


4 


6 


4 


5 


9 


5 


2 


5 


1 


10 


5 


7 


5 


6 


20 


11 


5 


11 


3 


30 


17 


2 


17 





40 


23 





22 


6 


50 


28 


7 


28 


3 



6 

7 

8 

9 

10 

20 

8C 

40 

50 



30 

30 

3-5 

4.0 

4.5 

50 

10-0 

150 

20-0 

25-0 





39 


39 


6 


2-9 


2.91 


7 


3 


4 


3 


4 


8 


3 


9 


3 


8 


9 


4 


4 


4 


3 


10 


4 


9 


4 


8 


20 


9 


8 


9 


6 


30 


14 


7 


14 


5 


40 


19 


6 


19 


3 


50 


24 


6 


24 


1 



38 

2-8 

3.3 

3-8 

4-3 

4.7 

9.5 

14.2 

19 Q 

23-7 



6 



60 
70 

8 

9 
10 1 
20 2 
30 3 
40 4 
50 5 



5_ 

0-5 



.5 
•6 
•6 

• 7 

•8 

• 6 
■ 5 

• 5 

• 1 



P.P. 



iA3° 



686 



66" 



TABLE VIl— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



155* 



Log. Sin. 


9-60 931 


9 


60 959 


9 


60 988 


9 


61 016 


9 


61 044 


9 


61073 


9 


61 101 


9 


61 129 


9 


61 157 


9 


61 186 


9- 


61 214 


9- 


61 242 


9- 


61 270 


9 


61 298 


9. 


61 326 


9- 


61354 


9. 


61 382 


9- 


61410 


9 


61 438 


9 


61466 


9 


61 494 


9 


61 522 


9. 


61 550 


9. 


61 578 


9. 
9. 


61 606 


61 634 


9. 


61 661 


9. 


61 689 


9. 


61 717 


9. 


61 745 


9. 


61 772 


9 


61 800 


9 


61 828 


9 


61 856 


9 


61 883 


9 


61 911 


9 


61 938 


9 


61 968 


9 


61 994 


9 


62 021 


9 


62 049 


9 


62 076 


9 


62 104 


9 


62 131 


9 


62 158 


9 


62 186 


9 


62 213 


9 


62 241 


9 


62 268 


9 


62 295 


9 


62 323 


9 


62 350 


9 


62 377 


9 


62 404 


9 
9 


62 432 


62 459 


9 


62 486 


9 


62 513 


9 


62 540 


9 
9 


62 567 


■ 62 595 


Log. Cos. 



d. 



28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
27 
28 
28 

28 

27 
28 
27 
28 

27 
28 
27 
28 
27 

27 
27 
27 
28 
27 

27 
27 
27 
21 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 



Log. Tan. 



858 
892 
926 
960 
994 

028 
062 
096 
129 
163 



197 
231 
265 
299 
332 

366 
400 
433 
467 
501 



535 
588 
602 
635 
669 



703 
736 
770 
803 
837 

870 
904 
937 
971 
004 



037 
071 
104 
137 
171 



204 
23? 
271 
304 
337 

370 
404 
437 
470 
503 



536 
5 '7.) 
603 
636 
M? 
702 
735 
768 
801 
834 



d. [Log. 



c.d. 

34 
34 
33 
34 

34 

34 
34 
33 
34 

34 
33 
34 
34 
33 

34 
33 
33 
34 
33 

34 
33 
31 
33 
33 

34 
33 
33 
35 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 

33 
33 

8P.7 ^^ 
Cot. c.d 



Log. Cot. 



35 141 
35 107 
35 073 
35 040 
35 006 



34 972 
34 938 
34 904 
34 870 
34 836 



34 802 
34 769 
34 735 
34 701 
34 667 



34 633 
34 600 
34 566 
34 532 
34 499 



34 485 
34 431 
34 398 
34 364 
34 331 



34 297 
34 263 
34 230 
34 196 
34 163 



34 129 
34 096 
34 062 
34 029 
33 996 



33 962 
33 929 
33 895 
33 862 
33 829 



33 
33 
33 
33 
33 

33 
33 
33 
33 
33 



795 
762 
729 
696 
662 



629 
596 
563 

528 
496 



33 463 
33 430 
33 

33 364 
33 331 



Log. Cos. 



397 9 
9 
9 

33 298|9 
33 265 9 



33 232 
33 198 
33_1A5 

33 132 9 



96 073 
96 067 
96 062 
96 056 
96 050 

96 045 
96 039 
96 033 
96 028 
96 022 



96 016 
96 011 
96 005 
95 999 
9 5 994 

95 988 
95 982 
95 977 
95 971 
95 965 



95 959 
95 954 
95 948 
95 942 
95 937 



95 931 
95 925 
95 919 
95 914 
95 908 



95 902 
95 896 
95 891 
95 885 
95 879 



95 873 
95 867 
95 862 
95 856 
95 850 



95 844 
95 838 
95 833 
95 827 
95 82l 



95 815 
95 809 
95 804 
95 798 

95 792 



95 786 
95 780 
95 774 
95 768 
95 763 

95 757 
95 751 
95 745 
95 739 
95 733 



95 727 



Log. Tan.jLog. Sin. 
087 



d. 



d. 



GO 

59 
58 
57 
56 



50 

49 
43 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 
29 
28 
27 
26 



25 
24 
23 
22 
21 

20 

19 
18 
17 
16 



15 
14 
13 

12 
11 

10 
9 
8 
7 
6 



P. P. 





34 


33 


3, 


6 


3 4 


3-3 


3. 


7 


3 


9 


3 


9 


3- 


8 


4 


5 


4 


4 


4. 


9 


5 


1 


5 





4. 


10 


5 


6 


5 


6 


5- 


20 


11 


3 


11 


1 


11. 


30 


17 





16 


7 


16. 


40 


22 


6 


22 


3 


22. 


50 


28 


• 3 


27 


9 


27 



6 
7 
8 
9 

10 
20 
30 
40 
50 



38 38 



6 

7 

8 

9 

10 

20 

30 

40 

50 



37 
2.1 



37 
2-7 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 5 






6;o. 





7,0. 





8'0. 





9!0 


1 


010. 


2 


Oil. 


3 





2. 


4 





3. 


5 





4. 



P.P. 



2 


8 


2-8 


3 


3 


3 


2 


3 


8 


3 


7 


4 


3 


4 


2 


4 


7 


4 


6 


9 


5 


9 


3 


14 


2 


14 





19 





18 


6 


23 


7 


23 


3 



05^ 



^S*" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



154*" 



Log. Sin. I d. 



595 
622 
649 
676 
703 



62 
62 
62 
62 
62 



730 
757 
784 
811 
838 




63 
63 
63 
63 
63 



132 
159 
186 
212 
239 



63 
63 
63 
63 
63 



266 
292 
319 
345 
372 



63 
63 
63 
63 
63 



398 

425 
451 
478 

504 



63 
63 
63 
63 
63 



530 
557 

583 
609 
R3R 



63 
63 
33 
63 
63 



662 
688 
715 
741 
767 



63 
63 
63 
63 
63 



793 
819 
846 
872 
898 



63 
63 
63 
64 
64 



92'! 
950 

976 
002 
028 



64 
64 
64 
64 
64 



9-64 



054 
080 
106 
132 
158 

184 



Log. Cos. d. 



27 
27 
27 
27 

27 
27 
27 
27 
27 

26 
27 
27 
27 
26 

27 
26 
27 
26 
27 

26 
27 
26 
26 
26 

27 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

26 
26 
26 
26 

26 
26 
26 
26 
26 

25 



Log. Tan. 



66 
66 
66 
66 
66 



867 
900 
933 
966 
999 



67 
67 
67 
67 
67 



032 
065 
097 
130 
163 



67 
67 
67 
67 
67 



196 
229 
262 
294 
327 



67 
67 
67 
67 
67 



360 
393 

425 
458 
491 



67 
67 
67 
67 
67 



523 
556 
589 
621 
654 



67 
67 
67 
67 
67 



687 

719 
752 
784 
817 



67 
67 
67 
67 
67 



849 
882 
914 
947 
979 



68 
68 
68 
68 
68 



012 
044 
077 
109 
141 



174 
206 
238 
271 
303 



335 
368 

400 
432 
464 



68 
68 
68 
68 
68 



68 



497 
529 
561 
593 
62_5 

657 
690 
722 
754 
786 

818 



C, d. 



Log. Cot. c. d 



32 
33 
33 
33 

33 
33 
32 
33 
33 

33 
32 
33 
32 
33 

32 
33 
32 
33 
32 

32 
33 
32 
32 
33 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 

32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 



Log. Cot. 



33 132 
33 100 
33 067 
33 034 
83001 

32 968 
32 935 
32 902 
32 869 
32 83b 



Log. Cos. 



32 803 9 
32 771 9 
32 738 9 
32 705 9 
32 672 9 



32 640 9 
32 607 9 
32 574 9 
32 541 9 
32 509 9 

32 476 9 
32 44c; 9 
32 411 9 
32 37£ 9 
32 34519 



32 31389 
32 280 9 
32 248 9 
32 215 9 
32 18319 



32 15019 
32 118 9 
32 085 9 
32 053 9 
32 020 9 



31 98819 
31 955 9 
31 923 9 
31 891 9 
31 8frj9 



82e 

79! 
761 
729 
696 



31 
31 
31 
31 
31 



664 
632 
60C 
567 
535 



31 503 
31 471 
31 439 
31 406 
31 374 



31 342 
31 310 
31 278 
31 246 
31 214 

31 I82I9 



95 727 
95 72l 
95 716 
bo 710 

Sa 704 

95 698 
95 692 
95 686 
95 680 

95J74 

95 661 
95 662 
95 656 
95 650 
95 644 



95 638 
95 632 
95 627 
95 621 
95 615 

95 609 
95 603 
95 597 
95 591 
95 585 



95 579 
95 573 
95 567 
95 561 
95 555 



95 549 
95 543 
95 537 
95 530 
95 524 



95 518 
95 512 
95 506 
95 500 
95 494 

95 488 
95 482 
95 476 
95 470 
95 464 



95 458 

95 452 
95 445 
95 439 
95 433 



95 427 
95 421 
95 415 
95 409 
95 403 



95 397 
95 390 
95 384 
95 378 
9 5 372 

95 366 



115* 



Log, Tan.jLog. Sin- 
688 



d. 



d. 



60 

59 
58 
57 

55 
54 
53 
52 
_§i 
50 
49 
48 
47 
46_ 

45 
44 
43 
42 
41 



40 

39 
38 
37 
36 

35 
34 
33 
32 
11 
30 
29 
28 
27 

25 
24 
23 

22 
2] 



20 
19 
18 
17 
16 
15 
14 
13 
12 

n 

10 

9 
8 
7 
6 



5 
4 
3 

2 
1 



P. P. 





33 


33 


33 


e 


3-3 


3-2 


3-2 


7 


3 


8 


3 


8 


3 


7 


8 


4 


4 


4 


3 


4 


2 


8 


4 


9 


4 


9 


4 


8 


10 


5 


5 


5 


4 


5 


3 


2C 


11 





10 


L 


10 


6 


3C 


1-6 


5 


16 


2 


16 





40 


22 





21 


fc 


21 


3 


50 


27 


5 


27 


1 


26 


6 





27 


6 


2.7 


7 


3 


1 


8 


3 


6 


9 


4 





10 


A 


5 


20 


9 





30 


13 


5 


40 


18 





50 


22 


5 



6 
7 
8 
9 

IC 
2C 
3C 
4C 
50 















36 36 35 


2-6 


2.6 


2.5 


3 


1 


3 


C 


3 





3 


5 


3 


A 


3 


4 


4 





3 


9 


3 


8 


4 


4 


4 





4 


2 


8 


8 


8 


C 


8 


5 


13 


2 


13 


c 


12 


7 


17 


6 


17 


3 


17 





22 


I 


21 


6 


21 


?. 



7 

8 

9 

IC 

20 
3C 
40 
50 



6_ 

0-6 





1 

1 

2 

3 

4 

5 



6 5 

0.6|0.i 

70- 



P. P. 



64* 



26° 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS 
AND COTANGENTS. 



153* 



Log. Sin. 



64 184 
64 210 
64 236 
64 262 
64 237 



64 313 
64 339 
64 365 
64 391 

64 4.16 



64 442 
64 468 
64 493 
64 519 
64 545 



64 570 
64 596 
64 622 
64 647 
64 673 



64 698 
64 724 
64 749 
64 775 
64 800 



64 828 
64 851 
64 876 
64 902 
64 927 



64 952 

64 978 

65 003 
65 028 
65 054 



65 079 
65 104 
65 129 
65 155 
65 180 



65 205 
65 23g 
65 255 
65 280 
65 305 



65 331 
65 356 
65 381 
65 406 
65 431 



65 456 
65 481 
65 503 
65 530 
65 555 



65 580 
65 605 
65 630 
65 655 
65 680 



9-65 704 



Log. Cos. 



d. 

26 
26 
26 
25 

26 
26 
25 
26 
25 

26 

25 
25 
28 
25 

25 
25 
26 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
24 
25 

25 
25 
24 
25 
25 

24 



Log. Tan. 



68 818 
68 850 
68 882 
68 914 
68 946 



68 978 

69 010 
69 042 
69 074 
69 106 



69 138 
69 170 
69 202 
69 234 
69 265 

69 297 
69 329 
69 361 
69 393 
69 425 

69 456 
69 488 
69 520 
69 552 
69 583 



69 615 
69 647 
69 678 
69 710 
69 742 



89 773 
69 805 
69 837 
69 868 
69 900 



69 931 
69 963 

69 994 

70 026 
70 053 

70 089 
70 121 
70 152 
70 183 
70 215 



70 246 
70 278 
70 309 
70 341 
70 372 



70 403 
70 435 
70 466 
70 497 
70 529 



70 560 
70 591 
70 623 
70 654 
70 685 



70 716 



d. |Log. Cot. 



C.d. 

32 
32 
32 
32 

32 
32 
32 
31 
32 

32 
32 
32 
32 
3l 

32 
32 
3i 
32 
32 

3l 
32 
3l 
32 
31 

32 
31 
31 

32 

3i 

31 

32 



Log. Cot. 



0.31 
0.31 
0-31 
0.31 
0.31 

0.31 
0.30 
0.30 
0.30 
0.30 



Log. Cos. 



182 
150 
117 
085 
058 9 

021 9 
989 9 
957 9 
926 9 
894 9 



95 366 
95 360 
95 353 
95 347 
95 341 



95 335 
95 329 
95 323 
95 316 
95 310 



0.30 
0.30 
0-30 
0.30 
0^30 

0.30 
0.30 
0-30 
0.30 
0.30 

0.30 
0.30 
0.30 
0.30 
0.30 



862 9 
8309 
798 9 
766 9 
734 9 



95 304 
95 298 
95 292 
95 285 
95 279 



0.30 
0-30 
0.30 
0.30 
0.30 



384 
353 
321 
289 
258 



0.30 

,T lO-30 

^^10.30 

0-30 

0.30 



31 
31 

3i 
31 
31 
32 
31 

31 
31 
31 
31 
31 

3l 
3l 
31 
31 
31 

31 

31 
3l 
31 
31 

31 

31 
3l 
31 
3l 

31 



226 
194 
163 
131 
100 



0.30 
0.30 
0.30 
0.29 
0.29 

0.29 
0.29 
0.29 
0.29 
0.29 



0.29 
0.29 
0.29 
0.29 
0.29 



068 
037 
005 
973 

942 

910 
879 
847 
816 
785 



753 
722 
690 
659 
628 



0.29 
0.29 
0.29 
0.29 
0.29 



596 
565 
533 
502 
471 



029 
0.29 
0-29 
029 
029 



439 
408 
377 
346 
314 



0.29 



283 

c. d. Log. Tan 



702 
670 
639 
607 

575 

543 

511 

480,. 

448i9 

416 



.95 273 

.95 267 

95 260 

95 254 

95 248 

•95 242 
95 235 
95 229 
95 223 
95 217 



■ 95 210 
95 204 
95 198 

.95 191 
95 185 



.95 179 

.95 173 

.95 166 

95 160 

95 154 



95 147 
95 141 
95 135 
95 128 
■95 129 

.95 116 
.95 109 
.95 103 
.95 097 
• 95 090 



95 084 
95 078 
95 071 
95 065 
95 058 



9-95 052 
9.95 046 
9. 95 039 
9. 95 033 
9.95 026 



9.95 020 
9-95 014 
9-95 007 
9. 95 001 
9-94 994 



9.94 988 
Log. Sin. 



6 

6 
6 
6 

6 
6 

6 

1 
6 

6 
6 
6 
6 
6 

6 

T 



60 

59 
58 
57 
56^ 

55 
54 
53 

52 

50 

49 
48 
47 
48 

45 
44 
43 
42 
41 

40 

39 
38 

37 
38 

35 
34 
33 
32 

11 

30 
29 
28 
27 
26 



25 
24 
23 
22 
21 

30 
19 
18 
17 
16 



15 
14 
13 
12 

10 

9 
8 

7 
6 
5 
4 
3 
2 
1 





P.P. 













33 3» 


6 


3.2 


3.2 


7 


3 


8 


3 


7 


8 


4 


3 


4 


2 


9 


4 


9 


4 


8 


10 


5 


4 


5 


3 


20 


10 


8 


10 


6 


30 


16 


2 


16 





40 


21 


6 


21 


3 


50 


[27 


1 


26 


6 





31 


31 


6 


3.1 


3.1 


7 


3 


7 


3 


6 


8 


4 


2 


4 


T 


9 


4 


7 


4 


6 


10 


5 


2 


5 


1 


20 


10 


5 


10 


3 


30 


15 


7 


15 


5 


40 


21 





20 


6 


50 


26 


2 


25 


d 





2Q 


3. 


5 


25 


6 


2.6 


2.5 


2.t 


7 


3 





3 





2 


9 


8 


3 


4 


3 


4 


3 


3 


9 


3 


9 


3 


8 


3 


^ 


10 


4 


3 


4 


2 


•4 


1 


20 


8 


6 


8 


5 


8 


3 


30 


13 





12 


7 


12 


5 


40 


17 


3 


17 





16 


6 


50 


21 


6 


21 


2 


20 


& 



6 
7 
8 
9 

10 
20 
30 
40 
50 



24 

2-4 

2 

3 

3 

4 

8 
12 
16 
20 



6 

0.6 



6 

08 



P.P. 



689 



«af* 



97" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



152« 



JLog. Sin. d. 





1 

2 9 

3 9 

5 9 

6 9 

7 9 

8 9 

-M- 

10 9 

11 9 

12 9 

13 9 

14 89 



15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 9 

26 

27 

28 

29 



30 

31 
32 

33 9 

34 9 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 9 



55 
56 
57 
58 

60 



65 704 
65 729 
65 754 
65 779 
65 803 



65 828 
65 853 
65 878 
65 902 
65 927 



65 951 

65 976 

66 001 
66 025 
66 050 

66 074 
66 099 
68 123 
66 148 
66 172 

66 197 
66 221 
66 246 
66 270 
66 294 

66 319 
66 343 
66 367 
66 392 
66 416 



66 440 
66 465 
66 489 
66 513 
66 537 

66 581 
66 586 
66 610 
66-634 
66 658 

66 632 
66 708 
66 730 
66 754 
66 778 



68 802 
68 826 
66 350 
66 874 
66_898 

68 922 
66 948 
66 970 

66 994 

67 018 



67 042 
67 066 
67 089 
67 113 
87 137 



9.87 IRl 



Log. Cos. 



25 
24 
25 
24 

25 
24 
25 
24 
24 

24 
25 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
23 
24 

24 
24 
23 
24 
23 

24 
T 



Loff. Tan. 



70 716 
70 748 
70 779 
70 810 
70 841 



70 872 
70 903 
70 935 
70 966 
70 997 



71028 
71 059 
71 090 
71 121 
71 152 



71 183 
71 214 
71 245 
71 276 
71 307 

71 338 
71 369 
71 400 
71431 
71 462 



71 49.3 
71 524 
71 555 
71 586 
71 617 

71 647 
71 678 
71 709 
71 740 
71 771 

71 801 
71 832 
71 883 
71 894 
71 925 

71 955 

71 986 

72 017 
72 047 
72 078 



72 109 
72 139 
72 170 
72 201 
72 231 



72 282 
72 292 
72 323 
72 354 
72 384 



72 415 
72 445 
72 476 
72 506 
72 537 

72 5fi7 
Log. Cot- 



c.d. 

3l 
31 
31 
31 

31 
31 
3l 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 

31 
31 
31 

31 
30 
31 
31 
31 

30 
31 
31 
30 
31 

30 
31 
31 
30 
31 

3S 
30 
31 
30 
31 

30 
30 
30 
31 
30 

30 
30 
30 
31 
30 

30 
30 
30 
30 
30 

30 



Log. CotlLog. Cos. 



0-29 283 
0.29 252 
0.29 221 
0.29 190 
0.29 158 



0.29 127 
0.29 096 
0.29 065 
0-29 034 
0.29 003 



28 972 
28 940 
28 909 
28 878 
28 847 



0-28 816 
0.23 785 
0.28 754 
0.28 723 
0.28 692 



0.28 661 
0.28 630 
0.28 599 
0.28 568 
0.28 537 



28 506 
28 476 
28 445 
28 414 
28 383 




0-28 044 
0.28 014 
0.27 983 
0.27 952 
0.27 92l 



0-27 891 
0.27 860 
0.27 830 
0.27 799 
0.27 768 



0.27 738 
0.27 707 
0.27 677 
0-27 646 
0.27 615 



0.27 585 
0.27 554 
. 27 524 
0.27 493 
0-27 463 
0-27 43^ 
Log. Tan 



94 988 
94 981 
94 975 
94 969 
94 962 



94 956 
94 949 
94 943 
94 936 
94 930 



94 923 
94 917 
94 910 
94 904 
94 897 



94 891 
94 884 
94 878 
94 871 
94 865 

94 858 
94 852 
94 845 
94 839 
94 832 



94 825 
94 81? 
94 812 
94 806 
94_799 

94 793 
94 786 
94 779 
94 773 
94 766 



94 760 
94 753 
94 746 
94 740 
94 733 

94 727 
94 720 
94 713 
94 707 
94 700 



94 693 
94 687 
94 680 
94 674 
94 667 



94 660 
94 654 
94 647 
94 640 
94 633 



94 627 
94 620 
94 61 3 
94 607 
94 60 

9 ■ 94 59^ 
Log. Sin. 



d. 



60 

59 
58 
57 

55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



45 
44 
43 
42 

JlL 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 

22 

£L 

30 

19 
18 
17 
16 



15 
14 
13 
12 
11 



10 

9 
8 

7 
6 



P. P. 





31 




31 


30 


6 


3.1 


31 3-0 


7 


3 


7 


3 


6 3 


5 


8 


4 


2 


4 


1 


4 





9 


4 


7 


4 


6 


4 


6 


10 


5 


2 


5 


1 


5 


1 


20 


10 


5 


10 


3 


10 


1 


30 


15 


7 


15 


5 


15 


2 


40 


21 





20 


6 


20 


3 


50 


2b 


2 


25 


8 


25 


4 



6 

7 

8 

9 
10 
20 
30 12 
40 16 
50 20 



35 

2-5 

2 

3 

3 

4 

8 



34 



6 


2 


4' 2 


7 


2 


8 2 


8 


3 


2 3 


9 


3 


7| 3 


10 


4 


l' 4 


20 


8 


1 8 


30 


12 


2 12 


40 


16 


3 16 


50 


20 


4 20 



34 

4 
3 
2 
6 








33 

2-3 





7 


6 


G 




6 


0-7 


0.8 


0.6 


7 





8 





7 





7 


8 





9 





8 





8 


9 


1 





1 








9 


10 


1 


1 


1 


1 


1 





20 


2 


3 


2 


1 


2 





30 3 


5 


3 


2 


3 





40 4 


6 


4 


3 


4 





50 


5 


8 


5 


4 


5 






P.P. 



117° 



690 



Qa" 



28" 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



151* 



Log. Sin. 



9-67 
67 



161 
184 
208 
232 
256 

279 
303 
327 
350 
374 

397 
421 
445 
468 
492 



67 
67 
67 
67 
67 



515 
539 
562 
586 
609 



67 
67 
67 
67 
67 



633 
656 
679 
703 
726 



750 
773 
796 
819 
843 

866 
889 
913 
936 
959 



982 
005 
029 
052 
075 

098 
121 
144 
167 
190 



213 
236 
259 
282 
305 



328 
35T 
374 
397 
4.9.0 



443 
46fi 
48R 
5lT 
534 



557 



609_68 

(Log. Cos. d. 

lis' 



23 
24 
23 
24 

23 

23 
24 
23 
23 

23 
24 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
23 
23 

23 
23 
23 
22 
23 

23 
23 
22 
23 
23 

22 



Log. Tan. 



72 567 
?'>. 598 
72 628 
72 659 
72 689 

72 719 
72 750 
72 780 
72 811 
72 841 



72 871 
72 902 
72 932 
72 962 
72 993 



73 023 
73 053 
73 084 
73 114 
73 144 



73 174 
73 205 
73 235 
73 265 
73 295 



73 325 
73 356 
73 386 
73 416 
73 446 



73 476 
73 506 
73 536 
73 567 
73 597 



73 627 
73 657 
73 687 
73 717 
78 747 

73 777 
73 807 
73 837 
73 867 
73 897 



73 927 
73 957 

73 987 

74 017 
74 047 



74 076 
74 106 
74 136 
74 166 
74 196 



c.d. 



74 226 
74 256 
74 286 
74 315 
74 345 



9-74 375 



Log, Cot. 



80 
30 
30 

30 

36 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
80 

30 
30 
80 
30 
3,0 

30 
30 
30 
30 
80 

29 
30 
30 
30 
29 

30 
30 
30 
29 
30 

29 



27 432 
27 402 
0.27 37l 
0.27 341 
0.27 311 



Log. Cot. 



Log. Cos, 



9.94 593 
9.94 587 
9.94 580 
9.94 573 
9.94 566 



d. 



0.27 280 
0.27 250 
0.27 219 
0.27 189 
0.27 159 



27 128 
27 098 
27 067 
27 037 
27 007 



0.26 976 

. 26 946 

26 916 

26 886 

.26 855 



26 825 
26 795 
0.26 765 
. 26 734 
0.26 704 



.26 674 
.26 644 
.26 614 
.26 584 
.26 553 



0.26 523 
0.26 493 
0.26 463 
0.26 433 
0.26 403 



26 373 
26 343 
26 313 

26 283 
26 253 



0.26 223 
0.26 193 
0.26 163 
0.26 133 
0.26 1G3 



0.26 073 
0.26 043 
0.26 013 
0.25 983 
25 953 

25 92? 
25 893 
25 863 
25 833 



94 560 
94 553 
94 546 
9.94 539 
9. 94 533 



94 526 
94 519 
94 512 
94 506 
94 499 



9.94 492 
9.94 485 
9.94 478 
94 472 
94 465 



94 458 
94 451 
94 444 
94 437 
94 431 



94 424 
9.94 417 
9.94 410 
9 . 94 403 
9.94 396 



9.94 390 
9.94 383 
9-94 376 
9.94 369 
9-94 362 



9-94 355 
9 . 94 348 
9.94 341 
9-94 335 
9-94 328 



94 321 
94 314 
94 307 
9.94 300 
9^94 293 

9.94 286 
9-94 279 
9-94 272 
9.94 265 
9-94 258 



9.94 251 
9.94 245 
9-94 238 
9. 94 231 
25 86419. 94 224 



0.25 774 9-94 217 
0-25 744 9.94 210 
0.25 71419.94 203 
0.25 68419-94 196 
0-25 65419.94 189 



0-25 62519-94 182 



60 

59 

58 

57 

56. 

55 

54 

53 

52 

51 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
30 

35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 

20 

19 
18 
17 
16 



10 

9 
8 
7 
6 



c, d,JLog, Tan,jLog. Sin,| d 



-m 



P.p. 





30 


f?0 


6 


3.0 


s.o 


7 


3.5 


3.5 


8 


4-0 


4.0 


9 


4.6 


4.5 


10 


5.l' 5.01 


20 


10.1 


10.0 


30 


15.2 


15-0 


40 


20.3 


20.0 


50 


25.4 


25.0 



89 

2.9 

3.4 

3.9 

4.4 

4.9 

9.8 

14.7 

19-6 

24.6 





S4 


6 


2.4 


7 


2.8 


8 


3.2 


9 


3.6 


10 


4.0 


20 


8.0 


30 


12.0 


40 


16-0 


50 


20.0 





23 


23 


6 


2.3 


2.3 


7 


2.7 


2-7 


8 


3.1 


3-0 


9 


3.5 


3-4 


10 


3.9 


3-8 


20 


7.8 


7-6 


30 


11.7 


11.5 


40 


15.6 


15.3 


50 


19.6 


19.1 



22 

2.2 

2.6 

3.0 

3.4 

3.7 

7.5 

11.2 

15-0 

18. 7 



6 
7 
8 
9 

10 
20 
30 
40 
50 



'' . 


0.7I 





8 





9 


1 





1 


1 


2 


3 


3 


5 


4 


6 


5 


8 



0.6 
0.7 
0.8 
10 
1.1 
2-1 
3-2 
4.3 
5.4 



P.P. 



691 



er 



29" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGP^NTS. l§(f 



Log. Sin. 



109 

n|9 

9 
9 

9 
9 
9 

9 
9 

9 
9 
9 
9 
9 



68 557 
68 580 
68 602 
68 625 
68 648 



68 671 
68 693 
68 716 
68 739 
68 761 



68 784 
68 807 
68 829 
68 852 
68 874 



68 897 

68 920 
68 942 
68 965 

68 987 

69 010 
69 032 
69 055 
69 077 
69 099 



69 122 
69 144 
69 167 
69 189 
69 211 



69 234 
69 256 
69 278 
69 301 
69 323 



69 34'5 
69 387 
69 390 
69 412 
69 434 

69 456 
69 478 
69 500 
69 523 
69 545 



69 567 
69 589 
69 6ll 
69 633 
69.655 



69 677 
69 699 
69 721 
69 743 
69 765 



69 787 
69 809 
69 831 
69 853 
69 875 



9-69 897 
Log. Cos, 



d. 



23 
22 
23 
22 

23 
22 
23 
22 
22 

23 
22 
22 
22 
22 

22 
23 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 



Log. Tan. 



9 



22 9 

22 
22 ^ 
22 ^ 



22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
2l 
22 
■22 



74 370 
74 405 
74 435 
74 464 
74 494 



74 524 
74 554 
74 583 
74 613 
74 643 



74 672 
74 702 
74 732 
74 761 
74 791 



74 821 
74 850 
74 880 
74 909 
74,939 

74 969 

74 998 

75 028 
75 057 
75 087 



75 116 
75 146 
75 175 
75 205 

75 234 



75 264 
75 293 
75 323 
75 352 
75 382 



75 411 
75 441 
75 470 
75 499 
75 529 



75 558 
75 588 
75 617 
75 646 
75 676 



75 705 
75 734 
75 764 
75 793 
75 822 



75 851 
75 8.81 
75 910 
75 939 
75 96S 



75 998 

76 027 
76 056 
76 085 
76 115 



76 144 



d. Log, Cpt. c, d 



c.d, 

30 
30 
29 
30 

29 
30 
29 
29 
30 

29 
30 
29 
29 
30 

2i 
29 
29 
29 
30 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 
29 



Log. Cot. 



25 625 
25 595 
25 565 
25 535 
25_505 

25 476 
25 446 
25 416 
25 387 
25 357 



Log. CoSf 



25 327 9 
25 297 9 
25 268 
25 238 
25 208 



25 179 
25 149 
25 120 
25 090 
25 060 

25 031 
25 001 
24 972 
24 942 
24 913 



24 888 
24 854 
24 824 
24 795 
24 765 



24 
24 
24 
24 
24 

24 
24 
24 
24 
24 



736 
706 
677 
647 
61| 
588 
559 
529 
500 
471 

24 441 
24 412 
24 383 
24 353 
24 324 



9 
9 

24 29519 
24 265 9 
24 236 9 
24 207 9 
24 177 9 



24 148J9 
24 119 9 



24 090 
24 060 
24 031 



24 002 
23 973 
23 943 
23 914 
23 886 



0.23 856 



94 182 
94 175 
94 168 
94 161 
94 154 



94 147 
94 140 
94 133 
94 126 
94 118 



94 111 
94 104 
94 097 
94 090 
94 083 



94 076 
94 069 
94 062 
94 055 
94 048 

94 041 
94 034 
94 026 
94 019 
94^12 

94 005 
93 998 
93 991 
93 984 
93 977 



93 969 
93 962 
93 955 
93 948 
93 941 



93 934 
93 926 
93 919 
93 912 
93 905 



93 898 
93 891 
93 883 
93 876 
93 869 



93 862 
93 854 
93 847 
93 840 
93 833 



93 826 
93 818 
93 811 
93 804 
93 796 



93 7?9 
93 782 
93 775 
£3 767 
93 760 



9-93 753 



Log. Tan, Log. Sin. 



7 
7 
7 
7 

7 
7 
7 
7 

7 

7 
7 
7 
7 
7 

7 
7 
7 
7 
7 

7 
7 
7 
7 
7 

7 
7 
7 
7 

7 

7 
7 
7 
7 
7 

7 
7 
7 
7 

7 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 

7 

7 
7 
7 
7 

7 
7 
7 
7 
7 

7 



60 

59 
58 
57 
56 



55 
54 
53 

52 
51 

50 

49 
48 
47 
j46 

45 
44 
43 
42 

M 
40 

39 
38 
37 
36 

35 
34 
33 
32 
_31 

30 

29 

28 
27 
-26 
25 
24 
23 
22 
21 

'50 
19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 
2 
J^ 





p.p. 





30 


S9_ 


SI 


6 


3.0 


2.9 


2 


7 


8.5 


3 


4 


3 


8 


4.0 


3 


9 


3 


9 


4-5 


4 


4 


4. 


10 


5.0 


4 


9 


4. 


20 


10. 


9 


8 


9. 


30 


15-0 


14 


7 


14. 


40 


20.0 


19 


6 


19. 


50 


25.0 


24 


6 


24. 



6 


2 


7 


2 


8 


3 


9 


3 


10 


3 


20 


7 


30 


11 


40 


15 


§0 


19 



2B 

3 
7 


I 
8 
6 
5 
3 
I 





22 


22 


a 


6 


2.2 


2.2 


2. 


7 


2 


6 


2.5 


2. 


8 


3 





2.9 


2. 


9 


3 


4 


3.3 


3. 


10 


3 


7 


3.6 


3. 


20 


7 


5 


7.3 


7. 


30 


11 


2 


11.0 


10. 


40 


15 





14.. 6 


14. 


50 


18 


7 


18.3 


17. 



7 
8 
9 
10 
20 
30 
40 
50 



7 
.6 0.7 
0.9 
1.0 
11 
1.2 
2.5 
3.7 
5.0 
6.2 



.0.7 
0.8 
0.9 
1.0 
l.I 
2.3 
3.5 
4.6 
5-8 



P. P. 



119" 



692 



ao* 



30" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 149* 



L(>g. Sin. 





1 
2 
8 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



69 897 
69 919 
69 9^.0 
69 962 
69 984 



d. 



70 006 
70 028 
70 050 
70 071 
70 093 

70 115 
.70 137 
.70 158 
•70 180 
.70 202 



30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

4i 

60 

51 

52 

53 

54 



.70 223 
.70 245 
.70 267 
.70 288 
70 310 



•70 331 
.70 353 
.70 375 
70 396 
.70 418 



.70 439 
.70 461 
.70 482 
.70 504 
.70 525 



•70 547 
.70 568 
■70 590 
.70 611 
.70 632 



.70 654 
.70 675 
•70 696 
.70 718 
. 70 739 

•70 765 

70 782 

.70 803 

.70 824 

70 846 



•70 867 
70 888 
70 909 
70 930 

•70 952 



55 
56 
57 
58 
59 



70 973 
.70 994 
.71 015 

71 036 
•71 057 



.71 078 
•71 099 
•71 121 
.71 142 l\ 

71 IRS "^l 



22 
2l 
22 
22 

2l 
22 
22 
21 
22 

2l 
22 
2l 
2l 
22 
2l 
21 
22 
2l 
21 

21 
22 
2l 
2l 
21 

2l 
2l 
2l 
2l 
21 

2l 
2l 
2l 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
2l 

21 

21 
21 
21 
2l 

21 
21 
21 
21 
21 

21 
21 
21 
21 



971 163 



60L9.71 184 ^1 



Log. Tan. c.d. Log. Cot. Log. Cos. 



Log. Cos. d. Log. Cot. 



76 144 
76 173 
76 202 
76 231 
76 260 

76 289 
76 319 
76 348 
76 377 
76 406 



76 435 
76 464 
76 493 
76 522 
76J5I 

76 580 
76 6G9 
76 638 
76 667 
76 696 



76 725 
76 754 
76 783 
76 812 
76 841 



76 870 
76 899 
76 928 
76 957 
76 986 



77 015 
77 043 
77 072 
77 101 
77 130 



77 159 
77 188 
77 217 
77 245 
77 274 



77 303 
77 332 
77 361 
77 389 
77 418 



77 447 
77 476 
77 504 
77 533 
77 562 



77 591 
77 619 
77 648 
77 677 
77 705 



77 734 
77 763 
77 791 
77 820 
77 849 



9-77 877 



0.23 
0.23 
0.23 
0.23 
0.23 



856 
827 
797 
768 
739 
710 
681 
652 
623 
594 



0.23 
0.23 
0.23 
0.23 
0-23 



0-23 
0-23 
0.23 
0.23 
Q-23 

0-23 
0.23 
0.23 
0.23 
0.23 



93 753 
93 746 
93 738 
93 731 
93 724 



93 716 
93 709 
93 702 
93 694 
93 687 



93 680 
93 672 
93 665 
93 658 
93 650 



9.93 643 
9-93 635 
9.93 628 
9.93 621 
9-93 613 

9.93 606 
9.93 599 
9.93 591 
93 584 
158 9-93 576 



0-22 
0-22 
0-22 
0.22 



0.22 
0.22 
0-22 
0.22 



9-93 569 
9-93 562 
9.93 554 
9. 93 547 
993 539 



93 532 
93 524 
93 517 
93 509 
93 502 



9. 



93 495 
93 487 
93 480 
93 472 
93 465 



696 9 
668 " 
639S9 
610 
581 



0.22 
0.22 
0.22 
0.22 
0-22 

0.22 
0.22 
0.22 
0.22 
0.22 



93 457 
93 450 
93 442 
93 435 
93 427 



553 9. S3 420 
52^ 9.93 412 
495 9.93 405 
466 9-93 397 
4.38 9. 93 390 

409 9-93 382 
380 9-93 374 
352 9.93 367 
323 9.93 359 
29419.93 352 



0.22 
0.22 
0.22 
0.22 
0.22 
0.22 



26619-93 344 
237 9-93 337 
208 9-93 329 
180 9.93 321 
151 9.93 314 



122 



C.d. Log. Tan. 



9-93 306 
Log. Sin. 



d. 

7 
7 
7 
7 

7 
7 
7 
7 
7 

7 
7 
7 
7 
7 

7 
? 
7 
7 
7 

7 
7 
7 
7 
7 

7 
7 
7 
7 
7 

7 
7 
7 
7 
7 

7 
7 

7 
7 
7 

7 
7 
7 
7 
7 

7 
7 
7 
7 
7 

8 

7 
7 
7 
7 

7 
7 
7 
8 
7 
7 

"d7 



l60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 

21, 
30 

19 
18 

17 
16 



15 
14 
13 
12 
11 



10 

9 
8 
7 







P.P. 





29 


29 


6 


2.9 


2.91 


7 


3 


4 


3 


4 


8 


3 


9 


3 


8 


9 


4 


4 


4 


3 


10 


4 


9 


4 


6 


20 


9 


8 


9 


6 


30 


14 


7 


14 


5 


40 


19 


6 


19 


3 


50 


24 


6 


24 


1 



28 

28 

33 

3-8 

4.3 

4-7 

9.5 

14.2 

19.0 

23-7 





22 


21 


2 


6 


2-2 


2.1 


2. 


7 


2 


5 


2 


5 


2. 


8 


2 


9 


2 


8 


2. 


9 


3 


3 


3 


2 


3. 


10 


3 


6 


3 


6 3. 


20 


7 


3 


7 


11 7. 


30 


11 





10 


7,10. 


40 


14 


6 


14 


3,14. 


50 


18 


3 


17 


9ll?'. 



• 1 

• 3 
8 

• 1 
5 

5 


• 5 





? 


[ 


f 


6 


0-8 


0.7 


7 





9 


0.9 


8 


1 





1.0 


9 


1 


2 


1.1 


10 


1 


3 


1.2 


20 


2 


6 


2.5 


30 


4 


C 


3-7 


40 


5 


3 


5.0 


50 


6 


6 


6.2 



7 

0-7 
0.8 

I 




4.6 



P.P. 



t8P° 



693 



59* 



«r 



TABLE Vir.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



f48*= 



Lc°:. Sin. 



30 

31 
32 

33 9 
34^9 

9 
9 
9 
9 
9 



71 184 
71 205 
71226 
71247 
71 268 

71 289 
71 310 
71 331 
71351 
71 372 

71 393 
71414 
71435 
71 456 
71 477 



71498 
71 518 
71 539 
71 560 
71 581 



71 601 
71 622 
71 C43 
71 684 
71 684 



71 705 
71 726 
71746 
71 767 
71 788 



71 803 
71 829 
71 849 
71 870 
71 891 



71 911 
71 932 
71 952 
71 973 
71 993 



72 014 
72 034 
72 055 
72 075 
72 09 S 



72 lis 
72 136 
72 157 
72 177 
72 198 



72 218 
72 238 
72 259 
72 279 
72 299 



72 319 
72 340 
72 360 
72 380 
72 400 



72 421 



Log. Cos, 



21 
21 
21 
21 

21 
21 
21 
20 
21 

21 
21 
20 
21 
21 

21 
25 
21 
25 
21 

25 

21 

20 
21 
20 

21 
20 
20 
21 
20 

20 
2Q 
20 
21 
20 

20 
20 
20 
20 
20 

25 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
T 



Log. Tan. 



77 877 
77 906 
77 934 
77 963 
77 992 



78 020 
78 049 
78 077 
78 106 
78 134 



78 163 
78 191 
78 220 
78 248 
78 277 



78 305 
78 334 
78 362 
78 391 
78 419 



78 448 
78 47 b 
78 505 
78 533 
78 561 



78 590 
78 618 
78 647 
78 675 
78 703 



78 732 
78 780 
78 788 
78 817 
78 845 



78 873 
78 902 
78 930 
78 958 
78 987 



79 015 
79 043 
79 07l 
79 100 
79 128 



79 156 
79 184 
79 213 
79 241 
79 269 



79 297 
79 325 
79 354 
79 382 
79 410 



79 438 
79 466 
79 494 
79 522 
79 551 



79 579 



Log. Cot. 



c.d. 

28 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 

28 
28 
28 
28 

28 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 



Log. Cot. 



Log. Cos. 



22 122 
22 094 
22 065 
22 037 
22 008 

2 19 79 9 
21 951 9 
21 922 9 
21 894 " 
21 865 

21 837 
21 808 
21 780 
21 751 
21 723 



21 694 
21 666 
21 637 
21 60'9j9 
21 580 9 



21 552i9 
21 523 9 
21 495|9 
21 467 
21 438 



9 



21 410 
21 381 
21 353 
21 325 
21 296 



21 
21 
21 
21 
21 



268 
239 
211 
183 
154 



21 
21 
21 
21 
21. 

20 
20 
20 
20 
20 



126 
098 
070 
04l 
013 9 






985 
956 
928 
900 
872 9 



20 843 
20 815 
20 787 
20 759 
20 731 



20 702 
20 674 
20 646 
20 618 
20 590!9 









0-20 421 

Log. Tan. 



20 561 
20 53^19 
20 505 9 
20 477 9 
20 449 9 



93 306 
93 299 
93 29 
93 284 
93 276 

93 268 
93 261 
93 253 
93 245 
93 238 



93 230 
93 223 
93 215 
93 207 
93 200 



93 192 
93 184 
93 177 
93 169 
93 16l 



93 153 
93 146 
93 138 
93 130 
93 123 

93 115 
93 107 
93 100 
93 092 
93 084 



93 076 
93 069 
93 061 
93 053 
93 045 



93 038 
93 030 
93 022 
93 014 
93 006 



92 999 
92 991 
92 983 
92 975 
92 967 



92 960 
92 952 
92 944 
92 936 
9^ 9^8 

92 920 
92 913 
92 905 
92 897 
92 889 



92 881 
92 873 
92 865 
92 858 
92 850 



181" 



9 92 842 
Log. Sin, 

694 



d. 

7 
7 
7 
8 

7 
7 
7 
8 
7 
7 
7 
8 
7 
7 

8 

7 
7 
8 
7 

8 
7 

7 
8 
7 

8 
7 
7 
8 
7 

8 
7 
8 
7 
8 

7 
8 
7 
8 
8 

7 
8 
7 
8 
8 

7 
8 
8 
7 
8 
8 
7 
8 
8 
7 

8 
8 
8 
7 
8 

8 

T 



60 

59 
58 
57 
_56. 
55~ 
54 
53 
52 
51 

.'SO 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
33 
37 

35 
34 
33 

32 
31 
30 

29 
28 
27 
M. 
25 
24 
23 
22 
21_ 

30 
19 
18 
17 

il 
15 
14 
13 
12 
11 

10 

9 
8 
7 

_6^ 

5 
4 
3 
2 

_x 





P. P. 



7 
8 
9 
10 
20 
30 
40 
50 



39 

2-9 



38 
2-8 



38 
28 
3 2 



3 

4 

4 

9 

14 

18 

23 



6 

7 

8 

9 

10 

20 

30 

40 

50 



31 

2 

2 

2 

3 

3 

7 
10 
14 
17 



30_ 

2-0 



30 

2.Q 



2 

2 

3 

3 

6 
10. 
13 



16-6 





8 


''. 


6 


0-8 


o.-^ 


7 





9 


0.9 


8 


1 





10 


9 


1 


2 


1.1 


10 


1 


3 


1.2 


20 


2 


6 


2.5 


30 


4 





3.7 


40 


5 


3 


5.0 


50 


6 


6 


6.2 



P. P. 



68" 



33° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 14^ 



Log. Sin. 



25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 



72 421 
72 441 
72 461 
72 481 
72 501 



72 522 
72 542 
72 562 
72 582 
72 602 



72 622 
72 642 
72 662 
72 682 
72 702 



72 723 
72 743 
72 763 
72 783 
72 802 



72 822 
72 842 
72 862 
72 882 
72 902 

72 922 
72 942 
72 962 

72 982 

73 002 



73 021 
73 041 
73 061 
73 081 
73 101 



73 120 
73 140 
73 160 
73 180 
73 19n 

73 219 
73 239 
73 258 
73 278 
73 298 



73 317 
73 337 
73 357 
73 376 
73 396 



73 415 
73 435 
73 455 
73 474 
73 494 



73 513 
73 533 
73 552 
73 572 
73 591 



9. 73 611 
Log. Cos. 



d. 

20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
19 

20 
20 
20 
20 
20 

20 
19 
20 
20 
20 

19 
20 
20 
19 
20 

19 
20 
19 
20 
19 

20 
19 
19 
20 
19 

19. 
20 

19 
19 
19 

19 
20 
19 
19 
19 

19 
19 
19 
19 
19 

19 

T 



Log. Tan. 



79 579 
79 607 
79 635 
79 663 
79 691 



79 719 
79 747 
79 775 
79 803 
79 83l 



79 859 
79 887 
79 915 
79 943 
79 971 



79 999 

80 027 
80 055 
80 083 
80 111 



80 139 
80 167 
80 195 
80 223 
80 25l 



80 279 
80 307 
80 335 
80 363 
80 391 



80 418 
80 446 
80 474 
80 502 
80 530 



80 558 
80 586 
80 513 
80 641 
80 6B9 



80 697 
80 725 
80 752 
80 780 
80 808 



80 836 
80 864 
80 81*1 
80 919 
80 947 



80 975 

81 002 
81 030 
81 058 
81 085 



81 113 
81 141 
81 168 
81 196 
81 224 



81 251 



Log. Cot. C. d. 



c.d. 



28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

27 
28 
28 
28 
28 

27 
28 
28 
28 

27 

28 
28 
27 
28 
28 

27 
2R 
27 
28 
28 

27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
27 



Log. Cot.jLog. Cos. 



20 421 
20 393 
20 365 
20 337 
20 308 



20 288 
20 252 
20 224 
20 196 
20 168 

20 140 
20 112 
20 084 
20 056 
20 028 



20 000 
19 972 
19 944 
19 916 
19 888 



19 860 
19 832 
19 804 
19 776 
19 748 



19 721 
19 693 
19 665 
19 637 
19 609 



19 581 
19 553 
19 525 
19 497 
19 470 



19 442 
19 414 
19 386 
19 358 
19 330 

19 303 
19 27F 
19 247 
19 219 
19 191 



19 164 
19 136 
19 108 
19 080 
19 053 



19 025 
18 997 
18 970 
18 942 
18 914 



18 886 
18 859 
18.831 
18 803 
18 776 



0-18 74R 



Log. Tan. 



92 842 
92 834 
92 826 
92 818 
92 810 



92 802 
92 794 
92 786 
92 778 
92 771 



92 763 
92 755 
92 747 
92 739 
92 731 



92 723 
92 715 
92 707 
92 699 
92 691 



92 683 
92 675 
92 667 
92 659 
92 651 



92 643 
92 635 
92 627 
92 619 
92 611 



92 603 
92 595 
92 587 
92 579 
92 570 



92 &62 
92 554 
92 546 
92 538 
92 530 



92 522 
92 514 
92 506 
92 498 
92 489 



92 481 
92 473 
92 465 
92 457 
92 449 



92 441 
92 433 
92 424 
92 416 
92 408 



92 400 
92 392 
92 383 
92 375 
92 367 



92 359 



133" 



Log. Sin, 
695 



d. 



60 

59 
58 
57 
56 



55 

54 

53 

521 

51 



50 

49 
48 

47 
46 



45 
44 
43 
42 

il 
40 
39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 

30 

19 
18 
17 
16 



15 
14 
13 
12 
11 



10 

9 
8 
7 



P.P. 



28 28 



6 


2 


8 


2.8 


7 


3 


3 


3.2 


8 


3 


8 


3-7 


9 


4 


3 


4-2 


10 


4 


7 


4.6 


20 


9 


5 


9-3 


30 


14 


2 


14-0 


40 


19 





18.6 


50 


23 


7 


23-3 



7 
8 
9 

10 
20 
30 
40 
50 



20 

2-0 



20 

2 

2-3 

2-6 

30 

3-3 

6.6 

10.0 

13-3 

16-6 



9 
10 
20 
30 
40 
50 



e 
1 
1 
1 
1 

2 
4 
5 



7.1 



2.7 

3.2 

36 

4.1 

4.8 

9.1 

13-7 

18-3 

22-9 



19 



1 
2 
2 
2 
3 
6 
9 

13 

16' 



0.? 
0-9 
1.0 
1.1 
1.2 
2-5 
3.7 
5.0 
3-2 



P.P. 



Bf* 



sat" 



TABLE Vll— LOGARITHMIC SINES, COSINES. TANGENTS. 

AND COTANGENTS. 14^8 



Log. Sin, 





1 
2 
3 

A 
5 
6 
7 
8 

^ 

It 

11 

12 

13 

14 

15 
16 
17 
18 
H 

21 
22 
23 
24 

25 
26 
27 
2f 
29 

3( 

31 
32 
33 
31 
35 
36 
37 
88 

II 

4( 

41 

42 

43 

44 

45 

46 

47 

48 

49 

5V 

51 

52 

53 

54 

55 

56 

57 

58 

59. 

60 



73 611 
73 630 
73 650 
73 669 
73 688 



73 708 

73 727 
73 74b 
73 766 
73 785 

73 805 
73 824 
73 843 
73 862 
73 882 

73 901 
73 92b 
73 940 
73 959 
73 978 

73 997 

74 01'6 
74 036 
74 055 
74 074 

74 093 
74 112 
74 131 
74 151 
74 170 



74 189 
74 208 
74 227 
74 246 

74 26F 



74 28'' 
74 303 
74 322 
74 341 
74 3 HO 

74 379 
74 398 

74 417 
74 436 
74 455 



74 474 
74 493 
74 511 
74 530 
74 549 

74 568 
74 587 
74 606 
74 62P 
74 643 



74 662 
74 681 
74 700 
74 718 
74 737 



74 756 



Log. Cos, 



d. 



19 
19 
19 
19 

19 
19 
19 
19 
19 

19 

19 
19 
19 
19 

1§ 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
18 

19 
19 
19 
19 

19 

19 
19 
18 
19 
]9 

19 
18 
19 
19 
18 
19 
18 
19 
18 
19 

18 



Log. Tan. 



81 251 
81 279 
81 307 
81 334 
81 362 



81 390 
81 417 
81 445 
81 473 
81 500 

81 528 
81 555 
81 583 
81 610 
81 638 



81 666 
81 693 
81 721 
81 748 
81 776 



81 80'3 
81 831 
81 858 
81 886 
81 913 



81 941 
81 968 

81 996 

82 023 
82 051 



82 078 
82 105 
82 133 
82 160 
82 188 



82 215 
82 243 
82 270 
82 297 
82 325 



82 352 
82 380 
82 407 
82 434 
82 462 



82 489 
82 516 
82 544 
82 571 
82 598 



82 626 
82 653 
82 680 
82 708 
82 735 



82 762 
82 789 
82 817 
82 844 
82 871 



82 898 



Log. Cot. c. d 



c. d. Log. Cot. Log. Cos. 



28 

27 
27 
28 

27 
27 
27 
28 
i27 

27 
27 
27 
27 
27 

28 

27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27' 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 



18 748 
18 720 
18 69S 
18 665 
18 63?' 



18 610 
18 582 
18 555 
18 527 
18 499 



18 472 
18 444 
18 417 
18 389 
18 362 



18 334 
18 306 
18 279 
18 251 
18 224 



18 1S6 
18 169 
18 141 
18 114 
18 086 



18 059 
18 031 
18 C04 
17 976 
17 949 



17 921 
17 694 
17 867 
17 839 

17 812 

17 784 

17 757 
17 729 
17 702 
17 675 

17 647 
17 620 
17 59? 
17 565 
17 538 



17 510 
17 483 
17 456 
17 428 
17 401 



17 374 
17 347 
17 319 
17 292 
17 265 



17 237 
17 210 
17 183 
17 156 
17 128 



1-7 101 



Log, Tan, 



92 359 
92 351 
92 342 
92 334 
92 326 

92 318 
92 310 
92 30l 
92 293 
92 285 



92 277 
92 268 
92 260 
92 252 
92 244 



92 235 
92 227 
92 219 
92 210 
92 202 

92 194 
92 185 
92 177 
92 169 
92 160 



92 152 
92 144 
92 135 
92 127 
9 2 119 

92 110 
92 102 
92 094 
92 085 
92 077 



92 069 
92 060 

92 052 
92 043 
92 035 



92 027 
92 018 
92 010 
92 001 
91 993 



91 984 
91 976 
91 967 
91 959 
91 951 



91 942 
91 934 
91 925 
91 917 
91 908 



91 900 
91 891 
91 888 
91 874 
91 866 



9-91 857 
Log. Sin. 



60 

59 
58 
57 
-M. 
55 
54 
53 
52 
^L 
50 
49 
48 
47 
46 



45 
44 
43 
42 
41 

40 
39 
38 
37 
_3^ 

35 
34 
33 
32 
31 

30 

29 
28 
27 
-2i 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 



15 
14 
13 
12 
JJ^ 

10 

9 
8 

7 
_6^ 

5 
4 
3 
2 
1 



P.P. 





. 28 


27 


27 


6 


2-8 


2-7 2- 


7 


3 


2 


3 


2 3. 


8 


3 


7 


3 


6 3- 


9 


4 


2 


4 


Ij 4- 


10 


4 


6 


4 


e! 4. 


20 


9 


3 


9 


1 


9. 


30 


14 





13 


7 


13 


40 


18 


6 


18 


3 


18. 


50 


23 


3 


22 


9 


22. 





19 


19 


18 


6 


1-9 


1-9 


1- 


7 


2 


3 


2 


2 


2. 


8 


2 


6 


2 


5 


2. 


9 


2 


9 


2 


8 


2. 


10 


3 


2 


3 


1 


3. 


20 


6 


5 


6 


3 


6. 


30 


9 


7 


9 


5 


9. 


4013 





12 


6 


12- 


50 


16 


2 


15 


8 


15- 





8 


F 


6 


08 


0. 


7 


1.0 


0. 


8 


1.1 


1. 


9 


1.3 


1- 


10 


1.4 


1- 


20 


2.8 


2. 


30 


4-2 


4. 


40 


5-6 


5- 


50 


7.1 


8. 



8 

.9 
.S 

• 2 
-3 

• 6 


.3 

• 6 



P. P. 



i^3° 



696 



66* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
8*9 AND COTANGENTS. I45® 



t 


Log. Sin. 


- 







9.74 758 


1 


9.74775 


2 


9.74793 


3 


9-74 812 


4 


9.74 831 


5 


9-74 849 


6 


9.74 868 


7 


9-74 887 


8 


9-74 905 


9 


8 . 74 924 


10 


9.74943 


11 


9-74 961 


12 


9-74 980 


13 


9-74 998 


14 


9-75017 


15 


9-75 036 


26 


9-75 054 


17 


9-75 073 


18 


9-75 091 


19 


9.75 110 


20 


9.75 128 


21 


9.75 147 


22 


9.75 165 


23 


9-75 184 


24 


9.75 202 


25 


9.75 221 


26 


9 75 239 


27 


9 75 257 


28 


9 75 276 


29 


9.75 294 


30 


9 75 313 


31 


9 75 331 


32 


9.75349 


33 


9 75 368 


34 


9 75 386 


35 


9 75 404 


3b 


9.75 423 


37 


9 75 441 


38 


9 75 459 


39 


9.75 478 


40 


9 75 496 


41 9.75 514 


42 9 75 532 


43 9.75 551 


44 


9.75 569 


45 


9.75 587 


46 


9.75 605 


47 


9-75 623 


48 


9-75 642 


49 


9.75 660 


50 


9 75 678 


51 


9.75 896 


52 


9 75 7-14 


53 


9 75 732 


54 


■^•75 750 


55 


9 75 789 


56 


9-75 787 


57 


9-75 805 


58 


9-75 823 


by^ 


9 - 75 841 


60 


9 7R 8R9 




Log, Cos. 



19 
18 
19 
18 

18 
19 
18 
18 
19 

18 
18 
18 
18 
18 

1? 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 



d. Log. Tan. c. d. Log. Cot. Log, Cos. 



-82 898 

82 926 
-82 953 
-82 980 
- 83 007 

-83 035 
-83 062 
-83 089 
-83 116 
-83 143 

'83 171 

83 198 
• 83 225 

83 252 
J8^279 

83 307 
83 334 
83 361 
83 338 
83 415 



83 442 
83 468 
83 496 
83 524 
83 551 



83 578 
83 605 
83 632 
83 659 
83 686 



83 713 
83 740 
83 767 
83 794 
83 821 



83 848 
83 875 
83 902 
83 929 
83 flsi? 



9- 



83 984 

84 011 
84 038 
84 065 
84 091 



9. 



84 118 
84 145 
84 172 
84 199 

84 226 



84 253 
84 280 
84 307 
84 334 
84 361 



84 388 
84 415 
84 442 
84 469 
84 496 



SA r,22 



27 
27 
27 
2J 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
26 

27 

27 
27 
27 
27 

27 
9.7 
27 
?7 

26 

27 
27 
27 
27 
27 

26 



0.17 iUl 
0-17 074 
0-17 047 
0-17 019 
0-16 992 



16 965 
16 938 9 
16 910 
16 883 
16 85b 



16 829 
16 802 
16 774 
16 747 
16 720 



91 857 
91 849 
91 840 
91 832 
91 823 

91 814 
91 806 
91 797 
91 789 
91 780 



9_ 

0.16 693 9. 
0.16 666 9. 
0.16 639 9- 
0-16 612 9. 
0-16 584 9- 



■91 772 
•91 763 
•91 755 
• 91 746 
91 737 



16 557 
16 530 
16 503 
16 476 
16 449 



91 729 
91 720 
91 712 
91 703 
91 694 



9.91 686 
9-91 677 
9.91 668 
9.91 660 
9-91 651 



d. 



0-16 422 9. 
0-16 395 9 
0.16 368 9. 
0-16 340 9. 
0.16 313 9 



16 286 
16 259 
16 232 
16 205 
16 170 



0-16 151 
0.16 124 
0.16 097 
0.16 070 
0.1.6 04? 



16 016 
15 989 
0.15 962 
0.15 935 
0-15 90P 



91 642 
91 634 
91 625 
91 616 
91 608 



91 599 
91 590 
91 582 
91 573 
• 91 564 



91 556 

91 547 

91 538 

•91 529 

• 91 521 



91 512 
91 50§ 
91 495 
91 486 
91 477 



0-15 88T 
0-15 851 
0-15 8P7 
0-15 ?i(^f^ 
0-15 77 8 

0-15 746 
0-15 71P 
0-15 692 
0-15 BP?^ 
0-15 639 



91 468 
91 460 
91451 
91 449 
91 43? 



d. Log. Cot.ic.d, 



9 
9 
9 
9 

0.15 ei?|9 

0-15 585 9 
0-15 558 9 
0.15 53] 9 
0.15 504 9 



91 424 
91 416 
91 407 
91 398 
91 389 



• 91 380 
91 372 
91 363 
91 354 
91 345 



0-15 477 9 91 3^6 
I og, Tan.lLog. Sin. 



d. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
^ 
50 
49 
48 
47 
18 
45 
44 
43 
42 
41 

40 
39 
38 
37 

35 
34 
33 
32 
31_ 

30 

29 
28 
27 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
]A 

10 

9 

8 

7 

_6 

5 
4 
8 
2 

O 



P. P. 





S7 


27 


26 


6 


2-7 


2-71 2-6 


7 


3 


2 


3 


1 


81 


8 


3 


6 


3 


6 


3-5 


9 


4 


1 


4 


c: 4-0 


10 


4 


6 


4 


5 


4 4 


20 


9 


1 


9 





8-8 


30 


13 


7 


13 


5 


13 2 


40 


18 


3 


18 





i7 6 


50 


22 


9 


22 


5 


'22-1 





19 


18 


18 


6 


1 


9 


1-8 


18 


7 


2 


2 


2-1 


2 


1 


8 


2 


5 


2-4 


2 


4 


9 


2 


8 


2 8 


2 


7 


10 


3 


1 


31 


3 





20 


6 


3 


6-1 


6 





30 


9 


5 


9-2 


9 





40 


12 


6 


12-3 


12 





50 


15 


8 


15.4 


15 






6 

7 
8 

9 

10 
20 
30 
40 
50 



9 S 






9 





1 





1 


1 


2 


1 


1 


3 


i- 


1 


\j 


1 . 


3 


C 


-> 


4 


g 


4- 


6 





5- 


7 


5 


7- 



P. p. 



&»4° 



697 



65" 



36° 



TABLE Vir.— LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



144" 



Log. Sin. 



75 859 
75 877 
75 895 
75 913 
75 931 



75 949 
75 967 

75 985 

76 003 
76 021 



76 039 
76 057 
76 075 
76 092 
76 llO 



76 128 
76 146 
76 164 
76 182 
76 200 

76 217 
76 235 
76 253 
76 271 
76 289 



78 306 
76 324 
76 342 
76 360 
76 377 



76 395 
76 413 
76 431 
76 448 
76 466 



78 484 
76 501 
76 519 
70 536 
76 554 



70 572 
76 589 
76 607 
76 624 
76 642 



76 660 
76 677 
76 695 
76 712 
76 730 



76 747 
76 765 
76 782 
76 800 
76 817 



76 835 
76 852 
13 869 
76 887 
76 904 



9-76 922 



Log. Cos. 



d. 



18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
17 
18 

18 
18 
17 
18 
18 

17 
18 
18 
17 
18 

17 
18 
17 

18 

17 

18 
17 
18 

17 
17 

18 

17 
17 
17 
18 

17 
17 

17 
17 
18 

17 
17 
17 
17 
17 

17 
17 
17 

17 
17 

17 
17 
1^ 
17 
17 

17 
T 



Log. Tan. 



84 522 
84 549 
84 576 
84 603 
84 630 



84 657 
84 684 
84 711 
84 737 

84 764 



84 791 
84 818 
84 845 
84 871 
84 898 



84 92§ 
84 952 

84 979 

85 005 
85 032 

85 059 
85 086 
85 113 
85 139 
85 166 



85 193 
85 220 
85 246 
85 273 
85 300 



85 327 
85 353 
85 380 
85 407 
85 433 



85 460 
85 487 
85 513 
85 545 
85 567 



85 594 
85 620 
85 647 
85 673 
85 700 



85 727 
85 753 
85 780 
85 807 
85 833 



85 860 
85 887 
85 913 
85 940 
85 966 



85 993 

86 020 
86 046 
86 073 
86 099 



86 126 



Log. Cot. 



,d. 



27 
27 
27 
26 

27 
27 
27 
26 
27 

27 
26 
27 
26 
27 

27 
26 
27 
26 
27 

27 

26 
27 
26 
27 
26 
27 
26 
27 
26 

27 
26 
26 
27 
26 

27 
26 
2G 
27 
26 

27 
26 
26 
2G 
27 

26 

26 
27 
26 
26 

26 
27 
26 
26 
26 

26 
27 
26 
26 
26 

26 



Log. Cot, 



15 477 
15 450 
15 423 
15 396 
15 370 



15 343 
15 316 
15 289 
15 262 
15 235 



15 208 
15 182 
15 155 
15 128 
15 101 



15 074 
15 048 
15 021 
14 994 
14 967 

14 940 
14 914 
14 887 
14 860 
14 83.3 



14 807 
14 780 
14 753 
14 726 
14 7C0 



14 673 
14 646 
14 620 
14 593 
14 566 



14 
14 
14 
14 
14 



539 
513 
486 
459 
433 9 



14 
14 
14 
14 
14 



4C6 
379 
35S 
326 
299 



Log. Cos, 



14 273 9 
14 246 9 
14 219 9 
14 193 9 
14 166 9 



14 140 
14 113 
14 086 
14 060 
14 033 



14 007 
13 980 
13 953 
13 927 
13 900 



13 874 9 



91 336 
91 327 
91318 
91310 
91 301 



91 292 
91 283 
91 274 
91 265 
91 256 



91 247 
91 239 
91 230 
91 221 
91212 



91 203 
91 194 
91 185 
91 176 
91 167 

91 158 
91 149 
91 140 
91131 
91 122 



91 113 
91 104 
91 095 
91 086 
91 077 



91 068 
91 059 
91 050 
91 041 
91 032 



91 023 
91014 
91005 
90 996 
90 987 



90 978 
90 969 
90 960 
90 951 
90 942 



90 933 
90 923 
90 914 
90 905 
90 896 



90 887 
90 878 
90 869 
90 860 
90 850 



90 841 
90 832 
90 823 
90 814 
90 805 



90 796 



135° 



Log. Tan.jLog. Sin. 
698 ~ 



60 
59 
58 
57 

'56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 
34 
33 
32 

11 
30 

29 
28 
27 
26 



25 
24 
23 
22 
21 



20 

19 
18 

17 
16 



15 
14 
13 
12 
JJ^ 

10 

9 
8 
7 



P. P. 





27 


26 


6 


2-7 


2-6 


7 


3 


1 


3 


1 


8 


3 


6 


3 


5 


9 


4 





4 





10 


4 


5 


4 


4 


20 


9 


C 


8 


8 


30 


13 


5 


13 


2 


40 


18 





17 


6 


50 


22 


5 


122 


1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



18 
1-8 
2-1 
2.4 
2-7 
3-0 
6.C 
9.0 
12-0 
15.0 



17_ 

1-7 



11 

1.7 



6 

7 

8 

9 

10 

20 

30 

40 

50 



9 

0-9 

1-1 
1.2 
1.4 
1.6 
3.1 
4.7 
6.3 
7.9 



9 

0.9 
1.0 
1-2 
1-3 
1.5 
3.0 
45 
6.0 
7.5 



8 
0-8 



P.P. 



54? 



36° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 143* 



O 

1 
2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 , 

27 9 

28 9 

29 9 



Log. Sin. 



9.76 922 
9-76 939 
9-76 956 
76 974 
9-76 991 



77 008 
77 026 
77 043 
77 060 
77 078 



77 095 
77 112 
77 130 
77 147 
77 164 



77 181 
77 198 
77 216 
77 233 
77 250 

77 267 
77 284 
77 302 
77 319 
77 336 



d. 



30 

31 

82 

S3 

3£ 

85 

86 

37 

38 

39_ 

40 
41 
'42 
43 
44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
56 
57 
58 
59 
60 



9-77 439 
9-77 456 
77 473 
77 490 
77 507 



77 353 
77 370 
77 387 
77 404 
77 421 



9-77 524 

77 541 

9-77 558 

9-77 575 

77 592 

77 609 
77 626 
77 643 
77 660 
77 677 



77 693 
77 710 
77 727 
77 744 
77 761 



77 778 
77 795 
77 812 
77 82§ 
77 845 



77 862 
77 879 
9.77 896 
9-77 913 
9-77 929 



Log. Tan. 



9 . 77 9A6 



Log. Cos. 



17 
17 
17 
17 

17 

17 
17 
17 
17 

17 
17 
17 

17 5 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

16 
17 
17 
17 
17 

16 
17 
17 
16 
17 

17 
16 
17 
17 
16 
17 



9-86 126 
9-86 152 
9.86 179 
9.86 206 
9-88 232 



9-86 259 
9.86 285 
9.86 312 
9.86 338 
9.8 6 365 

9. 
9. 



86 391 
86 418 
86 444 
86 471 
86 497 



86 524 

9.86 550 

86 577 

86 603 

9. 86 630 



36 656 
86 683 
86 709 
86 736 
86 762 



9.86 788 
9.86 815 
9«86 84l 
9.86 868 
9. 86 894 



86 921 
86 947 

9.86 973 

9.87 000 
9jJ7jD26 

9-87 053 
9.87 079 
9-87 105 
9-87 132 
9.87 158 



9.87 185 
9.87 211 
9.87 237 
9.87 264 
87 290 



87 316 
9-87 343 
9-87 369 
9-87 395 
9-87 422 

9.87 44§ 
9-87 474 
9-87 501 
9.87 527 
9-87 553 



9-87 580 
9-87 60P 
9- 87 63? 
9-87 659 
9. 87 685 



9-87 71] 



c.d. 

26 
26 
27 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 



13 874 
0-13 847 
0-13 821 
0-13 794 
0-13 767 



Log. Cot. 



13 741 
13 714 
0-13 688 
0-13 661 



13 608 
13 582 



Log. Cos. 

90 796 

90 786 

-90 777 

-90 768 

90 759 



d. 



9-90 750 
9 . 90 740 
9.90 731 
9.90 722 
9-90 713 



9. 



d. Logi Cot. 



90 703 
90 694 
13 55'5!9.90 685 
13 529 9.90 676 
13 502 9-90 666 



0.13 476 9.90 657 
0.13 449 9-90 648 



0.13 423 



13 396 
13 370 
13 343 
13 317 
13 290 
13 264 
13 237 



90 639 
9-90 629 
9-90 620 



9.90 611 
9.90 602 
9.90 592 
9-90 583 
9-90 574 



13 211 
13 185 
13 158 
13 132 
13 105 



9.90 564 
9.90 555 
90 546 
9.90 536 
9-90 527 



13 079 
13 052 
0.13 026 
0.13 OOC 
0.12 973 



12 947 
12 920 
. 12 894 
0.12 868 
12 841 

12 815 
12 78? 
0.12 762 
0.12 73C 
0.12 7G9 



0.12 68? 
0.12 657 
0.12 63C 
0.12 604 
0.12 57£ 



C.d. 



0.12 55] 
0.12 525 
0.12 49P 
0.12 472 
0-12 446 



9-90 518 
9-90 508 
9-90 499 
9-90 490 
9 - 90 480 



9-90 471 
9-90 461 
90 452 
90 443 
90 433 

90 424 
9.90 414 
9- 90 405 
9-90 396 
9-90 386 

9-90 377 
9-90 367 
9-90 358 
9.90 348 
9- 90 339 



9- 90 330 
9.90 320 
9-90 311 
9-90 30l 
9 - 90 292 



0-12 420 9-90 282 
0-12 393 9-90 273 
0-12 367 9-90 263 
0-12 341 9.90 254 
0-12 315 9-90 244 



0-12 288 9 90 ?35 



i^e" 



Log. Tan-JLog. Sin, 
699 



60 

59 

58 

57 

56_ 

55 

54 

53 

52 

51 

50 
49 
48 
47 
46 



45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



30 

29 
28 
27 
26 

25 
24 
23 
22 
21. 
20 
19 
18 
17 
16 



_ 6 

Q — — 



15 
14 
13 
12 
11 

10 

9 
8 
7 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



2*? 

2-7 

3-1 

3-6 

4-0 

4-5 



9. 

13. 



18.0 
22.5 



S6„ 

2-6 
3-1 
3-5 
4-0 

4.4 

8-i 

13 2 
17.6 
22-1 



36 

2^6 

8.5 

3.1 

39 

4-3 

8-6 

lS-0 

IV -3 

21.6 





IT 


17 


6 


1-7 


1-7 


7 


2-0 


2.0 


8 


2-3 


2.2 


9 


2-6 


2.5 


10 


2.9 


2-8 


20 


5-8 


5.6 


80 


8.7 


8.5 


40 


11.6 


11-3 


50 


14.6 


14.1 



16^ 

1.6 
1.9 
2.2 
2.5 
2.7 
5.5 
8.2 
11.0 
13. f 





9 


9 


6 


0.9 


0-9 


7 


1.1 


l.Q 


8 


1.2 


1.2 


9 


1.4 


1.3 


10 


1.6 


1.5 


20 


3.1 


3.0 


30 


4-7 


4.5 


40 


6.3 


6.0 


50 


7.9 


7-5 



P.P. 



£3" 



37* 



TABLE VII.— LOGARITHMIC SINES, COSINE^, TANGENTS, 

AND COTANGENTS. 14!^ 



JLog. Sin, 



77 946 
77 963 
77 980 

77 996 

78 013 



78 030 
78 046 
78 063 
78 080 
78 097 



78 113 
78 130 
78 147 
78 163 
78 180 



78 196 
78 213 
78 230 
78 246 
78 263 



78 279 
78 296 
78 312 
78 329 
78 346 



78 362 
78 379 
78 395 
78 412 
78 428 



78 444 
78 461 
78 477 
78 494 
78 510 



78 527 
78 543 
78 559 
78 576 
78 592 



78 609 
78 625 
78 641 
78 658 
78 674 



78 690 
78 707 
78 723 
78 739 
78 755 



78 772 
78 788 
78 804 
78 821 
78 837 



78 853 
78 869 
78 885 
78 902 
78 918 

9:78 934 

Log. Cos. 



d. 
16 

12 

16 
17 

16 
16 

17 
16 
17 

16 
1'6 
17 
16 
16 

16 
16 
17 
16 
16 

16 
16 
16 
16 
17 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 

16 

16 
16 
16 

16 
T 



Log. Tan. c.d. 



87 
87 
87 
87 
87 



711 
737 
764 
790 
816 



87 
87 
87 
87 
87 



843 
869 
895 
921 
948 



87 
88 
88 
88 
88 



974 
000 
026 
053 
079 



105 
13l 
157 
184 
210 



88 
88 
88 
88 
88 



236 
262 
288 
315 
341 



88 
88 
88 
88 



367 
393 
419 
445 
472 



88 
88 
88 
88 
88 



498 
524 
550 
576 
602 



88 
88 
88 
88 
88 



629 
655 
681 
707 
733 



88 
88 
88 
88 
88 



759 
785 
811 
838 
864 



88 
88 
88 
88 
88 



890 
916 
942 
968 
994 



89 
89 
89 
89 
89 



020 
046 
072 
098 

124 



89 
89 
89 
89 
89 



150 
177 
203 
229 
255 



89 281 
Log. Cot. 



26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 

26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 



Log. Cot. 



0.12 288 
0-12 l;62 
0.12 236 
0.12 209 
0.12 183 



0-12 157 
0-12 131 
0.12 104 
0-12 078 
0-12 052 



12 026 
11 999 
11973 
11 947 
11 921 



Cll 8 
0.11 868 
0.11 842 
Cll 816 

Cll 790 



0-11 763 
Cll 737 
0.11 711 
0.11 685 
0.11 659 



0-11 633 
0-11 606 
Cll 580 
0-11 554 
0.11 528 



11 502 
11 476 
0-11 449 
0.11 423 
Cll 397 



Log, Cos. 



90 235 
90 225 
90 216 
90 206 
90 196 



90 187 
90 177 
90 168 
90 158 
90 149 



90 139 
90 180 
90 120 
90 110 
90 101 



90 091 
90 082 
90 072 
90 062 
90 053 



9. 



90 043 
90 033 
90 024 
90 014 
90 004 



9. 89 995 
9. 89 985 
9-89 975 
9-89 966 
9-89 956 

9 
9. 



89 946 
89 937 
89 927 
89 917 
89 908 



11 371 
11 345 
0.11 319 
0.11 293 
0-11 266 9 

0-11 240 9 
0.11 214 9 
0.11 188 9 
0.11 162 9 
0.11 136 9 



0.11 110 
0.11 084 
0.11 058 
0-11 032 
0-11 005 



89 898 
89 888 
89 878 
89 869 
89 859 

89 849 
89 839 
89 830 
89 820 
89 810 



- 10 979 
0.10 953 
0-10 927 
0-10 901 
0-10 875 



0-10 849 
0-10 823 
0-10 797 
0-10 771 
Old 745 

0-10 719 
Log. Tan. 



89 800 
89 791 
89 781 
89 771 
89 761 



89 751 
89 742 
89 732 
89 722 
89 712 



89 702 
89 692 
9-89 683 
9-89 673 
9-89 663 

9-89 653 
Log. Sin. 



d. 



9 

9 

9 

10 

9 
9 
9 
9 
9 

9 
9 
10 
9 
9 

9 
9 
10 
9 
9 

9 

10 

9 

9 

10 

9 
9 

IQ 
9 
9 

IQ 
9 
9 

10 
9 

10 
9 

10 
9 

10 

9 
10 

9 
10 

9 

10 

9 

10 

10 

9 

10 

9 

10 

10 

9 

10 

10 

9 

10 

10 

10 

T 



60 

59 

58 

57 

_56 

55 
54 
53 
52 
_51 

50 

49 

48 

47 

J6 

45 
44 
43 
42 
41 

40 

39 
38 
37 
_36 

35 
34 
33 
32 
31 

30 
29 
28 
27 

-2i 

"25 
24 
23 
22 

_21 

30 

19 
18 

17 
16 

15 
14 
13 
12 
11 



10 

9 
8 
7 



P.P. 





26 


26 


6 


2-6 


2-6 


7 


3 


1 


3 





8 


3 


5 


3 


4 


9 


4 





3 


9 


10 


4 


4 


4 


3 


20 


8 


8 


8 


6 


30 


13 


2 


13 





40 


17 


6 


17 


g 


50 


22 


1 


21 


6 





17 


16 


1( 


6 


1-7 


1.6 


1- 


7 


2 





1 


9 


1- 


8 


2 


2 


2 


r 


2. 


9 


2 


5 


2 


5 


2. 


10 


2 


8 


2 


7 


2- 


20 


5 


6 


5 


5 


5- 


30 


8 


5 


8 


2 


8- 


40 


11 


3 


11 


C 


10- 


50 


14 


1 


13 


7 


13. 



•6 
-8 
.1 
.4 
•6 
-3 

•Q 
-6 
.3 



10 9 



61 1 

71 

81 

9 1 

101 

20 3 

30 5 

40 6 

50 8 



-0 


0- 


-1 


1- 


-3 


1- 


.5 1- 


-6 1- 


-33- 


.0 4- 


-6 6- 


-3 


7. 



1.9 
-1 

-2 

-4 

-6 

1-1 

•Z 

i-3 
.9 



P. P. 



700 



i5^ 



38° 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS. 
AND COTANGENTS. 



1^1' 



Log. Sin. 



60 9 



78 934 
78 950 
78 966 
78 982 
78 999 



79 015 
79 031 
79 047 
79 063 
79 079 



79 095 
79 111 
79 127 
79 143 
79 159 



79 175 
79 191 
79 207 
79 223 
79 239 



79 255 
79 271 
79 287 
79 303 
79 319 



79 335 
79 351 
79 367 
79 383 
79 399 



79 415 
79 431 
79 446 
79 462 
79 478 



79 494 
79 510 
79 526 
79 541 
79 557 



79 573 
79 589 
79 605 
79 620 
79 636 



79 652 
79 668 
79 683 
79 699 
79 715 



79 730 
79 746 
79 762 
7^777 
79 793 



79 809 
79 824 
79 840 
79 856 
79 871 



Log. 



79 887 
Cos. 



d. 



16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
15 
16 

16 
16 
15 
16 
16 

15 
16 
16 
15 
16 

16 
15 
16 
15 
16 

15 
16 
15 
16 
15 

15 
16 
15 
15 
16 

15 
15 
16 
15 
15 

15 
T 



Log. Tan. 



89 281 
89 307 
89 333 
89 359 
89 385 



89 411 
89 437 
89 463 
89 489 
89 515 



89 541 
89 567 
89 593 
89 619 
89 645 



89 671 
89 697 
89 723 
89 749 
89 775 
89 801 
89 827 
89 853 
89 879 
89 905 



89 931 
89 957 

89 982 

90 008 
90 034 



90 060 
90 086 
90 112 
90 138 
90 164 



90 190 
90 216 
90 242 
90 268 
90 294 



90 319 
90 345 
90 371 
90 397 
90 423 

90 449 
90 475 
90 501 
90 526 
90 552 



90 578 
90 604 
90 630 
90 656 
90 682 



90 707 
90 733 
90 759 
90 785 
90 811 



90 837 



Log. Cot, c.d. 



c.d. 



26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
25 
26 
26 
26 
26 
26 
26 
25 

26 
26 
26 
26 
26 

25 
26 
26 
26 
25 

26 
26 
26 
25 
26 

26 
26 
25 
26 
26 

25 
26 
26 
26 

25 

26 



Log. Cot. 



10 719 
10 693 
10 667 
10 641 
10 615 



10 589 
10 563 
10 537 
10 511 
10 485 



10 459 
10 433 
10 407 
10 381 
10 355 



10 329 
10 303 
10 277 
10 251 
10 225 



Log. Cos. 



10 199 
10 173 
10 147 
10 121 
10 095 



10 069 
10 043 
10 017 
09 991 
09 965 



09 939 
09 913 
09 887 
09 86l 
09 836 



09 810 
09 784 9 
09 758 
09 732 
09 706 



09 680 
09 654 
09 628 
09 602 
09 577 



09 551 
09 525 
09 499 
09 473 
09 447 



09 421 
09 395 
09 370 
09 344 
09 318 



09 292 
09 266 
09 240 
09 214 
09 189 



0.09 163 
Log. Tan, 



89 653 
89 643 
89 633 
89 623 
89 613 



89 604 
89 594 
89 584 
89 574 
89 564 



89 554 
89 544 
89 534 
89 524 
89 514 



89 504 
89 494 
89 484 
89 474 
89 464 



89 454 
89 444 
89 434 
89 424 
89 414 



89 404 
89 394 
89 384 
89 374 
89 364 



89 354 
89 344 
89 334 
89 324 
89 314 



89 304 
89 294 
89 284 
89 274 
89 264 



89 253 
89 243 
89 233 
89 223 
89 213 



89 203 
89 193 
89 182 
89 172 
89 162 



89 152 
89 142 
89 132 
89 121 
89 111 



89 101 
89 091 
89 081 
89 070 
89 060 



9-89 050 
Log. Sin, 



d. 



9 
10 
10 
10 

9 
10 
10 
10 
10 

10 
9 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 
10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 

10 
10 
10 
10 

10 

T 



GO 

59 
58 
57 
-56 
55 
54 
53 
52 

50 

49 

48 

47 

J6 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 

32 

_31 
30 

29 
28 
27 
_26 

25 
24 
23 
22 
21 



30 

19 
18 
17 
16. 
15 
14 
13 
12 
11 



10 

9 
8 
7 
6 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



36 



2 


6 


2. 


3 





3 


3 


4 


3. 


3 


9 


3. 


4 


3 


4. 


8 


6 


8- 


13 





12. 


17 


3 


17. 


21 


6 


21. 



35_ 

5 

4 
8 
2 
5 
7 

2 





16 


16 


11 


6 


1.6 


16 


.1. 


7 


1 


9 


1 


8 


1- 


8 


2 


2 


2 


1 


2. 


9 


2 


5 


2 


4 


2- 


10 


2 


7 


2 


6 


2. 


20 


5 


5 


5 


3 


5. 


30 


8 


2 


8 





7- 


40 


11 


10 


6 


10. 


50 


13 


7 


13 


i 


12. 



6 

7 

8 

•9 

10 

20 

30 

40 

50 



10 10 9 



00 

1 

3 
5 



9 
.1 
.2 
.4 

6 
.1 

7 
.3 
.9 



Pi P. 



138° 



701 



5r 



39* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



140"= 



Log. Sin. 



79 887 
79 903 
79 918 
79 934 
79 949 



79 965 
79 980 

79 996 

80 011 
80 027 



80 042 
80 058 
80 073 
80 089 
80 104 



80 120 
80 135 
80 151 
80 166 
80 182 



80 197 
80 213 
80 228 
80 243 
80 259 



80 274 
80 289 
80 305 
80 320 
80 335 



80 351 
80 366 
80 381 
80 397 
80 4li 

80 427 
80 443 
80 458 
80 473 
80 488 



80 504 
80 519 
80 534 
80 549 
80 564 



80 580 
80 595 
80 610 
80 625 
80 640 



80 655 
80 671 
80 686 
80 701 
80 716 



80 731 
80 746 
80 76l 
80 776 
80 791 



80 806 



Log, Cos. d. 



d. 



16 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 

15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 



Log. Tan, 



9- 90 
90 
90 



837 
863 
888 
914 
940 

966 
992 
017 
043 
069 



91 
91 
91 
91 
91 



095 
121 
146 
172 
198 



91 
91 
91 
91 
91 



224 
250 
275 
301 
327 



353 
378 
404 
430 
456 



91 
91 
91 
91 
91 



481 
507 
533 
559 
584 

610 
636 
662 
687 
713 



91 
91 
91 
91 

iL 
91 
91 
91 
91 
91 



739 
765 
790 
816 
842 

867 
893 
919 
945 
970 



996 
022 
047 
073 
099 

124 
150 
176 
201 
227 



9.92 



253 
278 
304 
33C 
355 

381 



Log. 



Cot. c. d. 



c.d. Log. Cot. 



26 
25 
26 
25 

26 
26 
25 
26 
26 

25 
26 
25 
26 
25 

26 
26 
25 
26 
25 

26 
25 
26 
25 
26 

25 
26 
25 
26 
25 

26 
25 
26 
25 
26 
25 
26 
25 
26 
25 

25 
26 
25 
26 
25 

25 
26 
25 
26 
25 

25 
26 
25 
25 
26 

25 
25 
26 
25 
25 

26 



09 163 
09 137 
09 111 
09 085 
09 060 



09 034 
09 008 
08 982 
08 956 
08 930 



08 905 
08 879 
08 853 
08 827 
08 802 



08 776 
08 750 
08 724 
08 698 
08 673 



08 647 
08 621 
08 59 
08 57C 
08 544 



08 518 
08 492 
08 467 
08 441 
08 41 1 



08 389 
08 364 
08 338 
08 312 
08 286 

08 261 
08 235 
08 209 
08 183 
08 158 



08 132 
08 106 
08 081 
08 055 
08 029 



08 004 
07 978 
07 952 
07 926 
07 901 



07 875 
07 849 
07 824 
07 798 
07 772 



07 747 
07 721 
07 695 
07 670 
07 644 



n. 07 63 8 



Log. Tan. 



Log. Cos, 



89 050 
89 040 
89 030 
89 019 
89 009 



88 999 
88 989 
88 978 
88 968 
88 958 



88 947 
88 937 
88 927 
88 917 
88 906 



88 896 
88 886 
88 875 
88 865 
88 855 



88 844 
88 834 
88 823 
88 813 
88 8C3 



88 792 
88 782 
88 772 
88 761 
88 751 



88 740 
88 73C 
88 720 
88 709 
88 608 



88 688 
88 678 
88 667 
88 657 
88 646 



88 636 
88 625 
88 615 
88 604 
88 594 



88 583 
88 573 
88 562 
88 552 
88 541 



88 531 
88 520 
88 51C 
88 4G9 

88 ^r9 



:88 478 

88 467 
88 457 
88 446 
88 436 



9. 88 425 



Log. Sin. 



d. 



10 
10 
10 
10 

10 
10 
10 
lO 
10 

lO 
10 
10 

10 
10 

10 
10 

10 
10 
10 

10 

10 
10 
10 
10 

10 

10 
10 
10 
lO 

10 
10 
10 
10 
10 

10 

IQ 

10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
lO 

10 
10 
10 
10 
lO 

lO 
11 

10 
10 
10 

10 



d. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 

45 
44 
43 
42 
41 



40 

39 
38 
37 
36 



35 

34 
33 
32 
11 
30 
29 
28 
27 
26 



25 
24 
23 
22 
21 

20 
19 
18 
17 
16 



15 
14 
13 
12 
_11 

10 

9 
8 
7 
6 






p. p. 



26 35 


6 


2.6i 2-5 


7 


3 


3 





8 


3 


41 3 


4 


9 


3 


9| 3 


8 


IC 


4 


3! 4 


2 


2C 


8 


6l 8 


5 


3C 


13 


C'l2 


7 


40 


17 


3|i7 





50 


21 


6!21 


2 



6 

7 
8 
9 
10 
2G 
30 
4C 
50 



16 



15_ 

1-5 



15 

1-5 



6 
7 
8 

9 

10 
2C 
30 
40 
50 



11 

1.1 



10 10 



6 1 
7ll 
513 
2 5 
06 



.0 
.1 
3 
•5 
•6 
■ 3 

Q 
6 



783 



P.P. 



1^9'' 



702 



60" 



40° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



139° 





1 

2 

3 

_4 

5 
6 
7 
8 
J^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



Log. Sin. 



30 9 
21 



9-80 806 
9-80 822 
9. 80 837 
9. 80 852 
9. 80 887 



9. 80 882 
9 80 897 
9. 80 912 
9.80 927 
9-80 942 



9.80 957 

9. 80 972 
80 937 

9-81001 

9. 81 016 



81 031 
81 046 
81081 
81 076 
81 091 



22 
23 
24 

25 

28 

27 

28 

2i 

30 

31 

32 

33 

34 

35 

38 

37 

38 

39 

40 

41 
42 
43 
44 
45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 

57 

58 

59_ 

60 



81 106 
81 121 
81 138 
81 150 
81 165 



81 180 
81 195 
81 210 
81 225 
81 239 



98 



81 254 
81 269 
81 284 
81 299 
1 3V6 



81 328 
81 343 
81 358 
81 372 
81 387 



81 402 
81 416 
81 431 
81 448 
81 430 



81 475 
81 490 
81 504 
81 519 
81 534 



9. 81 543 
9.81 583 
9. 81 578 
9. 81.592 
9-81 607 



981 821 
9- 81 636 
9.81 650 
9-81 665 
9- 81 680 



9. 81 694 



Log. Cos, 



d. 



15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
14 
15 

15 
15 
15 

15 
14 

15 
15 
15 
14 
15 

15 
14 
15 
15 
14 

15 
14 
15 
15 
14 

15 
14 
15 

14 
14 

15 
14 
15 
14 
14 

15 
14 
14 
15 

14 

14 
14 
15 
14 
14 

14 
14 

14 
14 
15 

14 



Log. Tan. 



92 381 
92 407 
92 432 
92 458 
92 434 



92 509 
92 53') 
92 581 
92 586 
92 612 



92 638 
92 683 
92 689 
92 714 
92^740 

92 788 
92 791 
92 817 
92 842 
92 818 



92 894 
92 919 
92 945 
92 971 
92 996 



93 022 
93 047 
93 073 
93 098 
93 124 



93 150 
93 175 
93 201 
93 226 
93 252 



93 278 
93 303 
93 329 
93 354 
93 380 

93 405 
93 431 
93 456 
93 482 
93 508 



93 533 
93 559 
93 584 
93 610 
93 635 



93 661 
93 886 
93 712 
93 737 
93 763 



93 788 
93 814 
93 840 
93 865 
93 891 



93 916 



c.d. 

25 
25 
26 
25 

25 
25 
28 
25 
25 

26 
25 
25 

25 
28 

25 
25 
25 
25 
26 

25 
25 
25 
28 
25 

25 
25 
25 
25 
26 

25 
25 
25 
25 
25 

26 

25 
25 
25 
25 

25 
25 
25 
25 
26 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
26 
25 
25 

25 



Log. Cot. [Log. Cos. 



0-07 618 9 
0-07 593|9 
0-07 537 
0-07 541 
007 516 



0-07 490 
0-07 435 
0. 07 439 
0.07 413 
007 388 



07 362 
007 336 
0-07 311 
007 285 
007 259 



0.07 234 
. 07 208 
007 183 
0-07 157 
007 131 



0-07 108 
0-07 030 
007 055 
0-07 029 
0-07 003 



0-06 978 
006 952 
006 927 
0-06 901 
006 875 



0.06 850 
006 824 
0.06 799 9 
0.06 773 
0.06 748 



0.08 722 
008 696 
006 671 
008 645 
0-06 620 



006 594 
006 569 
0.06 543 
0.08 518 
0.06 492 



0.06 486 
. 08 441 
008 415 
0.08 390 
008 3R2 



0-06 339 
0-06 313 
0.06 288 
0.08 262 
0-06 237 



0-06 211 
006 186 
0-06 160 
0-06 134 
0-06 109 



n.Ofi 083 



d. Log. Cot. c.d. Log. Tan. 



88 425 
88 415 
88 404 
88 393 
88 383 



88 372 
88 381 
88 351 
88 340 
88 329 



88 319 
88 308 
88 297 
88 287 

88 276 



88 265 
88 255 
88 244 
88 233 
88 223 



88 212 
88 201 
88 190 
88 180 
88 169 



88 158 
88 147 
88 137 
88 126 
88 115 



88 104 
88 094 
88 083 
88 072 
88 061 



88 050 
88 039 
88 029 
88 018 
88 007 



87 996 
87 985 
87 974 
87 963 
87 953 



87 942 
87 931 
87 920 
87 909 
87 898 



87 887 
87 876 
87 865 
87 854 
87 844 



87 833 
87 822 
87 811 
87 800 
87 789 



9-87 778 



Log. Sin. 



d. 



10 
11 
10 
10 

10 
11 
10 
10 
11 

10 
10 
11 
10 
10 

11 

10 

10 
11 
10 

11 
1(5 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 
11 

10 

11 
11 

10 

11 
11 

10 

11 
11 
11 

10 

11 
11 
11 

10 

11 

11 
11 
11 
11 

10 

11 
11 
11 
11 
11 

11 



60 

59 
58 
57 
56_ 

55 
54 
53 
52 
51 

50 

49 

48 

47 

il 

45 

44 

43 

42 

41_ 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 



d. 



10 

9 
8 
7 
6 



P. P. 





26 


25 


6 


2-6 


2-5 


7 


3 





3 





8 


3 


4 


3 


4 


9 


3 


9 


3 


8 


10 


4 


3 


4 


2 


20 


8 


6 


8 


5 


30 


13 





12 


7 


40 


17 


3 


17 





50 


21 


6 


21 


2 





15 


15 


1^ 


6 


1.5 


1.5 


1. 


7 


1 


8 


1 


7 


1- 


8 


2 





2 





1. 


9 


2 


3 


2 


O 

£J 


2- 


10 


2 


6 


2 


5 


2- 


20 


5 


1 


5 





4- 


30 


7 


7 


7 


5 


7. 


40 


10 


3 


10 





9. 


50 


12 


9 


12 


5 


12- 





11 


10 


6 


1.1 


1-0 


7 




3 




2 


8 




A 




4 


9 




6 




6 


10 




8 




7 


20 


3 


6 


3 


5 


30 5 


5 5 


2 


40,7 


3 7 





50 


9 


1 


8 


7 



P. p. 



130° 



703 



49*= 



41' 



^ABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 133« 



7 

8 

_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 

ii 

20 

2J 

22 

23 

24 

25 
26 
27 
28 
29 

30 

31 
82 
83 
34 

35 
36 
37 
38 
39_ 

40 9 



Log. Sin. 



9-81 694 
9. 81 709 
9-81 723 
9-81 738 
9. 81 752 



9-81 767 
9. 81 78l 
9.81 798 
9.81 810 
9-81 824 



9. 81 839 
9. 81 853 
9. 81 868 
9-81 882 
9-81 897 



9-81 911 
9. 81 925 
9. 81 940 
81 954 
9. 81 969 



81 983 

81 997 

82 012 
82 026 
82 040 



9.82 055 
9.82 069 
9.82 083 
9. 82 098 
9-82 112 



9.82 126 
9.82 140 
9.82 155 
9.82 169 
9.82 183 



82 197 
82 212 
82 226 
82 240 
82 254 



41 
42 
43 
44 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



82 269 
82 283 
82 297 
82 311 
82 325 



■82 339 
82 354 
82 368 
82 382 
82 396 



9-82 410 
9.82 424 
9. 82 438 
9.82 452 
9.82 467 



82 481 
82 495 
9. 82 509 
9. 82 523 
9-82 537 

9-82 551 

Log. Cos. 



d. JLog. Tan, 



14 
14 

li 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 

14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

12 
14 

14 
14 
14 

14 
14 
14 
li 
14 

14 
14 
14 
14 
14 

14 
12 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 



93 916 

• 93 942 
93 967 

93 993 
.94 018 

. 94 044 
.94 0fi9 

94 095 

• 94 120 

• 94 146 



94 171 
94] 97 
94 222 
94 248 
.94 273 



• 94 299 
94 324 

.94 350 
•94 375 

• 94 400 



• 94 426 

• 94 45! 

• 94 477 

• 94 502 

• 94 528 



• 94 553 

• 94 579 
. 94 604 

• 94 630 

• 94 655 



• 94 681 
•94 706 
•94 732 
•94 757 

• 94 782 



C.d. 



• 94 808 

• 94 833 

• 94 859 

• 94 882 

• 94 910 



• 94 935 
.94 961 

■ 94 986 

■ 95 Oil 
95 037 

95 062 
95 088 
95 113 
95 139 
95 164 



■95 189 

■ 95 215 

• 95 240 

95 266 

95 291 



95 316 
95 342 
95 367 
95 393 
95 418 

95 443 



d. Log. Cot. c. d 



25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



Log. Cot. 



0^ 06 083 
006 058 
0^ 06 032 
0-06 007 
0-05 981 



0-05 956 
05 930 
005 905 
05 879 
05 854 



0-05 82P 
0^05 80S 
005 77? 
0.05 752 
0.05 726 



005 701 
0.05 675 
0-05 650 
0^05 625 
■ 05 599 

0^05 574 
05 548 
0^05 523 
0^05 497 
0^ 05 472 



0^ 05 446 
05 42] 
©•05 395 
005 37C 
0^05 344 



005 3ie 987 
0.05 293 9.87 



Log. Cos. 



9. 87 
9. 87 
9-87 
9.87 
9.87 



9.87 
9. 87 
9. 87 
9-87 
9 87 



778 
767 
756 
745 

73A 

723 
712 
701 
690 
679 



9 87 
987 
9. 87 
9. 87 
9 •87 



9 87 
9^87 
9-87 
987 
9. 87 

987 
987 
9^87 
9^87 
9 87 



9^87 
987 
987 
9^87 
9. 87 



668 
657 
645 
634 
623 
612 
601 
590 
579 
568 

557 
546 
535 
523 
512 

501 
490 
479 
468 
457 



05 268 
0^05 243 
005 217 



0^05 192 
05 166 
0^05 141 
005 115 
0^05 090 

0^05 064 
0^05 039 
©•05 014 
04 988 
04 963 



9. 87 
9.87 
9.87 



0^04 
0^04 
0^04 
0^04 
04 



87 
87 
87 
87 
87 



445 
434 
423 
412 
401 

38§ 
378 
367 
356 
345 



987 
9^87 
9. 87 
9^87 
987 



333 
322 
311 
300 
28R 



0.04 
0.04 
0.04 
0.04 
0.04 



937 
912 
886 
861 
836 

81C 
785 

759 
734 
708 



9. 87 
9.87 
9.87 
9. 87 
9. 87 



277 
266 
252 
243 
232 



004 683 
0.04 658 
004 632 
. 04 607 
04 581 

0-04 556 

Log. Tan. 



87 
9^87 
9 87 
9^87 
9^87 



987 
9. 87 
9. 87 
987 
9-87 

9 87 



221 
209 
198 
187 
175 

164 
153 
14l 
130 
11| 

107 



d.' 



Log. Sin, d. 



11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
ll 

11 
11 

11 
11 
11 

11 
11 
11 
11 
11 

ll 
11 
ll 

11 
11 

ll 
11 
11 
11 
11 

ll 
11 
11 
11 
11 

ll 
11 
11 
11 
11 
11 
11 
ll 
11 
11 

11 
11 
11 
11 
11 

11 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 



10 

9 
8 
7 
6 



P.P. 



6 

7 
8 
9 

10 
20 
30 
40 
50 



25 

2 

3 

3. 

3 

4- 

8 

12. 

17^ 

21. 



25 



•5 


2 


5 





2 


9 


• 4 


3 


3 


• 8 

• 2 


3 

4 


1 


•5 


8 


3 


•7 


12 


5 


• 


16 


6 


;^ 


20 


8 





11 


14 


6 


1-4 


1.4 


7 


1 


7 


1 


6 


8 


1 


9 


1 


8 


9 


2 


2 


2 


1 


10 


2 


4 


2 


S 


20 


4 


8 


4 


6 


30 


7 


2 


7 





40 


9 


6 


9 


3 


50 


12 


1 


11 


6 





11 


11 


6 


1^1 


l^] 


7 




3 


l-ii 


8 




5 


i.^:i 


9 




7 


l.Cj 


10 




9 


i.i 


20 


3 


8 


3.6 


30 


5 


7 5-5 


40 


7 


6 7.3 


50 


9 


6 


9 1 



P. v: 



131= 



704 



W 



TABLE VTI.— LOGARITHAJIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



137' 



Log. Sin. d. 



82 551 
82 565 
82 579 
82 593 
82 607 



82 621 
82 635 
82 649 
82 663 
82 677 



82 691 
82 705 
82 719 
82 733 

82 746 



82 760 
82 774 
82 788 
82 802 
82 816 

82 833 
82 844 
82 858 
82 871 
82 885 



82 899 
82 913 
82 927 
82 940 
82 954 



82 963 
82 932 

82 993 

83 009 
83 023 



83 037 
83 051 
83 064 
83 078 
83 092 



83 106 
83 119 
83 133 
83 147 
83 160 



83 174 
83 188 
83 20l 
83 215 
83 229 



83 242 
83 258 
83 269 
83 283 
83 297 



83 310 
83 324 
83 337 
83 351 
83 365 



9. 83 378 



14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
13 

14 
14 
14 

H 
13 

14 
14 
14 
13 
14 

14 
13 
14 
13 
14 

14 

13 
14 
13 
14 

13 
14 
13 
14 
13 

14 
13 
13 
14 
13 

13 
14 
13 
13 
14 

13 

13 
13 
14 
13 

13 
13 
13 
13 
14 

13 



Log. Tan. 



Log. Cos. d. 



443 
469 
494 
520 
545 

571 
596 
621 
647 
672 



95 
95 
95 
95 
95 



697 
723 
748 
774 
799 



824 
850 
875 
901 
926 



951 
977 
002 
027 
053 



96 
96 
96 
96 
96 



078 
104 
129 
154 
180 



96 
96 
96 
96 
96 



205 
230 
256 
281 
306 



332 
357 
383 
408 
433 



96 
98 
96 
96 
96 



459 
484 
509 
535 
560 



96 
98 
96 
96 
96 



585 
611 
636 
661 
687 



96 
96 
96 
96 
96 

96" 
96 
96 
96 
9_6_ 
96 



712 
737 
763 
788 
813 

839 
864 
889 
915 
940 
965 



c. d. Log. Cot. 



Log. Cot. c. d 



25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



04 556 
04 531 
04 505 
04 480 
04 454 



04 429 
04 404 
04 378 
04 353 
04 327 



04 302 
04 277 
04 251 
04 226 
04 200 



04 175 
04 150 
04 124 
04 099 
04 074 



04 048 
04 023 
03 997 
03 972 
03 947 



03 921 
03 896 
03 871 
03 845 
03 820 



03 795 
03 769 
03 744 
03 718 
03 693 



03 668 
03 642 
03 617 
03 592 
03 566 



03 541 
03 516 
03 490 
03 465 
03 440 



03 414 
03 389 
03 364 
03 3: 
03 313 



03 287 
03 262 
03 237 
03 211 
03 186 



03 161 
03 135 
03 110 
03 085 
03 059 



03 034 9 



Log. Cos. 



87 107 
87 096 
87 084 
87 073 
87 06 2 

87 050 
87 039 
87 027 
87 016 
87 004 



86 993 
86 982 
86 970 
86 959 
86 947 



86 936 
86 924 
86 913 
86 901 
86 890 



86 878 
86 867 
86 855 
86 844 
86 832 



86 821 
86 809 
86 798 
86 786 
86 774 

86 763 
86 75l 
86 740 
86 728 
86 716 



86 705 
86 693 
86 682 
86 670 
86 658 



86 647 
86 635 
86 623 
86 612 
86 600 



86 588 
86 577 
86 565 
86 553 
86 542 



86 530 
86 518 
86 507 
86 495 
86 483 



86 471 
86 460 
86 448 
86 436 
86 424 



86 41 2 



Log. Tah.jLog; Sin. 



11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

ll 
ll 
ll 
11 
11 

ll 
ll 
ll 
12 

11 

11 
11 
ll 
ll 

12 

11 
11 
ll 
ll 

12 

ll 
ll 
12 
11 
11 

12 
ll 
ll 
12 
11 

12 
11 
11 
12 
ll 
12 
ll 
12 
ll 
12 

12 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46. 
45 
44 
43 
42 
41_ 

40 

39 
38 
37 
36_ 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 

17 
16 
15 
14 
13 
12 

n 

10 

9 
8 

7 
6 

5 
4 
3 
2 
1 

O 



d. 



P. P. 



: 


25 25 


6 


2.5 


2.5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 


6 


50 


21 


2 


20 


8 





14 


13 


6 


1-4 


1-3 


7 


1 


6 


1 


6 


8 


1 


8 


1 


8 


9 


2 


1 


2 





10 


2 


3 


2 


2 


20 


4 


6 


4 


5 


30 


7 





6 


7 


40 


9 


3 


9 





50 


11 


6 


11 


2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



12 11 



1 


2 




1 




1 


4 




3 




1 


6 




5 




1 


8 




7 




2 







9 




4 





3 


8 


3- 


6 





5 


7 


5. 


8 





7 


6 


7- 


10 





9 


6 


9. 



11 

1 

3 

■ i 

6 
8 
6 
5 
■ 3 

I; 



p. P. 



133" 



705 



47" 



43° 



TABLE VII.— LOGARITHMIC SlNES, COSINES, TANGENTS, 
AND COTANGENTS. 





1 

2 

3 

_4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
1£ 

15 
16 
17 
18 
19_ 

30 

21 

22 

23 

21 

25 

26 

27 

28 

29, 

30 

31 

32 

33 

3i 

35 

36 

37 

38 

39 

40 

41 
42 
43 
4£ 

45 

46 

47 

48 

49L 

60 

51 

52 

53 

5£ 

55 

56 

57 

58 

5?_ 

fiO 



Log. Sin. 



83 378 
83 392 
83 405 
83 419 
83 432 



83 446 
83 459 
83 473 
83 486 
83 500 



83 513 
83 527 
83 540 
83 554 
83 567 

83 580 
83 594 
83 607 
83 621 
83 634 



83 647 
83 661 
83 674 
83 688 
83 701 



83 714 
83 728 
83 741 
83 754 
83 768 



83 781 
83 794 
83 808 
83 821 
83 834 



83 847 
83 861 
83 874 
83 887 
83 900 



83 914 
83 927 
83 940 
83 953 
83 967 



83 980 

83 993 

84 006 
84 019 
84 033 



84 046 
84 059 
84 072 
84 085 
84 098 



84 111 
84 124 
84 138 
84 151 
84 164 



84 177 



d. 



13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 



Log. Tan. 



Log. Cos. d. Log. Cot. 



96 985 

96 991 

97 016 
97 041 
97 067 



97 092 
97 117 
97 143 
97 168 
97 193 



97 219 
97 244 
97 269 
97 295 
97 320 



97 345 
97 370 
97 396 
97 421 
97 446 

97 472 
97 497 
97 522 
97 548 
97 573 



97 598 
97 624 
97 649 
97 674 
97 699 



97 725 
97 750 
97 775 
97 801 
97 826 



97 851 
97 877 
97 902 
97 927 
97 952 

97 978 

98 003 
98 028 
98 054 
98 079 



c.d. 



98 104 
98 129 
98 155 
98 180 
98 205 



98 231 
98 256 
98 281 
98 306 
98 332 



98 357 
98 382 
98 408 
98 433 
98 458 



9. 98 483 



25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 

25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



Log. Cot. 



03 034 
03 009 
02 984 
02 958 
02 933 



02 908 
02 882 
02 857 
02 832 
02 806 



02 781 
02 756 
02 730 
02 705 
02 680 9 



Log. Cos. 



02 654 
02 629 
02 604 
02 578 
02 553 

02 528 
02 502 
02 477 
02 452 
02 427 



02 401 
02 376 
02 351 
02 325 
02 300 



02 275 
02 249 
02 224 
02 199 
02 174 



02 148 
02 123 
02 098 
02 072 
02 047 



02 022 
01 996 
01 971 
01 946 
01 921 



01 89 F 
01 87C 
01 845 
01 819 
01 794 



01 769 
01 744 
01 718 
01 693 
01 668 



01 64? 
01 617 
01 592 
01 567 
01 541 



01 516 



86 412 
86 401 
86 389 
86 377 
86 365 



86 354 
86 342 
86 330 
86 318 
86 306 



86 294 
86 282 
86 271 
86 259 
86 247 



86 235 
86 223 
86 211 
86 199 
86 187 

86 176 
86 164 
86 152 
86 140 
86 128 



86 116 
86 104 
86 092 
86 080 
86 068 



86 056 
86 044 
86 032 
86 020 
86 008 



85 996 
85 984 
85 972 
85 960 
85 948 



85 936 
85 924 
85 912 
85 900 
85 887 



85 875 
85 863 
85 851 
85 839 
85 827 



85 815 
85 803 
85 791 
85 778 
85 766 



85 754 
85 742 
85 730 
85 718 
85 705 



d. 



85 693 



C.d. Log. Tan. Log. Sin. d. 



11 
12 
ll 
12 

ll 
12 
12 
11 
12 

12 
12 
11 
12 
12 

ll 
12 
12 
12 
12 

11 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 



60 

59 
58 
57 
56. 
55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 

27 
26 

25 
24 
23 
22 
21 

30 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 
7 
6 

5 
4 
3 
2 
1 

O 



P.P. 





25 


25 


6 


2.5 


2-5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 


6 


50 


21 


2 


20 


8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



13 

1-3 



13 



1 
1 
1 
1 
2 
4 
6 
8 
110 



■ 3 

•5 

• 7 
•9 

1 

• 3 
•5 

6 

• 8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



12 12 



?.\ 



09 



P. P. 



133° 



706 



C14° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



135° 



JLog. Sin. 





1 

2 
3 
± 

5 
6 
7 
8 

to 

LI 
L2 
L3 
L4 

L5 
L6 9 



9-84 177 
9.84 190 
9.84 203 
9. 84 216 
9.84 229 



9 . 84 242 
9. 84 255 
9.84 268 
9. 84 281 
9.84 294 



17 
L8 

LI 
30 

21 
22 
23 
2i_ 

25 

26 

27 

28 

21 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39_ 

40 9 



84 307 
84 320 
84 333 
84 346 
■ 84 359 

.84 372 
84 385 
84 398 
84 411 

.84 424 



84 437 
84 450 
84 463 
84 476 
84 489 



84 502 
84 514 
84 527 
84 540 
84 553 



84 566 
84 579 
84 592 
84 604 
84 617 



41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



84 630 
84 643 
84 656 
84 669 
84 681 

84 694 
84 707 
84 720 
84 732 
84 745 



84 758 
84 771 
84 783 
84 796 
84 809 



84 822 
84 834 
84 847 
84 860 
84 872 



84 885 
84 898 
84 910 
84 923 
84 936 



9. 84 948 



Log. Cos. 



d. 



13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

12 
13 
13 
13 
13 

13 
12 
13 
13 
13 

12 
13 
13 
12 
13 

13 

12 
13 
13 
12 

13 
12 
13 
12 
13 

12 
13 
12 
13 
12 

13 
12 
12 
J.3 
12 

12 
13 
12 
12 
13 

12 



Log. Tan. 



98 433 
98 509 
98 534 
98 559 
98 585 



98 610 
98 635 
98 660 
98 68& 
98 711 



98 736 
98 762 
98 787 
98 812 
98 837 



98 883 
98 888 
98 913 
98 938 
98 964 

98 989 

99 014 
99 040 
99 065 
99 090 



99 115 
99 141 
99 166 
99 191 
99 216 



99 242 
99 267 
99 292 
99 318 
99 343 



C.d. 



99 368 
99 393 
99 419 
99 444 
99 469 

99 494 
99 520 
99 545 
99 570 
99 595 



99 621 
99 846 
99 671 
99 697 
99 722 



99 747 
99 772 
99 798 
99 823 
99 848 



99 873 
99 899 
99 924 
99 949 
99 974 



00 onn 



d. Log. Cot. 



25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 

25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



Log. Cot. 



01 516 
01 491 
01 465 
01 440 
01 415 



01 390 

01 364 
01 339 
01 314 
01 289 



Log. Cos. 



85 693 
85 681 
85 669 
85 657 
85 644 



9. 



01 263 
01 238 
01 213 
01 187 
01 162 



01 137 
01 112 
01 086 
01 061 
01 036 



01 010 
00 985 
00 960 
00 935 
00 909 



00 884 
00 859 
00 834 
00 808 
00 783 



00 758 
00 733 
00 707 
00 682 
00 657 



85 632 
85 620 
85 608 
85 595 
85 583 



85 571 
85 559 
85 546 
85 534 
85 522 



9. 



85 509 
85 497 
85 485 
85 472 
85 460 

85 448 
85 435 
85 423 
85 411 
85 398 



85 386 
85 374 
85 361 
85 349 
85 336 



85 324 
85 312 
85 299 
85 287 
85 274 



00 631 
00 606 
00 581 
00 556 
00 530 9 

9. 
9 



00 505 
00 480 
00 455 
00 429 
00 404 



00 379 
00 353 
00 328 
00 303 
00 278 



00 252 
00 227 
00 202 
00 177 
00 151 



00 126 
00 101 
00 07P 
00 050 
00 025 
00 000 



85 262 
85 249 
85 237 
85 224 
85 212 

85 199 
85 187 
85 174 
85 162 
85 149 



85 137 
85 124 
85 112 
85 099 
85 C87 



85 074 
85 062 
85 049 
85 037 
85 024 



85 Oil 
84 999 
84 986 
84 974 
84 961 

84 948 



134' 



c.d. Log. Tan. Log. Sin. 
707 



d. 



12 
12 
12 
12 

12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 

12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
13 

12 
12 
12 
12 
13 
12 

T 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 

47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 

11 
35 
34 
33 
32 
IL 
30 
29 
28 
27 
2i 
25 
24 
23 
22 

21. 
20 

19 
18 
17 

li 

15 
14 
13 

12 

n 

10 

9 



P. p. 



25 



6 
7 
8 
9 

10 
20 
30 
40 
50 



2 


5 


2. 


3 





2. 


3 


4 


3 


3 


8 


3 


4 


2 


4. 


8 


5 


8. 


12 


7 


12. 


17 





16. 


21 


2 


20. 



25 

5 
? 
3 
7 
1 
3 
5 
6 
3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



13 



13 


1. 


1 


6 


1. 


1 


8 


1. 


2 





1. 


2 


2 


2. 


4 


5 


4. 


6 


7 


6. 


9 





8. 


111 


2 


10. 



13 

3 

5 
7 
p 

"i 

3 
5 
6 
8 



7 
8 
9 
10 
20 
30 
40 
50 



12_ 13 

2 
4 
6 
8 








1.2 


1. 


1 


4 


1. 


1 


6 


1. 


1 


9 


1. 


2 


] 


2. 


4 


1 


4- 


6 


2 


6. 


8 


3 


8. 


10 


4 


10. 



P. p. 



45' 



TABLE VIII .— LOGARitHMlC VERSED SINES AND EXTERNAL 
0° SECANTS. 1° 



Log. Vers. 



62642 
22848 
58066 
83054 



02436 
18272 
31682 
43260 
53490 



62842 
70920 
78478 
85431 
91868 



97860 
03468 
08732 
13696 
18393 



22848 
27086 
31126 
34987 
38884 



42230 
45636 
48915 
52073 
55121 



58066 
60915 
63872 
66344 
68937 



71455 
73902 
76282 
78598 
80854 



83053 

85198 

87291 

89335 

il332_ 

93284 

95193 

97061 

98890 

00680 



02435 
04155 
05842 
07496 
09120 



1071.4 
12Z79 
13816 
15327 
16811 



D 



6 18271 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 

9151 
8278 
7558 
6953 
6437 

5992 
5605 
5266 
4964 
4696 

4455 
4238 
4040 
3861 
3697 

3545 
3406 
3278 
3l58 
3048 

2944 
2848 
2757 
2672 
2593 

2518 
2447 
2379 
2316 
2256 

2199 
2145 
2093 
2044 
1996 

1952 
1909 
1868 
1829 
1790 

1755 
1720 
1686 
1654 
1623 

1594 
1565 
1537 
1511 
1484 
1460 



Log. Vers, x> 



Log. Exsec. 


— 00 

2.62642 
3.22848 
3.58066 
3. 83054 


4.0243g 
.18272 
•31662 
•43260 
•53491 


4.62642 
.70921 
.78478 
.85431 
.91868 


4.97861 

5.03466 

.08732 

.13697 

.18393 


5.22849 
•27087 
.31127 
•34988 
•38685 


5.42231 
.45638 
.48916 
.52075 
.55123 


5. 58068 
.60916 
.63674 
.66346 
.68940 


5.71457 
.73904 
.76284 
.78601 
•80857 


5.83056 
•85201 
•87295 
.89338 
•91335 


5 93288 
•95197 
.97065 

5. 98894 

6 00685 


8 . 02440 
.04160 
.05847 
.07501 
.09125 


6.10719 
.12284 
•13822 
•15333 
.16818 

6.18278 


Log. Exsec. 



J> 



60206 
35218 
24987 

19382 
15836 
13389 
1159§ 
10230 

9151 
8279 
7557 
6952 
6437 

5993 
5605 
5266 
4964 
4696 

4456 
.4238 
4040 
3861 
3697 

3545 
3407 
3278 
3159 
3048 

2945 
2848 
2758 
2672 
2593 

2517 
2447 
2380 
2316 
2256 

2199 
2145 
2093 
2043 
1997 

1952 
1909 
1868 
1829 
1791 

1755 
1720 
1687 
1654 
1623 

1594 
1565 
1537 
1511 
1485 

1460 



I) 



Log. Vers, 



6. 18271 

19707 

21119 

.22509 

23877 



6 



25223 
26549 
27856 
29142 
30410 



31669 
32892 
34107 
35305 
38487 



37653 
38803 
39938 
41059 
42165 



43258 
44337 
45403 
46455 
47498 



48524 
49539 
50544 
51536 
52518 



53488 
54448 
55397 
56336 
57255 



58184 
59093 
59993 
60884 
61706 

62639 
63503 
64359 
65206 
66045 



66876 
67700 
68515 
69323 
70124 



70917 
71703 
72482 
73254 
74019 



74777 
75529 
76275 
77014 
77747 



6. 78474 
Log. Vers, 



1435 
1412 
1389 
1368 

1346 
1326 
1306 
1286 
1268 

1250 
1232 
12l4 
1198 
1182 

1166 
1150 
1135 
1121 
1106 

1093 
1078 
1066 
1052 
1040 

1028 

1016 

1004 

992 

981 

970 
960 
949 
939 
929 

919 
909 
900 
891 
882 

872 
864 
855 
84? 
839 

831 
823 
815 
808 
800 

793 
786 
779 
772 
785 

758 
752 
745 
739 
733 

726 



Log. ExseCi 



18278 
19714 
21126 
22516 
2^884 

25231 
26557 
27864 
29151 
30419 



31669 
32901 
34116 
35315 
36497 



37663 
38814 
39949 
41070 
42177 



43270 
44349 
45415 
46468 
47509 



48537 
49553 
50557 
51550 
52532 



53503 
54463 
55413 
56352 
57281 



58201 
59110 
60011 
60902 
61784 



62657 
63522 
64378 
65226 
66065 



66897 
67720 
6853B 
69345 
70145 



70939 
71725 
72505 
73277 
74043 



74802 
75554 
76300 
77040 
77773 



.6.785P0 



Log. Exsec. 



708 



TABLE VIII —LOGARITHMIC VERSED SINES AND EXTERNAL 

3° . SECANTS. a° 



o 

1 

2 
3 

4 

5 
6 
7 
8 
J_ 

10 

u 

12 
13 
14 

15 
16 
17 
18 

iL 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 



Log. Vers. 



6-78474 
.79195 
.79909 
.80618 
.81322 



n 



6 



82019 
82711 
83398 
.84079 
l84755_ 

.85425 
.86091 
.86751 
.87407 
88057 



88703 
.89344 
.89980 
.90612 
.91239 



.91862 
.92480 
.93093 
.93703 
.9430a 



Log. Exsec. -O Log. Vers. 



30 

31 
S(2 
33 
34 



.94909 
.95506 
.96099 
.96688 
.9727? 



6.97853 

.98430 

.99004 

6. 99573 

7. 00139 



35 

36 

37 

38 

39 

40 

41 

42 

43 

U_ 

45 
46 
47 
48 
49_ 

50 

51 
92 
53 
5^ 

55 
56 
57 
58 
59 

60 



7.00701 
.01259 
.01814 
.02366 
.0^914 



7 03458 
.03999 
.04537 
.05071 
.n5R03 



706130 
.06655 
.07177 
.07695 
.08211 



7.n87:'.^ 

.09^32 
.09739 
.10242 
.10743 



7-11240 
.11735 
.12227 
.12716 
.]3203 



7-13687 



721 
714 
709 
703 

697 
692 
686 
681 
676 

670 
665 
660 
655 
650 

646 
641 
636 
631 
627 
622 
618 
613 
609 
605 

601 
597 
592 
589 
584 

581 

577 
573 
569 
565 

562 
558 
555 
551 
548 

544 
541 
537 
534 
531 

527 
525 
521 
518 
515 
51 C> 
509 
5nR 
50S 
500 

497 
495 
492 
489 
486 

484 



6-78500 
.79221 
.79937 
.80646 

^81350 

6 -82048 
.82740 
.83427 
.84109 
.84785 



Log. Vers. 



6-85457 
.86123 
.86783 
.87439 
.88090 



6-88737 
.89378 
.90015 
.90647 
■91275 

6-91898 
.92516 
.93131 
.93741 
.94346 

6-94948 
.95545 
.96139 
.96728 

_^97313 

6.97895 

.98472 

.99046 

6-99616 

7-00182 



7-00745 
.01304 
.01860 
.02412 
■02960 

7-03505 
.04047 
.04585 
.05120 
-05652 



7-06180 
.06706 
.07228 
.07747 
■08263 



708776 
■09286 
.09793 
.10297 
-10798 



I> 



7-11297 
.11792 
.12285 
.12775 
.13262 



721 
715 

709 
703 

698 
692 
687 
682 
676 

67i 



7-13746 



Log. Exsec 



D 



7-13687 
-14168 
.14646 
.15122 
.15595 



7> 



660 
656 
651 

646 
641 
636 
632 
628 

623 
618 
614 
610 
605 

601 
597 
593 
589 
585 

581 

577 
574 
570 
566 

563 
559 
555 
552 
548 

545 
541 
538 
535 
531 

528 
525 
522 
519 
516 
513 
509 
507 
503 
501 

498 
495 
493 
490 
487 

484 



7.16066 
.16534 
.17000 
.17463 

_a7923 

7.18382 
.18837 
.19291 
.19742 

_^20191 

7-20637 
.21081 
.21523 
.21963 
.22400 



7-22836 
.23269 
.23700 
.24129 
.245 55 

7-24980 
.25402 
.25823 
.26241 
■26658 

7-27072 
.27485 
.27895 
.28304 
-28711 



7-29116 
.29518 
.29919 
.30319 
■30716 

7-31112 
.31505 
.31897 
.32288 
■ 32676^ 

7 ■33^63^ 
■33448 
.33831 
.34213 
■34593 



7 34971 
■35348 
■35723 
■36097 
■36468 



7-36839 

-37207 
■37574 
■37940 
■38304 



7.386ft7 
Log. Vers. 



Log. Exsec. 



481 
478 
475 
473 

470 
468 
466 
463 
460 

458 
455 
453 
451 
448 

446 
444 
442 
440 
437 

435 
433 
431 

429 
426 

424 
422 
420 
418 
416 

414 
412 
410 
4G? 
406 

405 
402 
401 
399 
397 

395 
393 
392 
390 
388 

386 
385 
383 
382 
380 

378 
377 
375 
373 
371 

370 
368 
367 
368 
364 

362 

-^ 



7 ■ 13746 
.14228 
.14707 
.15183 
■15657 



7-16129 
.16598 
.17064 
.17528 

_^17989 

7-18448 
.18905 
.19359 
.19811 
.20260 



7-20707 
-21152 
-21595 
.22035 
■22473 



7-22909 
■23343 
■23775 
■24204 
■24632 



7 ■25057 
■25480 
.25902 
.26321 
■26738 



7^27153 
.27567 
.27978 
.28387 

_^879^ 

7 29200 
.29604 
■30006 
■30406 
.3rPC4 



7-31201 
-31595 
-31988 
■32379 
■32768 



733156. 
.33542 
.33926 
.34309 
.34689 



7-35069 
.35446 
.35822 
.36196 
.36569 



J> 



481 
479 
476 
474 

47l 



7-36940 
-37310 
■37678 
.38044 

_^3845i- 
"7^38773 



466 
464 
461 

459 
456 
454 
452 
449 

447 
445 
442 
440 
438 

436 
434 
431 
429 
427 
425 
423 
42l 
419 
417 

415 
413 
4ll 
409 
407 

405 
404 
402 
400 
398 

396 
394 
393 
391 
389 

388 
385 
384 
382 
380 

379 
377 
376 
374 
373 

371 
369 
368 
366 
365 

363 



Log. Exsec, 



D 





1 

2 

3 

_^ 

5 
6 
7 
8 
_9 

10 

11 

12 

13 

JA 

15 
16 
17 
18 

JJ 
20 

21 

22 

23 

_24 

25 
26 
27 
28 
J% 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 

45 
48 
47 
48 
j49 

50 

51 

52 

53 

_54 

55 
56 
57 
58 
J9 
60 



709 



TABLE VIII —LOGARITHMIC VERSED SINES ANb EXTERNAL SECANTS. 
4° 5° 



Lg. Vers, I> Log.'Exs, D Lg. Vers, J> 



38667 
39028 
39387 
39745 
40102 



40457 
40810 
41163 
41513 
41863 



42211 
42557 
42903 
43246 
43589 



43930 
44270 
44608 
44946 
45281 



45616 
45949 
46281 
46612 
46941 



47270 
47597 
47922 
48247 
48570 



48892 
49213 
49533 
49852 
50169 



50485 
50800 
51114 
51427 
51739 



52050 
52359 
52667 
52975 
53281 



53586 
53890 
54193 
54495 
54796 



55096 
55395 
55692 
55989 
56285 



56580 
5687^ 
57166 
57458 
57749 



58039 



' Lg. Vers. J^ 



361 
359 
358 
356 

355 
353 
352 
350 
349 

348 

346 
345 
343 
342 

341 
339 
338 
337 
335 

334 
333 
332 
330 
329 

328 
327 
325 
324 
323 

322 
321 
320 
318 
317 

316 
315 
314 
313 
311 

31.1 
309 
308 
307 
306 

305 
304 
303 
302 
300 

300 
299 
297 
297 
295 

295 
293 
293 
292 
290 

290 



7-38773 
39134 
39495 
39854 
40211 



40567 
40922 
41275 
41627 
41977 



42326 
42673 
43019 
43364 
43708 



44050 
44390 
44730 
45068 
45405 



45740 
46075 
46407 
46739 
47070 

47399 
47727 
48054 
48379 
48703 



49026 
49348 
49669 
49989 
50307 



50624 
50941 
51256 
51569 
51882 



52194 
52504 
52814 
53122 
53429 



53735 
54041 
54345 
54648 
54950 



55251 
55550 
55849 
56147 
56444 



56740 
57035 
57329 
57621 
57913 



58204 



Log. Exs, I> 



361 
360 
359 
357 

356 
354 
353 
352 
350 

349 
347 
346 
345 
343 

342 
340 
339 
338 
337 

335 
334 
332 
332 
330 

329 
328 
327 
325 
324 

323 
822 
321 
319 
318 

317 
316 
315 
313 
313 

311 
310 
309 
308 
307 

306 
305 
304 
303 
302 

301 
299 
299 
298 
296 

296 
295 
294 
292 
292 

291 



58039 
58328 
58615 
58902 
59188 



59473 
59758 
60041 
60323 
60604 



60885 
61164 
61443 
61721 
61998 



62274 
62549 
62823 
63096 
63369 



63641 
63911 
64181 
64451 
64719 



64986 
65253 
65519 
65784 
66048 



66311 
66574 
66836 
67097 
67357 



67617 
67875 
68133 
68390 
68647 



68902 
69157 
69411 
69665 
69917 



70169 
70421 
70671 
70921 
71170 



71418 
71666 
71913 
72159 
72404 



72649 
72893 
73137 
73379 
73621 



738R3 



Lg. Vers. 



289 
287 
287 
286 

285 
284 
283 
282 
281 

280 
279 
279 
277 
277 

276 
275 
274 
273 
272 

272 
270 
270 
269 
268 

267 
266 
266 
265 
264 

263 

263 
26i 
261 
260 

259 
258 
258 
257 
256 

255 
255 
254 
25S 
252 

252 
251 
25C 
25C 
249 

248 
247 
247 
246 
245 

245 
244 
243 
242 
242 

24l 
1i 



Log. Exs. D 



58204 
58494 
58783 
59071 
59358 



59645 
59930 
60214 
60498 
60780 



61062 
61342 
61622 
61901 
62179 



62456 
62733 
63C08 
63282 
63556 



63829 
64101 
64372 
64643 
64912 



65181 
65449 
65716 
65982 
66247 



66512 
66776 
67039 
67301 
67562 



67823 
68083 
68342 
68601 
68858 



69115 
69371 
69627 
698C1 
70135 



70388 
70641 
7C8G3 
71144 
71394 



71644 
71892 
72141 
72388 
72635 



72881 
73126 
73371 
73615 
73859 



74101 



Log. Exs, 



290 
289 
288 
287 

286 
285 
284 
283 
282 

281 
280 
280 
279 
278 

277 
276 
275 
274 
274 

273 
272 
271 
270 
269 

269 
268 
267 
266 
265 
264 
264 
263 
262 
261 

261 
260 
25S 
258 
25? 

257 
256 
255 
254 
254 

253 
252 
252 
251 
25C 

250 
248 
248 
247 
246 

246 
245 
245 
244 
243 

242 

17 





1 

2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



360 

36. 

42 



6 

7 

8 

9 

10 

20 

30 

40 

50 



48 
54 
60 
120 
180 
240 
300 



350 

35 

40 

46 

51 

58 
116 
175 
233 
291 





830 


330 


310 


6 


330 


32-0 


31. o: 


7 


38 


5 


37 


3 


36 


1 1 


8 


44 





42 


6 


41 


3| 


9 


49 


5 


48 





46 


5! 


10 


55 





53 


3 


51 


6 


20 


110 





106 


6 


103 


3 


30 


165 





160 





155 





40 


220 





213 


3 


206 


6 


50 


275 





266 


6 


258 


3 





300 


390 


38( 


6 


30.0 


29.0 


28. 


7 


35 





33 


8 


32- 


8 


40 





38 


6 


37- 


9 


45 





43 


5 


42. 


10 


50 





48 


3 


46. 


20 


100 





96 


6 


93. 


30 


150 





145 





140. 


40 


200 





193 


3 


186. 


50 


250 





241 


6 


233. 





370 


360 


35( 


6 


27.0 


280 


25. 


7 


31 


5 


30 


3 


29. 


8 


36 





34 


6 


33. 


9 


40 


5 


39 





37. 


10 


45 





43 


3 


41. 


20 


90 





86 


6 


83- 


30 


135 





130 


125. 


40 


180 





173 


3 166. 


50 


225 





216 


6 


208. 





340 


33 





33 


6 


24. 


23. 


22. 


7 


28 





26 


8 


25- 


8 


32 





30 


6 


29. 


9 


36 





34 


5 


33- 


10 


40 





38 


3 


36- 


20 


80 





7% 


6 


73. 


30 


120 





115 





110. 


40 


160 





153 


3 


146. 


50 


200 





191 


6 


183. 



310 

21.0 

24 

28 

31 

35 

70 
105 
140 
175 



300 


3 9( 


20-0 


19- 


23 


3 


22. 


26 


6 


25. 


30 





28. 


33 


3 


31- 


66 


6 


63. 


100 





95. 


133 


3 


126. 


166 


6 


158. 



P. p. 



710 



TA.BLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT3, 



6° 



O 

1 
2 
3 

5 
6 
7 
8 
J^ 

10 

11 
12 
13 
ii 
15 
16 
17 
18 
19 



Lg. Vers. -O 



30 

21 
22 
23 
24 



25 

26 

27 

28 

29. 

30 

31 

32 

33 

51 

35 

36 

37 

38 

39_ 

40 
4i 
42 
43 

§1 

45 

48 

47 

48 

49^ 

50 

51 
52 
53 
5£ 

55 

56 

57 

58 

59, 

60 



73863 
74104 
74344 
74583 
74822 



75060 
75297 
75534 
75770 
76006 

76240 
76475 
76703 
76941 
77173 



77405 
77636 
77867 
78097 
78326 



78554 
78783 
79010 
79237 
79463 



79689 
79914 
80138 
80362 
80586 



80808 
81031 
81252 
81473 
81694 



81914 
82133 
82352 
82570 
82788 



83005 
83222 
83438 
83653 
83868 



84083 
84297 
84510 
84723 
84935 



85147 
85359 
85570 
85780 
85990 



86199 
86408 
86616 
86824 
87031 



87238 



241 
240 
239 
239 

238 

237 
236 
236 
235 

234 
234 
233 
233 
232 

232 
231 
230 
230 
229 

228 
228 
227 
227 
226 

225 
225 
224 
224 
223 

222 
222 
22l 
221 
220 

220 
219 
219 
218 
217 

217 
217 
216 
215 
215 

214 
214 
213 
213 
212 

212 
2ll 
211 
^10 
210 

209 
209 
208 
208 
207 
206 



Log.Exs. 



■74101 
.74343 
.74585 
.74826 
•75066 



.75305 
.75544 
.75782 
.76019 
.76256 



'.76492 
.76728 
.76963 
.77197 
.77431 



'.77664 
.77897 
.78128 
•78360 
•78590 



2> 



•78820 
•79050 
.79279 
.79507 
79735 



•79962 

.80188 

.80414 

80639 

80864 



■81088 

.81312 

.81535 

81758 

81980 



7. 



82201 
82422 
82642 
82862 
83031 

83300 
83518 
83735 
83952 
84169 



7. 



84385 
84600 
84815 
85030 

85243 



7. 



85457 
85670 
85882 
86094 
86305 



86516 
•86726 
.86936 
.87146 

87354 



7-87563 



Lg. Vers. -OjLog.Exs. -£>|Lg. Vers 



Lg. Vers. D 



242 
241 
241 
240 

239 
239 
238 

237 
237 

236 
235 
235 
234 
233 

233 
232 
231 
231 
230 

230 
229 
229 
228 
228 

227 
226 
226 
225 
225 

224 
224 
223 
222 
222 

221 
221 
220 
219 
219 

219 
218 
217 
217 
216 

216 

215 
215 
214 
213 

213 
213 
212 
211 
211 

211 
210 
210 
209 
208 

208 



•87238 
• 87444 
.87650 
.87855 
•88060 

•88264 
.88468 
.88672 
.88875 
.89 077 

.89279 
.89481 
.89682 
.89882 
.'9008 2 

•90282 
.90481 
•90680 
•90878 
•91076 



.91273 
•91470 
.91667 
.91863 
•92058 



.92253 
. 92448 
•92642 
.92836 
.93029 



•93222 
•93415 
.93607 
.93799 
93990 



.94181 
.94371 
.94561 
.94751 
.94940 



.95129 
•95317 
•95505 
•95693 
•95880 



96066 

96253 

.96439 

.96624 

96809 



.96994 
.97178 
•97362 
•97546 
•97729 



.97912 
98094 
98276 

.98458 
98639 



7-98820 



206 
205 
2u5 
204 

204 
204 
203 
203 
202 

20? 
201 
201 
200 
200 

199 
199 
198 
198 
197 

197 
197 
196 
196 
195 

195 
195 
194 
194 
193 

193 
192 
192 
191 
191 

190 
190 
190 
189 
189 

189 
188 
187 
188 
187 

186 
186 
186 
185 
185 
184 
184 
184 
183 
183 

183 
182 
182 
182 
181 

181 



Log.Exs. 



•87563 
.87771 
.87978 
.88185 
^88391 
•88597 
.88803 
.89008 
.89212 
.89416 



•89620 
.89823 
.90025 
-90228 
.90429 



.90630 
.90831 
.91032 
.91231 
•91431 



•91630 
•91828 
•92027 
•92224 
92421 



•92618 
•92815 
•93010 
.93206 
.93401 



•93596 
•93790 
.93984 
.94177 
.94370 



D 



.94562 
.94754 
.94946 
•95137 
.95328 

.95519 
.95709 
95898 
.96088 
•96276 



•96465 
•96653 
•96841 
•97028 
.97215 



•97401 
•97587 
•97773 
•97958 
•98143 



7- 



98327 
98512 
98695 
98879 
99062 



7-99244 

I> JLog.Exs. 

7U 



208 

207 
207 
206 

206 
205 
205 
204 
204 

203 
203 
202 
202 
201 

201 
201 
200 
199 
199 

199 
198 
198 
197 
197 

197 
196 
195 
195 
195 

195 
194 
194 
193 
193 
192 
192 
192 
191 
191 

190 
190 
189 
189 
188 

188 
188 
188 
187 
187 

186 
186 
185 
185 
1841 

184 
184 
183 
183 
183 

182 



O 

1 
2 
3 
£ 

5 
6 
7 

8 
9 

10 

11 

12 

13 

-14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
3i 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



1> 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



P.P. 





180 


9 


6 


18-0 


0^9 


7 


21^0 


1^1 


8 


24^0 


1-2 


9 


27.0 


1-4 


10 


30-0 


1^6 


20 


60^0 


3.1 


30 


90.0 


4^7 


4C 


120 .C 


3-3 


51 


150. C 


7^9 



9 

0^9 
1^0 
1.2 





S 


; 


8 


6 


0.8 


0-81 


7 


1 








9 


8 


1 


1 


1 





9 


1 


3 


1 


2 


10 


1 


4 


1 


3 


20 


2 


8 


2 


6 


30 


4 


2 


4 





40 


5 


6 


5 


3 


50 


7 


1 


[a 


6 





7 


6 


6 


0.7 


0.61 


7 


0.8 





7 


8 


0-9 





8 


9 


1^0 


1 





10 


1^1 


1 


1 


20 


2-3 


2 


1 


30 


3-5 


3 


2 


40 


4.6 


4 


3 


50 


5.8 


5 


4 



7.5 

7 
0-7 
0-9 
1.0 

l-l 
1.2 
2.5 
3-7 
o.Q 
6.2 

6 

0.6 





o 


, 


? 


4 




6 


0^5 


0-5 


0-4 





7 


0-6 





6 


0-5 





8 


0.7 





6 


0-6 





9 


0^8 





7 


0^7 





10 


0.9 





8 


0.7 





20 


1-8 


1 


6 


1.5 


1 


30 


2.7 


3 


5 


2.2 


2 


40 


3^6 


3 


3 


3.0 


2 


50 


46 


4 


1 


3^7 


3 





3 


3 




6 


0^3 


0-3 





7 


0^4 





3 


0^ 


8 


0-4 





4 


0. 


9 


0-5 





4 


0. 


10 


0.6 





5 


0^ 


20 


1^1 


1 





0- 


30 


1-7 


1 


5 


1^ 


40 


2.3 


2 





1^ 


50 


2.9 


2 


5 


2. 



30 
4^0 
5^0 



.4 
5 
6 
6 
3 

6 
3 

2 

0-2 
0-2 
0.^ 
0-3 
0-3 



6,0 

7 

8,0 

90 

10 

20 

30 

40 1 

50'1 



P. P 



O 

0-0 



TABLE VIII.— LOGARITHMIC VEIISED SINES AND EXTERNAL SECANTg 
8° 9° 



Lg. Vers. I> Log.Exs. D 



o 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 

11 
20 

21 
22 
23 
24 

25 

26 

27 

28 

29. 

30 

81 

32 

33 

34 

35 
36 
37 
38 
39 

40 

41 I 
42 
43 
44 

45 

46 

47 

48 

49. 

50 

51 

52 

53 

5£ 

55 

56 

57 

58 

59_ 

60 



98820 
99000 
99180 
99360 
99539 



99718 
99897 
00075 
00253 
00431 

00608 
007B4 
009 dl 
01137 
01313 

01488 
01663 
01838 
02012 
02186 



7.99244 
.99427 
.99609 
.99790 

7-99971 



180 
180 
179 
179 

179 
178 
178 
177 
178 

jn 8.01050 
\ij-A .01229 
\i,l\ .01407 
\i,l\ .01585 
^■'^\ .01763 
175 



8.00152 
.00332 
.00512 
.00692 
.008-71 



02359 
02533 
02706 
02878 
03050 



03222 
03394 
03565 
03736 
03906 



04076 
04246 
04416 
04585 
04754 



04922 
05090 
05258 
05426 
05593 



C576C 
05926 
06093 
06259 
06424 



06589 
06754 
06919 
07C83 
07247 

07411 
07575 
07738 
07900 
08063 



08225 
08387 
08549 
08710 
08871 



8 09031 
Lg. Vers. 



8.01940 
.02117 
.02293 
.02469 
.02645 



8-02820 
.02995 
.03170 
.03345 
.03519 



8.03692 
.038.66 
•04039 
.04212 
-04384 



.04556 
-04728 
.04899 
.05070 
-05241 



175 
175 
174 
174| 

173 
173 
173 
172 
172 
172 
171 
171 
171 
170 
17C 
17C 
169 
169 
169 

168 
168 
168 
167 
167 

167 
166 
166 
166 
165 

165 
165 
165 
164 
164 

164 
163 
163 
162 
162 

}ftfl8. 08753 
\l^\ .08917 
\li\ .09081 
\l\\ .09244 
^°£| .09407 



8.05411 
.05581 
.05751 
.05921 
.OR090 



;. 06259 
-06427 
.06595 
-06763 
-06931 



8-07098 
.07265 
.07431 
.07598 
.07764 



8.07929 
08095 
08260 
08424 
08589 



^^^ 809569 
I •£> JLog.Exs. 



Lg. Vers. 



182 
182 
181 
181 
180 
18U 
180 
180 
179 

179 
178 
178 
178 
177 
177 
177 
176 
176 
175 

175 

175 
175 
174 
174 

173 

173 

173 

173 

17 

172 

171 

171 

171 

170 

17C 
17C 
17C 
169 
169 

169 
168 
168 
168 
167 

167 
167 
166 
166 
166 

165 
165 
165 
164 
164 

16^ 
163 
164 
16'? 
163 

162 



8-09031 
09192 
09352 
09512 
09671 



8-09830 
.09989 
.10148 
.10306 
.10484 



n 



Log.Exs. 



8.10622 
.10779 
.10936 
.11093 
.11250 



8.11406 
.11562 
.11718 
.11873 
•12029 

12184 
12338 
12492 
12647 
12800 



.12954 
.13107 
.13260 
.13413 
.13565 



13717 
13869 
14021 
14172 
.14323 



8 . 14474 
•14625 
.14775 
.14925 
-15075 

8 15225 
^5374 
-15523 
.15672 
-15820 



8-15968 
.16116 
.16264 
.16412 
-16559 



8-16706 
.16852 
.16999 
.17145 
•17291 



8.17437 
.17582 
•17728 
•17873 
•18017 



8. 18162 



160 
160 
160 
159 
159 
159 
158 
15b 
158 

157 
157 
157 
157 
156 

156 
156 
155 
155 
15'5 

155 

154 
154 
154 
153 
153 
153 
153 
152 
152 

15 

152 

151 

151 

151 

151 
150 
150 
150 
149 

150 

149 
149 
149 
148 

148 

148 
148 
147 
147 

147 
146 
146 
146 
146 

145 
145 
145 
145 
144 

144 



09569 
09732 
09894 
10056 
10217 



Lg. Vers. 



JD 



10378 
10539 
107C0 
10860 
11020 



11180 
11340 
11499 
11658 
11816' 



P. P. 



11975 
12133 
12291 
12448 
12605 

12762 
12919 
13075 
13232 
13387 



13543 
13698 
13854 
14008 
14163 



14317 
14471 
14625 
14778 
14932 



15085 
15237 
15390 
15542 
15694 



15846 
15997 
16148 
16299 
16450 



16600 
16750 
16900 
17050 
17199 



17349 
17497 
17646 
17795 
17943 



18091 
18238 
18386 
18533 
18680 



18827 



I> fLog.Exs. 



162 
162 
162 
16i 
161 
161 
160 
160 
160 

16C 
159 
159 
159 
158 

158 
158 
158 
157 
157 

157 
157 
156 
156 
155 

156 
155 
155 
154 
154 

154 
154 
153 
153 
153 

153 
152 
152 
152 
152 

15? 
151 
151 
151 
155 

15C 
150 
150 
149 
149 

149 
148 
149 
148 
148 

148 

147 
147 
147 
147i 



5 
6 
7 
8 
_9^ 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 



21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_. 59 



6 

7 

8 

9 

10 

20 

30 

40 

50 



180 

18-0 
21.0 
24-0 
27-0 
30-0 
60-0 
90-0 
120-0 
1150-0 



170 

17-C 
19 



22 
25 
28 
56 
85 
113 
141 



16< 

16 

18 

21 

24 

26 

53 

80 
106 
133 





150 


140 


6 


15-0 


140 


7 


17 


5 


16 


3 


8 


20 





18 


6 


9 


22 


5 


21 





10 


25 





23 


3 


20 


50 





46 


6 


30 


75 





70 


r; 


40 


ICO 





93 


3 


50 


125 





[life 


6 





9 


9 


S 


60-9 


0-9 


0-8 


7 


1-1 


1 





1 





8 


1-2 


1 


2 


1 


1 


9 


1.4 


1 


3 


1 


3 


10 


1.6 


1 


5 


1 


4f 


20 


3-1 


3 


C 


2 


s 


30 


4-7 


4 


5 


4 


3 


40 


6-3 


6 





5 


6 


50 


7.9 


7 


5 


7 


1 



6 

7 

8 

9 

10 

20 

30 

40 

50 






8 


0-7 


0- 





9 


0-9 


0- 


1 





1-C 


0- 


1 


2 


1-1 


1- 


1 


3 


1-2 


1- 


2 


6 


2-5 


2- 


4 





3-7 


3- 


5 


3 


5-C 


4 


6 


6 


6-2 


5- 



-7 
8 
-9 
-0 
-I 

■ 3 
5 

■ 6 

■ 8 





6 


6 


6 


0-6 


0-6 


7 





7 





7 


8 





8 





8 


9 


1 








9 


10 


1 


1 


1 





20 


2 


1 


2 





3C 


3 


2 


3 





40 


4 


3 


4 





50 


5 


4 


5 






6 

7 

8 

9 

10 

20 

30 

40 

50 



0-5 


0- 


0-6 


0- 


0-7 


0- 


0-8 


0. 


0-9 


0- 


1-8 


1- 


2 7 


2- 


3-6 


3- 


4.R 


4 



P.P. 



712 



[fABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
10° 11° 



Lg. Vers. 



8.18162 
18306 
18450 
18594 
18738 



18881 
19024 
19167 
19309 
19452 



19594 
19736 
19878 
20019 
20160 



20301 
20442 
20582 
20723 
20833 



21003 
21142 
21282 
21421 
21560 



21698 
21837 
21975 
22113 
22251 



22389 
22528 
22663 
22800 
22937 



23073 
23209 
23346 
23481 
23617 



23752 
23888 
24023 
24158 
24292 



24426 
24561 
24695 
24828 
24963 



25035 
25228 
25361 
25494 
25627 



2> 



Log.Exs. 



25759 
25891 
26023 
26155 
26286 



ftp 8-26417 
' Lg. Vers. 



144 
144 
144 
143 
143 
143 
142 
142 
142 

142 
142 
142 
141 
141 

141 
140 
140 
140 
140 

140 
139 
139 
139 

139 

138 
138 
138 
138 
137 
138 
137 
137 
136 
137 

136 
138 
136 
135 
136 

135 
135 
135 
135 
134 

134 
134 
134 
133 
133 

133 
133 
133 
-132 
133 

132 
132 
132 
132 
131 

131 



18827 
18973 
19120 
19266 
19411 



19557 
19702 
19847 
19992 
20137 



20281 
20425 
20569 
20713 
20857 

21000 
21143 
21286 
21428 
21571 



21713 
21855 
21996 
22138 
22279 
22420 
22561 
22701 
22842 
22982 



23122 
23282 
23401 
23540 
236_79 
23818 
23957 
24095 
24234 
2437!^ 

24509 
24647 
24784 
24922 
25059 



25195 
25332 
25488 
25604 
25740 



7> 



Lg. Vers. 



25876 
26012 
26147 
23282 
2^6417 
26552 
26686 
26821 
26955 
27089 



27223 



146 
146 
146 

145 

145 
145 
145 
145 
144 

144 
144 
144 
144 
143 

143 
143 
143 
142 
142 

142 
142 
141 
141 
141 

141 
140 
140 
140 
140 

140 
140 
139 
139 
139 

139 
138 
138 
138 
138 

137 
138 
137 
137 
137 

136 
136 
136 
136 
136 

136 
135 
135 
135 
135 

134 
134 
134 
134 
134 

134 



•26417 
•26548 
•26679 
.26810 
.28941 



■27071 
•27201 
•27331 
•27461 
•27590 



•27719 
.27849 
•27977 
•28106 
•28235 



•28363 
•28491 
.28619 
.28747 
.28875 



•29002 
•29129 
.29256 
.29383 
.29510 



•29636 
.29763 
.29889 
.30015 
•30140 



-O Log, Exs. 



•30266 
•30391 

30516 
•30642 

30766 



•30891 
•31015 
•31140 
31264 
.313RB 



,31511 
31635 
•31758 
.31882 
.32005 



32128 
•32250 
•32373 
.32495 
•32617 



J) 



32739 

32361 

•32983 

•33104 

.33225 



33347 
•33468 
•33588 
•33709 
^3829 



131 
131 
131 
130 

130 
130 
130 
130 
129 
129 
129 
128 
129 
128 

128 
128 
128 
128 
127 

127 
127 
127 
127 
126 

126 
126 
126 
126 
125 

125 
125 
125 
125 
124 

124 
124 
124 
124 
124 

123 
124 
123 
123 
123 

123 
122 
122 
122 
122 

122 

122j° 

121 

121 

121 

121 ^ 
120 
120 
120 

120 



J^ [Lg. Vers 



Log.Exs. 



27223 

27356 

•27490 

•27623 

27756 



•27889 
•28021 
.28153 
•28286 
L.2_8418 
•28550 
.28681 
.28813 
•28944 
•29075 



.29206 
•29336 
.29467 
.29597 
•29727 

•29857 
.29987 
.30117 
.30246 
.30375 



■30504 
.30633 
.30762 
.30890 
•31019 



•31147 
•31275 
.31402 
.31530 
•31657 



31785 
.31912 
•32039 
.32165 
.32292 



•32418 
.32544 
.32670 
.32796 
.32922 



•33047 
33173 
•33298 
.33423 
^33^47 

r33672 
.33797 
.33921 
•34045 
^34169 

■ 34293 
■34417 
■34540 
■34663 
.34786 



8.3490P 



l> Log.Exs. 
713 



D 

133 
133 
133 
133 

133 

132 
132 
132 
132 

132 
131 
131 
131 
131 

131 
130 
130 
130 
130 

130 
130 
129 
129 
129 

129 
129 
128 
128 
128 

128 
128 
127 
127 
127 

127 
127 
127 
126 
126 

126 
126 
126 
126 
125 

125 
125 
125 
125 
124 

125 
124 
124 
124 
123 

124 
124 
123 
123 
123 

123 



15 
16 
17 
18 
19 



30 

21 
22 
23 
24 



30 

31 
32 
33 
34 



40 

41 
42 
43 
44 



10 

11 
12 
13 

14 



45 
46 
47 
48 
Jt9 

50 

51 
52 
53 

55 
56 
57 
58 
b^ 

fiO 



P. P. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



130 



13 





12. 


15 


1 


14. 


17 


3 


16- 


19 


5 


18^ 


21 


6 


20 ■ 


43 


3 


40. 


65 





60. 


86 


6 


80. 


1108 


3 


100- 



130 















1 


4 


3 


6 


4 


4 


0-3 


7 





5 





4 





4 


8 





6 





5 





4 


9 





7 





6 





5 


10 





7 





6 





6 


20 


1 


5 


1 


3 


1 


1 


30 


2 


2 


2 





1 


7 


40 


3 





2 


6 


2 


3 


50 


3 


7 


3 


3 


2 


9 



6 

7 

8 

9 

10 

20 

30 

40 

50 






3 


0^ 





3 


0. 





4 


0- 





4 


0- 





5 


0- 


1 





0- 


1 


5 


1. 


2 





1. 


2 


5 


2. 



■ 2 
•3 

3 

■ 4 

• 4 

• 8 
.2 
.6 
.1 













2 1 


6 


0-2 


0-1 


7 





2 





2 


8 





2 





2 


9 





3 





2 


10 





3 





2 


20 





6 





5 


30 


1 








7 


40 


1 


3 


1 





50 


1 


6 


1 


2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



O 



P. P. 



TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
13° 13° 





1 

2 

3 

_4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

AO 

31 
32 
33 
3i 
35 
36 
37 
38 
39 

40 

41 
42 
43 
11 
45 
46 
47 
48 

60 

51 
52 
53 
5£ 

55 

56 

57 

58 

59_ 

60 



Lg. Vers. 



8-3395U 
34070 
34190 
3430S 
34429 



34549 
34668 
34787 
34908 
35025 



35143 
35262 
35380 
35498 
35616 



2> 



Log.Exs. 



35734 
35852 
35969 
36086 
36204 



36321 
36437 
36554 
36671 
36787 



36903 
37019 
37135 
37251 
37366 

37482 
37597 
37712 
37827 
37942 



38057 
38171 
38286 
38400 
3851.4 



38628 
38741 
38855 
38969 
39082 



39195 
39308 

39421 
3953^ 
3<)R4B 



39758 
39871 
39983 
40095 
40907 



40318 
40430 
40541 
40652 
40764 



40875 



Lg. Vers, 



120 
120 
119 
120 

119i„ 

119'** 

119 

119 

119 

118 
118 
118 
118 
118 

117 
118 
117 
117 
117 

117 
116 
117 
116 
116 

116 
116 
116 
115 
115 

115 ft 

115^ 

115 

115 

115 



114 
114 
114 
114 
114 

113"^ 
114 
113 
113 

113 
113 
113 
113 
112 

112 
112 
112 
112 
112 

111 
111 
111 
111 
111 

111 



34909 
35032 
35155 
35277 
35399 



35522 
35644 
35765 
35887 
36009 



36130 
36251 
36372 
36493 
36614 



36734 
36855 
36975 
37095 
37215 



37335 
37454 
37574 
37693 
37812 

37931 
38050 
38169 
38287 
38406 

38524 
38842 
38760 
38878 
38995 

39113 
39230 
39347 
39464 
39^81 

39698 
39814 
39931 
40047 
40163 



4027P 
40395 
40511 
40626 
4074? 



40857 
40972 
41087 
41202 
41317 



41431 
41546 
41660 
41774 
41888 



42009 



n 

123 
122 
122 
122 

122 
122 
I2I 
122 
12i 

121 
121 
121 
120 
121 

120 
126 
120 
120 
120 

120 
119 
119 
119 
119 

119 
118 
119 
118 
118 

118 
118 

118 
118 
117 

117 
117 
117 
117 
117 

116 
116 
116 
116 
116 

116 
116 
115 
115 
115 

115 
115 
115 
115 
114 

114 
114 
114 
114 
114 

114 



Lg. Vers. 



J> Log.Exs. I> Lg. Vers, 



40875 
40985 
41096 
41206 
41317 

41427 
41537 
41647 
41757 
41867 



D 



41976 
42086 
42195 
42304 
42413 



4252Z 
42630 
42739 
42847 
42956 



43064 
43172 
43280 
43388 
43495 



43603 
43710 
43817 
43924 
44031 



44138 
44245 
4435I 
44458 
44564 



44670 
44776 
44882 
44988 
45093 

45199 
45304 
45409 
4551? 
45619 



45724 
45829 
4593^ 
46038 
46142 



46247 
46351 
46455 
46558 
46662 

46766 
46869 
46972 
47076 
47179 



47282 



110 
110 
110 
110 

110 
110 

IIQ 
109 
110 

109 
109 
109 
109 
109 

10? 
108 
109 
108 
108 

108 
108 
108 
108 
107 

107 
107 
107 
107 
107 

106 
107 
106 
106 
106 

loe 

106 
105 
106 
105 

105 
105 
105 
105 
105 

105 

104 
105 
104 
104 

104 
104 
104 
103 
104 

103 
103 
103 
103 
103 

103 



Log. Exs. 



42002 
42116 
42229 
42343 
42456 



42569 
42682 
42795 
42908 
43021 



43133 
43246 
43358 
43470 
43582 



43694 
43805 
43917 
44028 
44139 



44251 
44362 
44473 
44583 
44694 



44804 
44915 
45025 
45135 
45245 



45355 
45465 
45574 
45684 
45793 



45902 
46011 
46120 
46229 
46338 
46446 
46555 
46663 
46771 
46879 



46987 
47095 
4720? 
47310 
47417 



47525 
47632 
4773P 
47846 
47953 



48060 
48166 
48273 
48379 
48485 



D 



48591 



113 
113 
113 
113 

113 
113 
113 
113 
112 

112 
112 
112 
112 
112 

112 
111 
111 
Hi 
111 

HI 
111 
111 

lie 
lie 

lie 
lie 
lie 
lie 

109 

110 
11c 

109 
109 
109 

109 
109 
109 
108 
109 

108 
108 
108 
108 
108 

108 
107 
108 
107 
107 

107 
107 
107 
107 
106 

107 
106 
106 
106 

106 

106 



10 

11 
12 
13 
14 



15 
16 
17 
18 
19 

30 

21 
22 
23 
24 



D Log.Exs, 
714 



D 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



60 



P.P. 



130 



6 


12 





11 


7 


14 





13 


8 


16 





15 


9 


18 





17 


10 


20 





19 


20 


40 





39 


30 


60 





59 


40 


80 





79 


50 


100 





99 



119 

9 





117 


116 


11 


6 


11.7 


11.6 


11. 


7 


13 


6 


13 


5 


13- 


8 


15 


6 


15 


4 


15. 


9 


17 


5 


17 


4 


17. 


10 


19 


5 


19 


3 


19. 


20 


39 





38 


6 


38. 


30 


58 


5 


58 


G 


57. 


40 


78 





77 


3 


76. 


50 


97 


5 


96 


6 


95. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



114 113 


11 


11.4 


11.3 


11. 


13 


3 


13 


2 


13. 


15 


2 


15 





14. 


17 


1 


16 





16. 


19 





18 


8 


18. 


38 





37 


6 


37. 


57 





56 


5 


56. 


76 





75 


3 


74. 


95 





94 


1 


93. 



118 

11.8 
13.7 
15.7 
17.7 
19.6 
39.3 
59.0 
78-6 
98.3 





111 


110 


109 


6 


11-1 


11.0 


10.9 


7 


12 


9 


12 


8 


12 


7 


8 


14 


8 


14 


6 


14 


5 


9 


16 


6 


16 


5 


16 


3 


10 


18 


5 


18 


3 


18 


1 


20 


37 





36 


6 


36 


3 


30 


55 


5 


55 


C 


54 


5 


40 


74 





73 


3 


72 


6 


50 


92 


5 


91 


6 


90 


8 



108 107 106 



7 
8 
9 

10 
20 
30 
40 
50 



7 
8 
9 
10 
20 
30 
40 
50 



10 


8 


10 


7 


10 


6 


12 


6 


12 


5 


12 


3 


14 


4 


14 


2 


14 


1 


16 


2 


16 





15 


9 


18 





17 


8 


17 


6 


36 





35 


6 


35 


3 


54 





53 


5 


53 





72 





71 


3|70 


6 


90 





89 


1 


88 


3 



105 

10.5 
12.2 
14.0 
15.7 
17.5 
35.0 
52-5 
70.0 
87-5 



P.P. 



104 





10.4 


0.0 


12 


1 


0.0 


13 


8 


0.0 


15 


6 


0.1 


17 


3 


0.1 


34 


6 


0.1 


52 





0.2 


69 


3 


0.3 


86 


6 


0.4 



*ABLE Vlll.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
14° 15° 



Lg.Vers. 



O 8 
1 



47282 
47384 
47487 
47590 
47692 



47795 
47897 
47999 
48101 
48203 



48304 
48406 
48507 
48609 
48710 



48811 
48912 
49013 
49114 
49215 



49315 
49415 
49516 
49616 
49716 



49816 
49916 
50015 
50115 
50215 

50314 
50413 
50512 
50611 
50710 



50809 
50908 
51006 
51105 
51!:!nR 



51301 
51399 
51497 
51595 
51693 



51791 
51888 
51986 
52083 
52180 



52277 
52374 
5247i 
52568 
52665 



52761 
52858 
52954 
53050 
53146 



53242 



D 

102 
103 
102 
102 

102 
102 
102 
102 
102 

lOlS 

101 

101 

101 

lOl 

101 
101 
101 
100 
101 

100 

log 

100 
100 
100 

lOOi 
1001 
99 
100 
99 
99 
99 
99 
99 
99 

98 
99 
98 
98 



Log. Exs. 



8 



98 
98 
98 
98 
97 
98 
97 
97 
97 
87 

97 
97 



96 
97 

96|fi 
96P 

96 
96 
96 

96 



48591 
48697 
48803 
48909 
49014 



49120 
49225 
49331 
49436 
49541 

49646 
49750 
49855 
49960 
50064 



50168 
50273 
50377 
50481 
50585 

50688 
50792 
50896 
50999 
51102 



51205 
51309 
51412 
51514 
51617 



51720 
51822 
51925 
52027 
52129 



52231 
52333 
52435 
52537 
52R?!'" 



5274G 
52841 
52943 
53044 
53145 



53246 
53347 
53448 
53548 
53649 



53749 
53850 
53950 
54050 
54150 



5425C 
54350 
54449 
54549 
54649 



54748 



' jLg.Vers. I^ Log. Exs. I>jLg. Vers 



106 
106 
105 
105 

105 
105 
105 
105 
105 

105 
104 
105 
104 
104 

104 
104 
104 
104 
104 

103 
104 
103 
103 
103 

103 
103 
103 
102 
103 

102 
102 
102 
102 
102 

102 
102 
102 
lOl 
lOl 

101 
101 
101 
101 
101 

101 
101 
101 
100 
100 

100 
100 
100 
100 
100 

100 
100 

99 
100 

99 

99 



Lg. Vers. 



53242 
53338 
53434 
53530 
53625 



53721 
53816 
53911 
54007 
54102 



54197 
54291 
54386 
54481 
54575 



54670 
54764 
54858 
54952 
55046 
55140 
55234 
55328 
55421 
55515 



55608 
55701 
55795 
55888 
55981 



56074 
56166 
56259 
56352 
56444 



56536 
56629 
56721 
56813 
56905 



56997 
57089 
57180 
57272 
57363 

57455 
57546 
57637 
57728 
57819 



57910 
58001 
58092 
58182 
58_273 

58363 
58453 
5854'! 
58634 
58724 



58814 



2> 



Log. Exs. 



96 
95 
96 
95 

95 
95 

95 
95 
95 

S5 
94 
95 
94 
94 

94 
94 
94 
94 
94 

94 
93 
94 
93 
93 

93 
93 
93 
93 
93 

93 
92 
92 
93 
92 

92 
92 
92: 
92 
92 

92 
92 

91 
91 
91 

91 
91 
91 
91 
91 

91 
90 
91 
90 
90 

90 
90 
90 
90 
90 

9Q 



54748 
54847 
54946 
55045 
55144 



552'!'3 
55342 
55:'41 
55539 
55638 



55736 
55834 
55933 
5603:. 
56129 



56226 
56324 
56422 
56519 
56617 

56714 
56812 
56909 
57006 
5710 3 
57200 
57296 
57393 
57490 
57586 



57682 
57779 
57875 
57971 
58067 



58163 
58259 
58354 
58450 
58546 



58641 
58736 
58832 
58927 
59022 

59117 
59211 
59306 
59401 
59495 



5959C 
59684 
59779 
59873 
59967 



60061 
60155 
60249 
60342 
6043 6 
60530 



~D Log. Exs, 
715 



J) 



99 
99 
99 
99 

99 
99 



98 

98 
98 
98 
98 
98 

97 
98 
97 
97 
97 
97 
97 
97 
97 
97 

97 
96 
97 
96 
96 

96 
96 
96 
96 
96 

95 
96 
95 
96 
95 

95 
95 
95 
95 
95 

95 
94 
95 
94 
94 
94 
94 
94 
94 
94 

94 
94 
94 
93 
94 

93 



O 

1 

2 
3 

4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
lA 

l5 
16 
17 
18 
]9 

30 

21 
22 
23 

-24 
25 
26 
27 
28 

-29 

30 

31 
32 
33 

35 
36 
37 
38 
39 

40 

41 
42 
43 

44 

45 
46 
47 
48 
_49 

50 

51 
52 
53 

55 
56 
57 
58 

-59 
60 



1> ' 



P.P. 



6 

7 
8 
9 

10 
20 
30 
40 
50 



103 

10.3 
12.0 



13 

15 

17 

34 

51 

68-6 

85. b 



102 

10.2 
11.9 
13.6 
15.3 
17.0 
34-0 
51-0 
68.0 
85.0 



101 

10.1 
11.8 
13.4 
15.1 
16-8 
33.6 
50-5 
67.3 
84.1 



iOO 

6 10-0 

7 11 

8 13 



9,15 
16 



20 
30 
40 
50 



99 98 

9.?i 9.8 
11-5 11. i 
13.2113.0 



14.8 
16-5 
33.0 
49.5 
56.0 



32.5J82 



14 
16 
32. 
49. 
65. 





97 


96 


6 


9.7 


9.6 


7 


11.3 


11.2 


8 


12.9 


12.8 


9 


14.5 


14.4 


10 


16.1 


16.0 


20 


32.3 


32.0 


30 


48.5 


48.0 


40 


64.6 


64.0 


50 


80.8 


80.0 





94 


93 


6 


9.4 


9.3 


7 


10 


9 


10.8 


8 


12 


5 


12.4 


9 


14 


1 


13.9 


10 


15 


6 


15.5 


20 


31 


3 


31.0 


30 


47 





46.5 


40 


62 


6 


62.0 


50 


78 


3 


77.5 



95 

9.5 

11.1 
12.6 
14.2 
15.8 
31.6 
47.5 
63.3 
79.1 



9» 

9.2 
10.7 
12.2 
13.8 
15.3 
30.6 
46.0 
61.3 
76.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



91 

9-1 
10.6 
12.1 
13.6 
15.1 
30.3 
45.5 
GO. 6 
75.8 



90 O 

9.C|0.0 
10.5 0.0 
12.C p.O 



13.5 
15.0 
30.0 



45.00.2 



60.0 
75.0 



0.1 
0.1 
O.I 



0.3 
0.4 



P.P. 



|*ABLE VIII.— LOGARITHMIC VEHSED SINES ANB EXTERNAL SECA?fTS. 



Lg. Vers. 

58814 
58904 
58993 
59083 
59173 
59262 
59351 
59441 
59530 
59619 

59708 
59797 
59886 
59974 
60063 



60152 
60240 
60328 
60417 
60505 



60593 
60681 
60769 
60857 
60944 



61032 
61119 
61207 
61294 
61381 



61463 
61556 
61643 
6173G 
61816 



61903 
61990 
62076 
62163 
62249 

62336 
62422 
62508 
62594 
62680 



62766 
62852 
62937 
63023 
63108 

63194 

63279 
63364 
63449 
63534 



63619 
63704 
63789 
63874 
63959 



8-64043 
Lg. Vers. 



2> 



Log.Exs, 



90 
89 
90 
89 

89 
89 

89 
89 
89 

89 
89 
89 
88 
89 

88 



88 

88 
88 
88 
88 
87 

87 
87 
87 
87 
87 

87 
87 
87 
87 
86 

87 
86 
86 
86 
86 

86 
86 
86 
86 
86 

86 
86 
85 
85 
85 

85 
85 
85 
85 
85 

85 
85 
85 
84 
85 
84 



60530 
60623 
60716 
60810 
60903 



60996 
61089 
61182 
61275 
61368 



61460 
61553 
61645 
61738 
61830 



61922 
62014 
62106 
62198 
62290 



62382 
62474 
62565 
62657 
62748 

62840 
62931 
63022 
63113 
63204 

63295 
63386 
63477 
63567 
63658 



63748 
63839 
63929 
64019 
64109 



64199 

64289 
64379 
64469 
64559 



64649 
64738 
64828 
64917 
6500_6 
65096 
65185 
65274 
65363 
65452 



65541 
65629 
65718 
65807 
65895 



65984 



1> Log.Exs. 



i> Lg. Vers. 



93 
93 
93 
93 

93 
93 
93 
92 
93 

92 
92 
92 
92 
92 

92 
92 

92 
92 
92 

91 

92 
91 
91 
9i 

91 
91 

91 
91 
91 

90 
91 
91 
90 
90 

90 
90 
90 
90 
90 

90 
90 
90 
90 
89 

90 
89 
89 
89 
89 

89 
89 
89 
89 
89 

89 
88 
88 
89 
88 

88 



64043 
64128 
64212 
64296 
64381 



64465 
64549 
64633 
64717 
64801 



64834 
64968 
65052 
65135 
65218 



65302 
65385 
65468 
65551 
65634 

65717 
65800 
65883 
65965 
M048 
66131 
66213 
66295 
66378 
66460 



66542 
66624 
66706 
66788 
66870 



66951 
67033 
67115 
67196 
67277 
67359 
67440 
67521 
67602 
67683 



67734 
67845 
67926 
68007 
68087 



68168 
68248 
68329 
68409 
68489 

68569 
68650 
68730 
68810 
68889 



68969 



g.Vers 



1> 

84 
84 
84 
84 

84 
84 
84 
84 
84 

83 

83 
84 
83 
83 

83 
83 
83 
83 
83 

83 
83 

82 
82 
83 
82 
82 
82 
82 
82 

82 
82 
82 
82 
82 

81 
81 
82 
81 
81 

81 
81 
81 
81 
81 

81 
81 
80 
81 
80 

80 
80 
80 
80 
80 

80 
80 
80 
80 
79 

80 



Log.Exs, 



65984 
66072 
66160 
66248 
66336 



66425 
66512 
66600 
66688 
63776 



66863 
66951 
67039 
67126 
67213 



67301 
67388 
67475 
67562 
67649 

67736 
67822 
67909 
67996 
68082 



68169 
68255 
68341 
68428 
68514 



68600 
63686 
68772 
68858 
68944 

69029 
69115 
69201 
68286 
69372 

69457 
69542 
68627 
69712 
69798 



69883 
69967 
70052 
70137 

70222 



70306 
70391 
70475 
7056C 
70644 



70728 
70813 
70897 
70981 
71065 



8.73149 



i> Log.Exs. 



JJ 



88 
88 
88 
88 

88 
87 
88 
88 
87 

87 
88 
87 
87 
87 

87 
87 
87 
87 
87 

87 
86 
87 
86 
86 

86 
86 
86 
86 
86 

86 
86 
85 
86 
86 

85 
86 
85 
85 
85 

85 
85 
85 
85 
85 
85 
84 
85 
85 
84 

84 
84 
84 
84 
84 

84 
84 
84 
84 
84 

84 

17 



10 

11 

12 
13 
li 
15 
16 
17 
18 
19 



20 
21 
22 
23 
24 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



60 



P.P. 





93 


93 


91 


6 


9.3 


9.2 


9. 


7 


10.8 


10.7 


10. 


8 


12.4 


12.2 


12. 


9 


13.9 


13-8 


13. 


10 


15.5 


15.3 


15. 


20 


31.0 


30.6 


30. 


30 


46.5 


46.0 


45. 


40 


62.0 


61.3 


60. 


50 


77.5 


76.6 


75. 





90 


89 


88 


6 


8.0 


89 


8. 


7 


10.5 


10.4 


10. 


8 


12. C 


11.8 


11. 


9 


13.5 


13.3 


13. 


10 


15. C 


14.8 


14. 


20 


30. C 


29.6 


29. 


3C 


45. C 


44.5!44. 


40 


60.0 


59.3|58- 


50 


75.0 


74-1 


73. 



1 - 

6 
1 
6 

I "■ 
5 

5' ■ 
6 
8 



6 

7 
8 

9 
10 
20 
30 
40 
50 



87 
8.7 

10.1 
11.6 
13.0 
14.5 
29.0 
43.5 
58.0 
72-5 



86 

86 



85 
8 5 





84 


83 


82 


6 


8.4 


88 


8. 


7 


9.8 


9.7 


9. 


8 


11.2 


11. C 


10. 


9 


12.6 


12.4 


12. 


10 


14.0 


13. 8 


13. 


20 


28.0 


27.6 


27. 


30 


42.0 


41.5 


41. 


40 


56.0 


55.3 


54- 


50 


70.0 


69.1 


68. 



7 
8 

9 
10 
20 
30 
40 
50 



81 
8.1 
84 
10.8 
12.1 
13.5 
27.0 
40-5 
54.0 
67-5 



80 
8.C 



79 

79 



6 

7 

i§ 

20 
30 
4C 

FT 



0.0 



O.Q 

0.1 
0.1 
0.1 
D.2 
0.3 
0.4 



P.P. 



716 



■TABLE VIII .— LOaAJlITHMIC VERSED SINES AND EXTERN^I. SEjCAlJT^. 



18° 



19° 



Lg. Vers. 



68969 
69049 
69129 
69208 
69288 



69367 
69446 
69526 
69605 
69634 



69763 
69842 
69921 
70000 
70079 



70157 
70236 
70314 
70393 
70471. 



70550 
70628 
70706 
70784 
708S2 



70940 
71018 
71096 
71174 
71251 



71329 
71406 
71484 
71561 
71639 
71716 
71793 
71870 
71947 
72024 



72101 
72178 
72255 
72331 
72408 



72485 
72581 
72637 
72714 
72790 



72886 
72942 
73018 
73094 
7317S 



73246 
73322 
73398 
73473 
73549 



7362! 



Ler. Vers. -D 



2> 

79 
80 
79 
79 

79 
79 
79 
79 
79 
79 
79 
79 
78 
79 

78 
78 
78 
78 
78 

78 
78 
78 
78 
78 

78 
78 
77 
78 

77 

77 
77 
77 
77 
77 

77 
77 
77 
77 
77 

77 

76 
77 
76 
77 

76 
76 
76 
76 
76 

76 
76 
78 
76 
76 

76 
76 
75 
75 
76 

75 



Log.Exs. 



8.71149 
71232 
71316 
71400 
714G4 



71567 
71651 
71734 
71817 
71901 



71984 
72067 
72150 
72233 
72316 



72399 
72481 
72564 
72647 
72729 



72812 
72894 
72977 
73059 
7314T 



73223 
73306 
73388 
73470 
73551 



73633 
73715 
73797 
73878 
73960 



74041 
74123 
74204 
74286 
74-;in7 



7444£ 
74528 
74610 
74691 
74772 



74853 
74934 
75014 
75095 
75175 



75256 
75336 
75417 
75497 
75577 



75658 
75738 
75818 
75898 
75978 



8.76058 



Log. Exsi 



8 



J> 

83 
84 
83 
84 

83 
83 
83 
83 
83 

83 
83 
83 
83 
83 

83 
82 
83 
82 
82 

Ih 

82 
82 
82 

82 
82 
82 
82 
81 

82 



Lg. Vers. 



82 
51 
81 
81 

81 
81 
81 
81 
81 

81 
81 
81 
81 
80 

81 
81 
80 
80 
80 

80 
80 
80 
80 
80 

80 
80 
80 
80 
80 
80 



73625 
73700 
73775 
73851 
73926 



74001 
74076 
74151 
74226 
74301 



74376 
74451 
74526 
74600 
74675 



74749 
74824 
74898 
74973 
75047 

75121 
75195 
75269 
75343 
75417 



75491 
75565 
75839 
75712 
75786 



75860 
75933 
76006 
76080 
76153 



76226 
76300 
76373 
76446 
76519 



76592 
76664 
^6737 
76810 
76883 



76955 
77028 
77100 
77178 
77245 



77317 
77390 
77462 
77534 
77606 



77678 
77750 
77822 
7789| 
77965 



8 78037 



I> Lg. Vers. 



D 

75 
75 
75 
75 
75 
75 
75 
75 
75 

75 
74 
75 
74 
74 

74 
74 
74 
74 
74 
74 
74 
74 
74 
74 
74 
73 
74 
73 
73 

74 
73 
73 
73 
73 
73 
73 
73 
73 
73 

73 

72 
73 
72 
73 

72 
72 
72 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
71 
72 

71 



Log.Exs. 



J> 



76058 
76137 
76217 
76297 
7C376 



76456 
76536 
76615 
76694 
76774 



76853 
76932 
77011 
77092 
77169 



77248 
77327 
77406 
77485 
77563 



77642 
77720 
77799 
77877 
77956 



78034 
78112 
78191 
78269 
78347 



78425 
78503 
78581 
78659 
78736 



78814 
78892 
78969 
79047 
79124 

79202 
79279 
79357 
79434 
79511 



79588 
79665 
79742 
79819 
79896 



79973 
80050 
80126 
80203 
80280 



80356 
80433 
80509 
80586 
80662 



8073« 
.og.Exs. 



7> 

79 
80 
79 
79 
80 
79 
79 
79 
79 

79 
79 
79 
79 
79 

79 
79 
7§ 
79 
78 

78 
78 
79 
78 
78 

78 
78 
78 
78 
78 
78 
78 
78 
78 
77 
78 
77 
77 
77 
77 

77 
77 
77 
77 
77 

77 
77 
77 
77 
77 

76 
77 
76 
77 
76 

76 

76 
76 
70 
76 

76 
■^ 



10 

11 
12 
13 
14 



20 

21 
22 
23 
24 



25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 

50 

51 
52 

53 
54 



55 
56 
57 
58 
19 
fiO 



P.P. 





84 


83 


6 


8-4 


8-3 


7 


9.8 


9.7 


8 


11.2 


11.0 


9 


12.6 


12.4 


10 


14.0 


13.8 


20 


28.0 


27-6 


30 


42. C 


41-5 


40 


56.0 


55.3 


50 


70.0 


69.3 





81 


8C 


6 


8.1 


8-0 


7 


9.4 


9.3 


8 


10.8 


10.6 


9 


12.1 


12.0 


10 


13-5 


13.3 


20 


27.0 


26.6 


30 


40.5 


40.0 


40 


54.0 


53.3 


50 


67-5 


66.6 





78 


77 


7( 


6 


7.8 


7-7 


7. 


7 


9.1 


9.0 


8. 


8 


10.4 


10.2 


10. 


9 


11.7 


11.5 


11. 


10 


13.0 


12.8 


12. 


20 


26.0 


25.6 


25. 


30 


39.0 


38.5 


38. 


40 


52.0 


51.3 


30. 


50 


65-0 


64.1 


63. 





75 


74 


6 


7.5 


7.4 


7 


8.7 


8.6 


8 


10.0 


9.8 


9 


11-2 


11.1 


10 


12.5 


12-3 


20 


25.0 


24-6 


30 


37-5 


i37.0 


40 


50-0 


49-3 


50 


62-5 


61-6 



83 
8-2 
9.5 
10. 9 
12.3 
13.6 
27.3 
41.0 
54.6 
68-3 



79 
7-9 
9.2 
10.. 5 
il.i 
13-1 
26-3 
39-5 
52-6 
85-3 



73 

7.3 
85 

9.7 
10-9 
12.1 
24.3 
36-5 
48.6 
60.8 



e 

7 
8 
9 
10 
20 
80 
40 
50 



73 


71, 


7.2 


7.]| 


8.4 


8 


8 


9.6 


9 


A. 


10.8 


10 


6 


12.0 


11 


8 


24.0 


23 


R 


36.0 


35 


5 


48.0 


47 


3 


80.0 


59 


1 



P.P. 



717 



TABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT 
30*' SI" 



Lg. Vers. 

78037 
78108 
78180 
78251 
78323 



.2> JLog.Exs 



78394 
78466 
78537 
78608 
78679 



78750 
78821 
78892 
78963 
79034 



79105 
79175 
79246 
79317 
79387 

79458 
79528 
79598 
79669 
79739 



79809 
79879 
79949 
80019 
80089 



80159 
80229 
80299 
80369 
80438 



80508 
80577 
80647 
80716 
80786 

80855 
80924 
80993 
81063 
81132 



81201 
81270 
81339 
81407 
81476 



81545 
81614 
81882 
81751 
81819 



81888 
81956 
82025 
82093 
82161 



8 -82229 
Lg. Vers. 



71 
71 
71 
71 

71 
71 
71 
71 
71 

71 
71 
71 
71 
70 

71 
70 
71 
70 
70 

70 
70 
70 
70 
70 

79 
70 
70 
70 
70 

70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 

69 
69 
69 
69 



69 
68 
69 

6§ 
69 
68 
68 
68 

68 

62 
68 
68 
68 
68 



8-80738 
.80814 
.80891 
.80967 
. 81043 



8.81119 
.81195 

.81271 
.81346 
•81422 



8.81498 
•81573 
.81649 
•81725 
•81800 



8.81876 
•81951 
•82026 
•82102 
.82177 

8-82252 
•82327 
•82402 
•82477 
•32552 



8^82627 
•82702 
•82776 
•82851 
•82926 



8.83000 
.83075 
.83149 
.83224 
.83298 



8.83373 
.83447 
•83521 
•83595 
.83670 



8.83744 

. .83818 

•83892 

•83966 

.84039 



8 . 84113 
.84187 
.84261 
.84334 
.84408 



8-84431 
.84555 
.84625 
.84'702 
. 8477 5 

8 . 8484S 
.84922 
.84995 
.85061 
.85141 

8. 85215 



I> 

76 
76 
76 
76 

76 
76 
76 
75 
76 

75 
75 
76 
75 
75 

75 
75 
75 
75 
75 

75 

75 
75 
75 
74 

75 

75 
74 
75 
74 

74 
74 
74 
74 
74 

74 
74 
74 
74 
74 

74 
74 
74 
74 
73 

74 
73 
74 
73 
7S 

n 

73 
73 
73 
73 

73 
73 
73 
73 
73 

73 



Lg.Vers. l>jLog.Exs. 



8 



82229 
82297 
82366 
82434 
82502 



82569 
82637 
82705 
82773 
82841 

8290§ 
82976 
83043 
83111 
83178 



83246 
83313 
83386 
83447 
83515 

83582 
83649 
83716 
83783 
83850 



83916 
83983 
84050 
84117 
84183 

84250 
84316 
84383 

84449 
84515 



84582 
84648 
84714 
84780 
R4-84J 

84912 
84978 
85044 
8511D 
8517B 



85242 
85308 
85373 
85439 
85506 



855''0 
85626 
85'='0i 

85832 



85214 
.85287 
■85360 
.85433 
.85506 



1> 



85579 
.85651 
.85724 
.85797 
.85869 



85942 
.86014 
.86087 
■86159 
.86231 



73 

72 
73 

73 
72 
73 

72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
72 
71 

72 
71 
72 
7i 
71 

71 
71 

71 
71 
71 

87739 l\ 
.87810 '^ 



■ 86804 
.86376 
.86448 
.86520 
•,86592 

86664 
86736 
■86808 
86880 
86952 



87024 
.87095 
.87167 
.87239 

87310 



87382 
•87453 
.87525 
•87096 



.87881 
.87953 
■P8024 

88095 
■88166 
■88237 
■88308 

88378 



88449 
■88520 
■88591 
.88661 
■88732 



85897 
85962 
86027 
86092 
86158 



88«03 
.888''3 

88944 
■88014 

89085 



86223 



■89155 
89225 
.89295 
.89366 
■89436 

■89506 



71 
71 
71 

71 
71 
71 
71 
70 

71 
71 
70 
70 
71 

79 
70 
70 
70 
70 

70 
70 
70 
70 
70 

70 



O 

1 

2 
3 

4 

5 
6 
7 
8 
__9 

10 

11 
12 
13 

M 

15 
16 
17 
18 
19 

20 

21 

22 

23 

,24 

25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



Log. Exs=| 



©fig. Vers.! -2>!Log.EKS.j ® 



35 
36 
37 
38 
_39 

40 

41 
42 
43 

M 

45 
46 
47 
48 
49 

50 

51 
52 
53 
_54 

55 
56 
57 
58 
59 



P.P. 





76 


75 


74 , 


6 


7.6 


7.5 


7.4 


7 


8.8 


8.7 


8.6 


8 


10.1 


10.0 


9^g 1 


9 


11-4 


11.2 


11.1 : 


10 


12.6 


12.5 


12.3 ,; 


20 


25.3 


25.0 


21.6 ■' 


SO 


38.0 


37.5 


37.0 


40 


50-6 


50.0 


49.3 


50 


63-3 


62.5 


61.6 





73 


72 


6 


7-3 


7.2 


7 


8.5 


8.4 


8 


9.7 


9-6 


9 


10.9 


10-8 


10 


12-1 


12-0 


20 


24 3 


24-0 


30 


36.5 


36.0 


40 


48.6 


48.0 


50 


60-8 


60-0 





70 


69 


6i 


6 7.0 


6-9 


6. 


7 


8.1 


8 





7- 


8 


9.3' 


9 


2 


9. 


9 


10-5 


10 


3 


10- 


10 


11-6 


11 


5 


11- 


20 


23-3 


23 





22. 


30 


35.0 


34 


5 


34. 


40 


46-6 


46 





45- 


50 


58-3 


57 


5 


56. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



67 

6-7 



66 

- 6-6 

7-7 

8-8 

9-9 

11-0 

22-0 

33-0 

44-0 

55.0 



65 

6-5 

7-6 

8-6 

9.7 

10-§ 

21-6 

32-5 

43.3 

54.1 



6 

7 

8 

S 

10 

20 

30 

40 

50 



P. P. 



718 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
33° 33° 



Lg. Vers. 



8.86223 
86287 
86352 
86417 
86482 



86547 
86612 
86676 
86741 
86805 
86870 
86934 
86999 
87063 
87127 



87192 
87256 
87320 
87384 
87448 



87512 
87576 
87640 
87704 
87768 



87832 
87895 
87959 
88023 
88086 



88150 
88213 
88277 
88340 
88404 



88467 
88530 
88593 
88656 
88720 



88783 
88846 
88909 
88971 
89034 



S9097 
89160 
89223 
89285 
89348 



89411 
89473 
89536 
89598 
89660 



89723 
89785 
89847 
89910 
89972 



00034 



g. Vers. 



64 
65 
65 
65 

64 
65 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
63 
64 
63 
64 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
62 
63 

63 
62 
63 
62 
62 

63 
62 
62 
62 
B2 

62 
62 
62 
62 
62 

62 



Log. Exs. 



•89506 
.89576 
.89646 
.89716 
.89786 



■89856 
.89926 
.89995 
.90065 
.90135 



•90205 
.90274 
•90344 
•90413 
.90483 



.90552 
.90622 
.90691 
•90760 
.90830 



90899 
.90988 

91037 
.91106 

91175 



■91244 
•91313 
•91382 
.91451 
•91520 



■91588 
•91657 
■91728 
■91794 
■91863 



91932 
■92000 
■92068 
■92137 

■92205 

.92274 
.92342 
.92410 
•92478 
.92546 



■92615 
■92683 
■92751 
■92819 
.92887 



.92955 
.93022 
.93090 
■93158 
■93226 



8- 



93293 
93361 
93429 
93496 
93564 

93631 



I* Log. Exs, 



i 8 



6? 
68 
68 
68 
68 

68 
68 
68 
68 
68 

68 
68 
68 
68 
68 

68 
67 
68 
67 
68 

67 
68 
67 
67 
67 

67 



70 
70 
70 
69 
70 
70 
69 
70 
69 

70 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
69 
68 
69 
69 

68 
69 
68 
68 
68 



Lg. Vers. 



8 



90034 
90096 
90158 
90220 
90282 



90344 
90406 
90467 
90529 
90591 

90652 
90714 
90776 
90837 
90899 



90960 
91021 
91083 
91144 
91205 

91267 
91328 
91389 
91450 
91511 



91572 
91633 
91694 
91755 
91815 



91876 
91937 
91997 
92058 
92119 



92179 
92240 
92300 
92361 
92421 



92487 
92542 
92602 
92662 
92722 



92782 
92842 
92902 
92962 
93022 

93082 
93142 
93202 
93261 
93321 



93361 
93440 
93500 
93560 
93619 

9RR79 



I> Lg. Vers, 



r> Log. Exs. J> 



62 
62 
62 
62 

62 
62 
61 
62 
61 

61 
62 
61 
61 
61 

6l 
61 
61 
6l 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
60 

61 
60 
60 
61 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 

59 
60 
59 
60 

59 
59 
60 

59 
59 

59 



93631 
93699 
93766 
93833 
93901 



93968 
94035 
94102 
94170 
94237 



94304 
94371 
94438 
94505 
94572 



94638 
94705 
94772 
94839 
94905 



94972 
9503? 
95105 
95172 
95238 



95305 
95371 
95437 
95504 
95570 



95636 
95703 
95769 
95835 
95901 



95967 
98033 
96099 
96165 
96231 

96297 
96362 
96428 
9G494 
96560 



96625 
96691 
98757 
96822 
98888 



96953 
97018 
97084 
9714g 
97214 



9-^280 
Jv345 
97410 
97475 
97540 



97606 
i> Log, Exs. 

n9 



10 

11 

12 
13 
14 



67 
67 
67 
67 

67 
67 
67 
67 
67 

67 
67 
67 
67 
67 
66 
67 
86 
67 
66 

66 
67 
66 
66 
66 

66 
66 
66 
66 
66 

66 
66 
86 
66 
66 

66 
66 
88 
66 
66 

66 
65 
66 
65 
68 

65 
65 
66 
65 
85 

65 
65 
61 
65 
65 

65 
85 
65 
65 
85 

_ff_60 



15 
16 
17 
18 
29 
30 
21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 
41 
42 
43 
44 



45 

46 

47 

48 

Ji 

50 

51 

52 

53 

_54 

55 
56 
57 
58 
59 



P.P. 





70 


69 


6? 


6 


7.0 


6.9 


6. 


7 


8.1 


8 





7. 


8 


9.3 


9 


2 


9. 


9 


10.5 


10 


3 


10. 


10 


11.6 


11 


5 


11- 


20 


23.3 


23 





22. 


30 


35.0 


34 


5 


34- 


40 


46-6 


46 





45- 


50 


58.3 


57 


5 


56. 



6 
7 
8 
9 

10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50J 



7 
8 
9 
10 
20 
30 
40 
50 



67 66 



6 


7 


8.6 


6. 


7 


8 


7.7 


7- 


8 


9 


8.8 


8- 


10 





9.9 


9. 


11 


1 


11-0 


10- 


22 


3 


22-0 


21. 


33 


5 


33-0 


32- 


44 


6 


44-0 


43. 


55 


8 


55.0 


54. 



64 

■ 4 

• 4 

• 5 
■6 
■6 

■ 3 

■Q 
•6 
.3 



7 
8 
9 
10 
21 
32 
42 
53 



63 

6.3 



65 

5 
6 
6 
7 
8 
6 
5 
3 

i 



63 

6.2 



61 

8.1 
7-1 
8.1 

• 1 

• 1 
■3 
•5 

• 6 
.8 



60 

6-0 

C 



c 







59 

5-9 

6 

7 

8 

9 
19 
29 
39 
49 



7 
8 
9 
10 
20 
30 
40 
50 



O 

0.5 

0.0 

0.5 

0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



P.P. 



fABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANfal 

34° 35° ' 



Lg. Vers. 



93679 
93738 
93797 
93857 
93916 



93975 
94034 
94094 
94153 
94212 



94271 
94330 
94389 
94448 
94506 



94505 
94624 
94683 
94742 
94800 



94859 
94917 
94976 
95034 
95093 



95151 
95210 
95268 
95326 
95384 



95443 
95501 
95559 
95617 
95675 



95733 
95791 
95849 
95907 
959B5 



96023 
96080 
96138 
96196 
96253 



96311 
96368 
96426 
96483 
96541 

9659i] 
96656 
96713 
96770 
96827 



96885 
96942 
96999 
97056 
97113 



8 97170 

Lg. Vers. 



D 

59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
58 

59 
59 
58 
59 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
57 
58 
58 

58 

57 
57 
58 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 



Log.Exs. 



97606 
97671 
97736 
97801 
97865 



97930 
97995 
98060 
98125 
98190 



98254 
98319 
98383 
98448 
98513 



98577 
98642 
98706 
98770 
98835 



98899 
98963 
99028 
99092 
99156 



99220 
99284 
99348 
99412 
99476 



99540 
99604 
99668 
99732 
99796 



99860 
99923 
99987 
00051 
00114 



00178 
00242 
00305 
00368 
00432 



00495 
00559 
00622 
00686 
00749 

008 li 
00875 
00938 
01002 
01065 



01128 
01191 
01254 
01317 
01380 



0144.9 
Log.Exs. 



j> 



65 
65 
65 
64 

65 
65 
64 
65 
65 

64 
64 
64 
65 
64 

44 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
63 
64 
63 
63 
64 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 

i7 



Lg. Vers; 



97170 
97227 
97284 
97341 
97398 



97455 
97511 
97568 
97625 
97681 



97738 
97795 
97851 
97908 
97964 



98020 
98077 
98133 
98190 
98246 



98302 
98358 
98414 
98470 
98527 



98583 
98639 
98695 
98750 
98806 



98802 
98918 
98974 
99030 
99085 



99141 
99197 
99252 
99308 
99363 



9941? 
99474 
■99529 
99585 
99640 



99695 
99751 
99806 
99861 
99916 



99971 
00026 
0008T 
00136 
00191 

00246 
0030T 
00356 
00411 
0046F 
nnH2r 
Lg. Vers. 



-D JLog.Exs. 



57 
56 
57 
57 

57 
56 
57 
56 
56 

56 
57 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
5o 
56 

56 
56 
55 
56 
55 

55 
56 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
54 
55 



01443 
01505 
01568 
01631 
01694 



01756 
01819 
01882 
01944 
02007 



02070 
02132 
02195 
02257 
02319 



02382 
02444 
02506 
02569 
02631 



02693 
02755 
02817 
02880 
02942 



03004 
03066 
03128 
03190 
03252 



03313 
03375 
03437 
03499 
03561 



03622 
03684 
03746 
03807 
03869 



03930 
03992 
04053 
04115 
04176 



04238 
04299 
04360 
04421 
04483 



04544 
04605 
04666 
04727 
0478r 



04850 
04911 
04972 
05033 
05098 



ORl.M 



i* Log.Exs. 



D 



62 
63 
62 
63 

62 
63 
62 
62 
63 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

61 

62 
62 
61 
62 

61 
6l 
62 
61 
6l 

61 
6l 
61 
61 
61 

61 
61 
6l 
61 
6l 

61 
61 
61 
61 
61 

61 
61 
61 
61 
60 

61 





1 
2 
3 

_^ 

5 
6 
7 
8 
_9_ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
,19 

20 

21 
22 
23 
2^ 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
M. 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 

52 

53 

_54 

55 
56 
57 
58 
A9 
60 



P.P. 





65 


64 


63 


6 


6-5 


6-4 


6-3 


7 


7 


6 


7 


4 


7-3 


8 


8 


6 


8 


5 


8-4 


9 


9 


7 


9 


6 


9.4 


10 


10 


8 


10 


6 


10.5 


20 


21 


6 


21 


3 


21.0 


30 


32 


5 


32 





31.5 


4C 


43 


3 


42 


6 


42.0 


5C 


54 


1 


53 


3 


52.5 





6? 


J 


61 


6 


6-2 


6-11 


7 


7 


2 


7 


1 


8 


8 


2 


8 


1 


9 


9 


3 


9 


1 


10 


10 


3 


10 


1 


20 


20 


6 


20 


3 


3C 


31 


C 


30 


5 


40 


4] 


3 


40 


6 


50 


51 


6 


50 


8 



60 

6.0 

7.0 

8.0 

9.0 

10. 

20.0 

30.0 

40.0 

50.0 





51 


i 


58 


57 


6 


5.9 


5-8 


5-7 


7 


6 


9 


6 


7 


6.6 


8 


7 


8 


7 


7 


7-6 


9 


8 


8 


8 


7 


8.5 


10 


9 


8 


9 


6 


9.5 


20 


19 


6 


19 


3 


19.0 


30 


29 


5 


29 





28.5 


40 


39 


3 


38 


6 


38.0 


50 


49 


1 


48 


3 


47.5 





56 


55 


54 


6 


5-6 


5.5 


5-4 


7 


6 


5 


6 


4 


6.3 


8 


7 


4 


7 


3 


7.2 


9 


8 


4 


8 


2 


8.1 


10 


9 


3 


9 


1 


9.0 


20 


18 


6 


18 


3 


18.0 


30 


28 





27 


5 


27.0 


40 


37 


8 


36 


6 


36.0 


50 


46 


6 


45 


8 


45.0 



6 

7 

8 

9 

10 

20 

30 

4P 

50 



O 

0.0 
0.0 
0.0 

d.l 

0.1 
O.I 
0.2 
0.3 
0.4 



p. P; 



720 



!PAfiLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
26° 27° 





1 

2 

3 

_4 

5 
6 
7 

I 8 
^ J> 
I 10 

I 11 
I 12 

I 13 

j 14 



9 



18 

19 

?0 

21 
22 

24 

2. 

gV 
2G 
ii 
30 9 



Lg. Vers. 



83 

S4 

35 

86 
87 
S8 

li 

40 

41 

42 

43 



45 9 



47 
48 
49 



60 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



00520 
00575 
00630 
00684 
00739 



0Q794 
00848 
00903 
00957 

Pioii 

01066 
01120 
01174 
01229 
01283 



01337 
01391 
01445 
01499 
01554 

01608 
01662 
01715 
01769 
01823 



01877 
01931 
01985 
02038 
02092 



02146 
02199 
02253 
02307 
02360 

02414 
02467 
02521 
02574 
02827 



02681 
02734 
02787 
02840 
02894 



02947 
03000 
03053 
03106 
03159 



03212 
03265 
03318 
03371 
03423 



03476 
03529 
03582 
03634 
03687 



9-03740 
Lg. Vers. 



J) 

55 
54 
54 
54 

55 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
53 
54 
54 

54 
53 
54 
53 

54 

53 
53 

H\ 

53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
-52 

53 
53 
52 
52 
53 
52 



Log.Exs 



05154 
05215 
05276 
05337 
05398 

"0545| 
05519 
05580 
05640 
05701 



05762 
05822 
05883 
05943 
06004 

06064 
06124 
06185 
06245 
06305 

06366 
06426 
06486 
06546 
06606 



06667 
06727 
06787 
06847 

069Q7 

06967 
07027 
07087 
07146 
07206 

07266 
07326 
07386 
07445 
07505 

0'.565 
07324 
07684 
07743 
07803 



T> 



07863 
07922 
07981 
08041 
08100 



08160 
08219 
08278 
08338 
08397 



08456 
08515 
08574 
08634 
8693 

08752 

Log.Exs. 



Lg. Vers. 



61 
61 
60 
61 

60 
61 
60 
60 
60 

61 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
59 
60 

60 .q 

59'^ 

60 

59 

60 

53 
59 
59 
59 
60 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 



I> 



03740 
03792 
03845 
03898 
03950 



04002 
04055 
04107 
04160 
04212 



04264 
04317 
04369 
04421 
04473 

04525 
04577 
04630 
04682 
04734 

04786 
04837 
04889 
04941 
04903 



05045 
05097 
05148 
05200 
05252 



05303 
05355 
05407 
05458 
05510 



D 



05561 
05613 
05664 
05715 
05767 

05818 
05869 
05921 
05972 
06023 



06074 
06125 
06176 
06227 
06279 



06330 
06380 
06431 
06482 
.06533 

06584 
06635 



06736 
0_67_87 

06838 
Lg. Vers. 



52 
52 
53 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 

52 
52 

52 
5l 
52 
52 
52 

5l 

52 
51 

52 
52 

5l 
52 
51 
51 
51 

51 
51 
51 
51 
51 

5l 
51 
5l 
51 
51 

51 
51 
51 
51 
51 

51 

50 
51 
51 
51 

51 
50 
51 
50 
51 

50 

IT 



Log.Exs. 



08752 
08811 
08870 
08929 
08988 

09047 
09106 
09164 
09223 
09282 

09341 
09400 
09458 
09517 
09576 



09634 
09693 
09752 
09810 
09869 



09927 
09986 
10044 
10102 
10161 



10219 
10278 
10o36 
10394 
10452 



10511 
10569 
10627 
10685 
10743 



10801 
10859 
10917 
10975 
11033 



11091 
11149 
11207 
11265 
11323 

11380 
11438 
11496 
11554 
11611 



11669 
11727 
11784 
11842 
1 1899 

11957 
12015 
12072 
12129 
12187 

12244 



Log.Exs. 



J> 



59 
59 

59 
59 

59 
59 
58 
59 
59 

58 
59 
58 
59 
58 

58 
58 
59 
58 
58 

38 
58 
58 
5C 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
57 
58 
58 

57 
58 
58 

57 
57 

58 

57 
57 
57 
57 

58 
57 
57 
57 
57 

57 



O 

1 

2 

3 

_4 

5 

6 

7 

8 

__9 

10 

11 

12 

13 

14 

15 
16 
17 
18 
19 



P.P. 



20 

21 
22 
23 
_24 
25 
26 
27 
28 
29 

30 

31 

32 
33 
34 

35 



37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
J9 
50 
51 
52 
53 
54 

55 
56 
57 
58 
-59 
60 



D 





61 


60 


6fl 


6 


6.1 


6.0 


5. 


7 


7 


1 


7.0 


6. 


8 


8 


1 


8.0 


7. 


9 


9 


1 


9.0 


8. 


10 


10 


1 


10.0 


9 


20 


20 


3 


20.0 


19. 


30 


30 


5 


30.0 


29 


40 


40 


6 


40.0 


39 


50 


50 


8 


50.0 


4&. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



58 



5 


8 


5 


6 


7 


6- 


7 


7 


7. 


8 


7 


8- 


9 


6 


9. 


19 


3 


19. 


29 





28. 


38 


6 


38- 


48 


3 


47. 



57 

? 
6 

5 



55 

5.5 

6.4 

7.3 

8.2 

9.1 
18.3 
27-5 
36.6 
45.8 45 -CI 



54 
5.* 
6.3 
7.8 
8-.1 
C.3 
18.0 
27.0 
8-0 



53, 

5.3 

6.2 

7.0 

7-9 

8.8 

17.S 

26.5 

35.3 

44.1 



17 
26 
34 
43 



6 

7 

8 

9 

10 

20 

30 

40 

50 



51 


Cl 


5.1 


0. 


5 


9 





6 


8 


0- 


7 


6 


0. 


8 


5 


0. 


17 





0. 


25 


8 


0. 


34 





0. 


42 


5 


0. 



p.p. 



721 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
38° 39° 



Lg.Vers. 



O 

1 
2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19, 

20 

21 
22 
23 
24 

25^9 

26 

27 

28 

29. 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39_ 

4019 

41 
42 
43 
44 

45 |9 

46 

47 

48 

49 

60|9 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



06838 
06888 
06939 
06990 
07040 

07091 
07141 
07192 
07242 
07293 



JD 



07343 
97393 
07444 
07494 
07544 



07594 
07644 
07695 
07745 
07795 

07845 
07895 
07945 
07995 
0804 



08095 
08145 
08195 
0C244 
08294 

08344 
08394 
08443 
08493 
08543 



08592 
08642 
08691 
08741 
08790 



08840 
08889 
08939 
08988 
09087 



09087 
09136 
09185 
09234 
09284 

09333 
09382 
09431 
09480 
09529 



09578 
09627 
09676 
09725 
09774 



9 •098':^.'^ 



Lg.Vers. 



Log.Exs. I> 



50 
51 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 I 
50 
50 

50 
50 

50 I 

49 

50! 

49 
50 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
48 
49 

49 



12244 
12302 
12359 
12416 
12474 



12531 
12588 
12645 
12703 
12760 

12817 
12874 
12931 
12988 
13045 

13102 
131.9 
13216 
13273 
13330 



13387 
13444 
13500 
13557 
13614 



13671 
13727 
13784 
13841 
13897 

13954 
14011 
14067 
14124 
14180 



14237 
14293 
14350 
14406 
14462 



14519 
14575 
14631 
14688 
14744 



14800 
14856 
14913 
14969 
15025 



15081 
15137 
15193 
15249 
15305 



15361 
15417 
15473 
15529 
15585 



15641 



I> Log.Exs. 



57 
57 
57 
57 

57 
57 
57 
57 
57 

57 1 

57 

57 

57 

57 

57 
57 
57 
56 
57 

57 
57 
56 
57 
56 

57 
56 
57 
56 
56 

57 |Q 

56"^ 

56 

56 

56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56' 

56 

56 

56 

55 

56 



Lg. Vers. 

09823 
09872 
09920 
09969 
10018 



10067 
10115 
10164 
10213 
10261 



10310 
10358 
10407 
10455 
10504 



10552 
10601 
10649 
10697 
1074 6 

10794 
10842 
10890 
10939 
10987 



-O Log.Exs. 



11035 
11083 
11131 
11179 
11227 



11270 
11323 
11371 
11419 
11467 



11615 
11562 
11610 
11658 
11706 

11754 
11801 
11849 
11897 
11944 



11992 
12039 
12087 
12134 
12182 

12229 
12277 
12324 
12371 
12419 



12466 
12513 
12560 
12608 
12655 



12702 



J> jLg. Vers. 



49 
48 
49 
48 
49 
48 
48 
49 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
47 
48 

48 
47 
48 
48 
47 

48 
47 
47 
48 
47 

47 
47 
47 
47 
47 



i 9 



47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 



^9 



15641 
15697 
15752 
15808 
15864 



15920 
15975 
16031 
16087 
16142 
16198 
16254 
16309 
16365 
16420 



16476 

16531 

16587 

1664. 

16698 



16753 
16808 
16864 
16919 
16974 



17029 
17085 
17140 
1719j 
17250 

17305 
17361 
17416 
17471 
17&26 

17581 
17636 
17691 
177'6 
17801 

17856 
17910 
17965 
18020 
18075 

18130 
18185 
18239 
18294 
18349 



18403 
18458 
18513 
18567 
18622 

18676 
18731 
18786 
18840 
18_894 
18949 



I> Log.Exs. 
! I 



56 
55 
56 
55 

56 
55 
56 
55 
55 

56 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
54 
55 

5;t 

55 
55 
54 
55 
54 

54 
55 
54 
54 
54 

54 
54 
55 
54 
54 

54 

IT 



5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 





57 


57 


6 


5.7 


5.71 


7 


6 


7 


6 


6 


8 


7 


6 


7 


6 


9 


8 


6 


8 


5 


10 


9 


6 


9 


5 


20 


19 


1 


19 





3. 


28 


7 


28 


5 


40 


38 


3 


38 





50 


47 


9 


47 


5 



15 
16 
17 
18 
]9 

20 

21 
22 
23 
2A 

25 
26 
27 
28 
29 



30 

31 
32 
33 
34 

35 
36 
37 

Ji 
40 

41 
42 
4? 
44 



P.P. 



4R 
46 
47 
48 

49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 





5( 


5 


55 


6 


5-6 


5.51 


7 


6 


5 


6 


5 


8 


7 


4 


7 


4 


9 


8 


4 


8 


3 


10 


9 


3 


9 


2 


20 


18 


6 


18 


5 


30 


28 





27 


7 


40 


37 


3 


37 





50 


46 


6 


46 


2 



56 

5-6 

6.6 

7.5 

8.5 

9.4 

18.8 

28.2 

37.6 

47.1 

55 

55 

6.4 

7.3 

8.2 

9.1 

18.3 

27 . Fj 

36. B 

45.8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



54 



5-4 


6 


3 


7 


2 


8 


2 


9 


1 


18 


1 


27 


2 


36 


3 


45 


4 



54 

5.4 

6.3 

7.2 

8.1 

3.0 

18.0 

27.0 

36.0 

45. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 
7 
8 
9 

10 
20 
30 
40 
50 



51 50 



5 


1 


5 





5. 


5 


9 


5 


9 


5. 


6 


8 


6 


7 


6. 


7 


6 


7 


3 


7. 


8 


5 


8 


4 


8. 


17 


C 


16 


8 


16. 


25 


5 


25 


2 


25. 


34 


C 


33 


6 


33. 


42 


5 


42 


1 


41. 



49_ 

9 
8 



49 

49 
7 
5 
3 
I 



50 


8 
6 
5 
3 
6 

3 
6 

48 
48 



48 



6 


4 


8 


4 


7 


5 


6 


5 


8 


6 


4 


6 


9 


7 


2 


7 


10 


8 





7 


20 


16 





IF 


30 


24 


C 


23 


40 


32 





31 


FP 


40 





39 



47 
7 
5 
3 
1 
9 
6 

2 

6 
6 



47 

4.7 



P.P. 



722 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTSi 



30° 



31' 



Lg. Verf. 



12702 
12749 
12796 
12843 
12890 



12937 
12984 
13031 
13078 
13125 



13172 
13219 
13266 
13813 
13359 



13406 
13453 
13500 
13546 
13593 

13639 
13686 
13733 
13779 
13826 



13872 
13919 
13965 
14011 
14058 



14104 
14151 
14197 
14243 
14289 



14336 
14382 
14428 
14474 
14520 



14566 
14612 
14658 
14704 
14750 



14796 
14842 
14888 
14934 
14980 



15026 
15071 
15117 
1516S1 
15209 



15254 
15300 
15346 
15391 
15437 
9 15483 
Lg. Vers. 



I) 



47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
46 
47 
47 
46 

47 
46 
47 
46 
46 

46 
47 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

46 
46 
48 
46 
45 

46 
45 
46 
46 
45 

45 
46 
45 
45 
45 

46 



Log. Exs. 



9-18949 
19003 
19058 
19112 
19167 



19221 
19275 
19329 
19384 
19438 



19492 
19546 
19601 
19655 
19709 



19763 
19817 
19871 
19925 
19979 



2003_ 
20087 
20141 
20195 
20249 

20303 
20357 
20411 
20465 
20518 



20572 
20626 
20680 
20733 
20787 



20841 
20894 
20948 
21002 
21055 



21109 
21162 
21216 
21269 
21323 



21376 
21430 
21483 
21537 
21590 



21643 
21697 
21750 
21803 
21857 



21910 
21963 
22016 
22070 
22123 
22176 
Log. Exs. 



J) 



Lg. Vers, 



54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 
54 
54 
54 
53 

51 
53 
54 
53 
54 

53 
53 
54 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 



15483 
15528 
15574 
15619 
15665 



15710 
15755 
15801 
15846 
15891 



15937 
15982 
16027 
16073 
16118 



16163 
16208 
16253 
16298 
16343 

16388 
16434 
16479 
16523 
16568 



16613 
1665G 
16703 
16748 
16793 



16838 
16882 
16997 
16972 
17017 



17061 
17106 
17151 
17195 
1724P 



17284 
17329 
17373 
17418 
17462 



17507 
17551 
17596 
17640 
17684 



17729 
17773 
17817 
17861 
17906 



17950 
17994 
1803R 
18082 
18126 
18170 



7> 



45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
44 
45 

45 
45 
45 
45 
44 

45 
44 
45 
44 
45 

44 
44 
45 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 



Log. Exs 



22176 
22229 
22282 
22335 
22388 



22441 
22494 
22547 
22600 
22653 



22706 
22759 
22812 
22865 
22918 



22971 
23024 
23076 
23129 
23182 

23235 
23287 
23340 
23393 
23446 



23498 
23551 
23603 
23656 
23709 



23761 
23814 
23866 
23919 
23971 



24024 
24076 
24128 
24181 
24233 

24285 
24138 
24390 
24442 
24495 



24547 
24599 
24651 
24704 
24756 



24808 
24860 
24912 
24964 
25016 



250.68 
25120 
25172 
25224 
25276 

25328 
Log. Exs, 



D 

53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
52 

53 
53 
52 
53 
52 

53 
52 
53 
52 
53 

52 
52 
52 
52 
53 

52 
52 
52 
52 
52 

53 

52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 

Id 



10 

11 

12 
13 

ii 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 



25 
26 
27 
28 
29 

30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

42 
43 
44 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 



60 



P.P. 





5l 


54 


6 


5.4 


5.4 


7 


6.3 


6.3 


8 


7.2 


7.2 


9 


8.2 


8-1 


10 


9.1 


9.0 


20 


18.1 


18.0 


30 


27.2 


27.0 


40 


36.3 


360 


50 


45.4 


45.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



53 

5.3 



53. 
5.2 



55 

5.3 

6.2 

7.1 

8.0 

8.9 

17.8 

26.7 

35.6 

44.6 



53 
5.2 



47_ 
7 
5 
3 
1 
9 
8 
7 
6 
6 



47 
4.7 



46 

4-6 



46 


45 


4^ 


4-6 


4-5 


4. 


5 


3 


5 


3 


5. 


6 


1 


6 





6. 


6 


9 


6 


8 


6. 


7 


6 


7 


6 


7. 


15 


3 


15 


1 


15. 


23 





22 


7 


22. 


30 


6 


30 


3 


30. 


38 


3 


37 


9 


37. 





44 


6 


4.4 


7 


5.2 


8 


5-9 


9 


6.7 


10 


74 


20 


14.8 


30 


22.2 


40 


29.6 


50 


37.1 



44 

4.4 

5.1 

5.8 

6-6 

7-3 

14.6 

22.0 

29.3 

36.6 



■pTT 



723 



^ABLE VIIT— LOQARITHMJC VERS^Pl SII^ES Al^D EXTERNAL SECANTS. 
33° 33° 



Lg. Vers. 



7 

8 

^ 

10 

11 
12 
13 
14 
15 
18 
17 
18 
19 

20 

21 
22 
23 
24 

25 

n 

28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

go 



18170 
18214 
18258 
18302 
18346 



J> 



18390 
18434 
18478 
18522 
18566 
18610 
18654 
18697 
18741 
18785 



18829 
18872 
18916 
18959 
19003 



19047 
19090 
19131 
19177 
19221 



19264 
19308 
19351 
19395 
19438 

19481 
19525 
19568 
19611 
196_54 
19698 
19741 
19784 
19827 
19870 



19914 
19957 
20000 
20043 
20086 



20129 
20172 
20215 
20258 
20301 



20343 
20386 
20429 
20472 
20515 



20558 
20600 
20643 
20686 
20728 

9.20771 
Lg. Vers. 



44 
44 
44 
44 

44 
44 
44 
44 
43 
44 
44 
43 
44 
43 

44 
4i 
43 
43 
44 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

42 
43 
43 
43 
42 

43 
42 
43 
43 
42 

43 



Log.Exs. 



25328 
25380 
25432 
25484 
25536 



25588 
25640 
25692 
25743 
25795 



25847 
25899 
25950 
26002 
26054 



26105 
26157 
26209 
26260 
26312 

26364 
26415 
26467 
26518 
26570 



26621 
26673 
26724 
26776 
26827 



26878 
26930 
26981 
27032 
27084 



27135 
27186 
27238 
27289 
27340 



27391 

27443 
27494 
27545 
27596 



27647 
27698 
27749 
27800 
27852 



27903 
27954 
28005 
28056 
28107 



28157 
28208 
28259 
28310 
28361 
28412 



i> Log.Exs. 



D 

52 
52 
52 
51 

52 
52 
52 
PI 
52 

5l 
52 
5l 
52 
51 

51 
52 
5l 
51 
51 

52 
51 
51 
5l 
51 

51 
51 
51 
5l 
51 

51 
51 
5l 
51 
5l 

5l 
51 
5l 
51 
5l 

51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

50 
51 
51 
51 
51 

50 



Lg. Vers. 



20771 
20814 
20856 
20899 

2_P?42 

20984 
21027 
21069 
21112 
21154 



21196 
21239 
21281 
21324 
21366 



21408 
21451 
21493 
21535 
21577 



21620 
21662 
21704 
21746 
21788 



21830 
21872 
21914 
21956 
21998 



22040 
22082 
22124 
2216(3 
22208 



22250 
22292 
22334 
22376 
22417 

22459 
22501 
22543 
22584 
22626 



22668 
22709 
22751 
22792 
22834 

22876 
22917 
22959 
23000 
23042 



23083 
23124 
23166 
23207 
23248 
23290 



I> Lg'Vers, 



li L 


og.Exs. 


42 9 
42 
42 
43 


•28412 
•28463 
•28514 
•28564 
•28615 


42 c 
42 " 
42 
42 
42 


•28666 
•28717 
•28768 
•28818 
•28869 


4^ < 
42 ^ 
42 
42 
42 


) -28920 
•28970 
-29021 
-29072 
•29122 


42 ( 
42 ' 
42 
42 
42 


)-29173 
•29223 
-29274 
-29324 
•?9375 


42 ■ 
42 ' 
42 
42 
42 


3 29426 
• 29476 
•29527 
.29577 
■29627 


42 , 
42 ^ 
42 
42 
42 


)• 29678 
•29728 
•29779 
-29829 
-29879 


42 ( 
42 ' 
42 
42 
42 


)■ 29980 
•29980 
•30030 
■30081 
•30131 


42 ( 
41 ^ 
42 
42 
41 


). 30181 
■30231 
•30282 
•30332 
■30382 


42 c 

41 ' 
42 
41 
41 

42 ( 
41 ' 
41 
41 

4i 


3 30432 
■30482 
■30533 
■30583 
•30633 

)■ 30683 
•30733 
•30783 
•30833 
•30883 


41 ( 
41 ^ 
41 
4J 
41 


)• 30933 
•30983 
•31083 
•31C8S 
•31133 


41 c 
41 ^ 
41 
4l 
41 

41 9 


K31183 
•31233 
•31283 
•31333 
•31383 

■31432 

.og.Exs. 



51 
51 
50 
51 

51 
50 
51 
50 
50 

51 
50 
50 
51 
50 

51 
50 
50 
50 
51 
50 
50 
50 
50 
5C 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 

50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
50 
50 
50 

49 





1 

2 
3 

_4_ 

5 
6 
7 
8 
_9 

10 

li 
12 
13 
14 

15 
16 
17 
18 
J9 

20 

21 
22 
23 
24 

25 
26 
27 
28 
-29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
A4 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
38 
59 

60 



P. P. 



6 

? 
8 
9 

10 
20 
30 
40 
50 



52 

5^2 



51 51 



• 1 


5. 


• 


5. 


8 


6^ 


•7 


I- 


■ 6 


8- 


■ 1 


17- 


•7 


25- 


■ 3 


34 • 


• 9 


42 • 





60 


50 


6 


5^0 


5^0| 


7 


5 


9 


5 


8 


8 


6 


7 


6 


6 


9 


7 


6 


7 


5 


10 


8 


4 


8 


3 


20 


16 


8 


16 


6 


30 


25 


2 


25 





40 


33 


6 


33 


3 


50 


42 


1 


41 


6 



.1 

• 9 
•8 

• 6 
-5 
-0 

5 

• 

• 5 



4.9 

5.8 

6^6 

7^4 

82 

16^5 

24-7 

33-Q 

41-2 



6 

7 
8 
9 
10 
20 
SO 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



44 

4-4 



43 

4.31 



43 

4^3 



42_ 



4'2 
4.-2 



41 

4-1 
4-8 
5-5 
fi.2 
6 9 

'20-7 
27-6 
34-6 





41 


6 


4.1 


7 


4-8 


8 


5-4 


9 


6.1 


10 


6.§ 


20 


13.6 


30 


20-5 


40 


27-3 


50 


34.1 



P. p. 



724 



TABLE Vni.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
34° 35° 



Lg. Vers. 



9.23290 
23331 
23372 
23414 
23455 



23496 
23537 
23579 
23620 
23661 



23702 
23743 
23784 
23825 
23866 



23907 
23948 
23989 
24030 
24071 



24112 
24153 
24194 
24235 
24275 



24316 
24357 
24398 
24438 
24479 



24520 
24561 
24601 
24642 
24682 



24723 
24764 
24804 
24845 
24885 



24926 
24966 
25007 
25047 
25087 



25128 
25168 
25209 
25249 
25289 



25329 
25370 
25410 
25450 
25490 



25531 
25571 
25611 
25651 
25691 



9. 25731 



Lg. Vers, 



JD 



41 
41 
41 
41 

4l 
41 
4l 
41 
41 

41 
41 
41 
41 
41 

41 
41 
41 
41 
41 

40 
41 
41 
41 
40 

41 
40 
41 
40 
41 

40 
41 
40 
40 
40 
41 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
4S 
40 
40 
40 

40 
40 
40 
40 
40 

40 



JD 



Log.Exs. 



31432 
31482 
31532 
31582 
31632 



31681 
31731 
31781 
31831 
31880 



31930 
31980 
32029 
32079 
32129 



32178 
32228 
32277 
32327 
32377 



32426 
32476 
32525 
32575 
32624 



32673 
32723 
32772 
32822 
32871 



32920 
32970 
33019 
33069 
33118 



33167 
33216 
33266 
33315 
33364 



33413 
33463 
33512 
33561 
33610 



33659 
33708 
33758 
33807 
33856 



33905 
33954 
34003 
340521 
34101 



34150 
34199 
34248 
34297 
34346 
34395 



Log.Exs. 



2> 



50 
50 
49 
50 

49 

50 

49 

50 I 

49 

50 

49 

49 

49 

50 

49 
49 
49 
49 
50 

49 
49 
49 
49 
49 

49 

49 
49 
49 
49 

49 

49 
49 
49 
49 

49 
49 
49 
49 

49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 



Lg. Vers, 



J> 



25731 
25771 
25811 
25851 
25891 



25931 
25971 
26011 
26051 
26091 



26131 
26171 
26210 
26250 
26290 



26330 
26370 
26409 
26449 
26489 

26528 
26568 
26608 
26647 
26687 



26726 
26766 
26806 
26845 
26885 



26924 
26964 
27003 
27042 
27082 



27121 
27161 
27200 
27239 
27278 

27318 
27357 
27396 
27435 
27475 



27514 
27553 
27592 
27631 
27670 



27709 
27749 
27788 
27827 
27866 



27905 
27944 
27982 
28021 
28060 



28099 



Lg, Vers, 



1) 



Log.Exs. 



40 
40 
40 
40 

40 
40 I 
401 
39 
40! 

40 i 

40 

39 

40 

40 

39 
40 
39 
40 
39 

39 
40 
39 
39 
39 

39 
40 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
38 
39 
39 

39 



34395 
34444 
34492 
34541 
34590 



34639 
34688 
34737 
34785 
34834 



34883 
34932 
34980 
34029 
35078 



35127 
35175 
35224 
35273 
35321 

35370 
35419 
35467 
35516 
35564 



35613 
35661 
35710 
35758 
35807 



35855 
35904 
35952 
36001 
36049 



36098 
36146 
36194 
36243 
36291 

36340 
36388 
36436 
36484 
36533 



36581 
36629 
36678 
36726 
36774 



36822 
36870 
36919 
36967 
37015 



37063 
37111 
37159 
37207 
37255 



3730_3 
.Exs 



I) 

49 
48 
49 
49 

49 
48 
49 
48 
49 

49 
48 
48 
i9 
48 

49 
48 
49 
48 
48 

48 
49 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
43 
48 

48 
48 
48 
48 
48 



48 
48 
48 

48 



i 
2 
3 
4 

5 
6 



9 

10 

11 

12 

13 

_14 

15 
16 
17 
18 
19 



30 

21 
22 
23 
24 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
li 
35" 
36 
37 
38 
39 



40 
41 
42 
43 
44 



45 
46 
47 
48 
49 

fso 

51 
52 
53 



54 
48' ^^ 



56 

57 

58 

_59 

160 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



50 

8 



49_ 

4.9 

5 

6 

7 

8 
16 
24 
33 
41 



49 

4.9 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 

41 

4 

5 

6 

6 
13 
20 
27 
34 



40 



4 


d 


1 . 


4 


7 


4. 


5 


4 


5. 


6 


1 


6. 


6 


7 


6- 


13 


5 


13- 


20 


2 


20. 


27 





26. 


33 


7 


33. 



39 


39 


3.9 


3.9 


4 


6 


4 


5 


5 


2 


5 


2 


5 


9 


5 


8 


6 


6 


6 


5 


13 


1 


13 





19 


7 


19 


5 


26 


3 


28 





32 


9 


32 


9 



48 48 


6 


4.8 


48 


7 


5 


6 


5 


6 


8 


6 


4 


6 


4 


9 


7 


3 


7 


2 


10 


8 


1 


8 





20 


16 


1 


16 





30 


24 


2 


24 





40 


32 


3 


32 





50 


40 


4 


40 






41 

4.1 



40 







38 


6 


3.S 


7 


4 


5 


8 


5 


I 


9 


5 


8 


10 


6 


4 


20 


12 


8 


30 


19 


2 


40 


25 


6 


50 


32 


1 



P.P. 



725 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
36° 37° 



Lg. Vers. 



28099 
28138 
28177 
28816 
28255 



28293 
28332 
28371 
28410 
28448 



28487 
28526 
28564 
28603 
28642 



28680 
28719 
28757 
28796 
28835 



28873 
28912 
28950 
28988 
29027 



29065 
29104 
29142 
29180 
29219 



29257 
29295 
29334 
29372 
29410 



29448 
29487 
23525 
29563 
29601 



29639 
29677 
29715 
29754 
29792 



29830 
29868 
29906 
29944 
29982 



30020 
30057 
30095 
30133 
30171 



30209 
30247 
30285 
30322 
30360 



9-30398 
Lg. Vers, 



l> 

39 
38 
39 
39 

38 
3? 
38 
39 
38 
39 
38 
38 
39 
38 

38 
38 
38 
38 
39 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
37 
38 
38 
38 

38 
37 
38 
37 
38 

38 



Log. Exs. 



9.37303 
37352 
37400 
37448 
37^.96 

9.37544 
•37592 
.37640 
.37687 
-37735 



9.37783 
.37831 
.37879 
.37927 
.37975 



9.38023 
.38U71 
.38119 
.38166 
•38214 



9.38262 
.38310 
•38357 
.38405 
-38453 



9.38501 
•38548 
.38596 
. 38644 
.38692 



9-38739 
.38787 
.38834 
.38882 
-38930 



9.38977 

.39025 

.39072 

.39120 

39168 



9.39215 
.39263 
.39310 
.39358 
.39405 



9-39453 
.39500 
.39548 
.39595 
.39642 



9.3969C 
.39737 
•39785 
.3983? 
.39879 



9.39927 
.39974 
.40021 
.40069 
.40116 



9-40163 
Loff.Exs. 



n 

48 
48 
48 
48 

48 
48 
48 
47 
48 

48 
48 
48 
48 

47 

48 
48 
48 
47 
48 

47 
48 
47 
48 
47 

48 
47 
48 
47 
48 

47 
47 
47 
48 
47 

47 
47 
47 
48 
47 

47 
48 

47 
47 
47 

47 
47 
47 
47 
47 

47 
47 

47 
47 
4? 

47 
47 
47 
47 
47 

47 

la 



Lg. Vers. 



30398 
30436 
30474 
30511 
30549 



30587 
30624 
30662 
30700 
30737 



J> 



30775 
30812 
30850 
30887 
30925 

30962 
31000 
31037 
31075 
31112 



31150 
31187 
31224 
31262 
31299 



31336 
31374 
31411 
31448 
31485 



3152a 
31560 
31597 
31634 
31671 



31708 
31746 
31783 
31820 
31857 



31894 
31931 
31968 
32005 
32042 



32079 
32116 
32153 
32190 
32227 



32263 
32300 
32337 
32374 
32411 



32447 
32484 
32521 
32558 
32594 



3263] 
Lg. Vers. 



37 
38 
37 
37 

38 
37 
37 
38 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

36 
37 
37 
36 
37 

36 
37 
36 
37 
SB 

37 



Log. Exs. 

9^40163 
.40210 
.40258 
.40305 
.4C352 



9 - 4C399 
.40447 
. 40494 
-40541 
-4C588 



9-40635 
-40682 
-40730 
-40777 
-40824 



9-40871 
.40818 
.40965 
.41012 
-41059 

9-41106 
-41153 
-41200 
-41247 
-41294 



9-41341 
-41383 
-41435 
-41482 
-41529 



9-41576 
-41623 
.41670 
.41717 
-41763 



9-41810 
-41857 
-41904 
-41851 
■4]898 



9 . 42044 
-42081 
.42138 
.42185 
.42231 



9.42278 
.42325 
.42372 
.42418 
.42465 

9.42512 
.42558 
.42605 
.42652 
.42688 



9.42745 
.42792 
.42838 
.42885 
.42931 



9-42978 



I> [Log. Exs. 



I> 

47 
47 
47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

46 
47 
47 
47 
46 

47 
47 
46 
47 
47 
46 
47 
46 
47 
46 

47 
46 
47 
46 
47 

46 
46 
47 
46 
46 

46 
47 

46 
46 
46 

46 



30 

21 
22 
23 
24 

25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
89 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59. 
60 



z>j. 



p. p. 





48 


48 


6 


4-8 


48 


7 


5 


6 


5.6 


8 


6 


4 


6-4 


9 


7 


3 


7.2 


10 


8 


1 


8.0 


20 


16 


1 


16-0 


80 


24 


2 


24.0 


40 


32 


3 


320 


50 


40 


.4 


40.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



47 



47 
4.7 
- 5 
2 




6 


4. 


7 


5- 


8 


6- 


9 


7- 


10 


7- 


20 


15- 


30 


23- 


40 


31- 


50 


38- 



46 

6 
4 
2 

7 
5 
2 

7 





39 


38 


6 


3-8 


3-8 


7 


4-5 


4-5 


8 


5-2 


5-1 


9 


5-8 


5-8 


10 


6-5 


6-4 


20 


13-0 


12-S 


30 


10-5 


19-2 


40 


26-0 


25.6 


50 


32-5 


32.1 









38 37 


6 


3-8 


3-7 


7 


4-4 


4-4 


8 


5-0 


5.0 


9 


5-7 


5-6 


10 


6-3 


6-2 


20 


12-6 


12-5 


SO 


19-0 


18.7 


40 


25-3 


25.0 


50 


31-6 


31.2 









37 


36 


6 


8-7 


3.6 


7 


4-3 


4-2 


8 


4-9 


4-8 


9 


5-5 


5.5 


10 


6-1 


6.1 


20 


12-3 


12-1 


80 


18-5 


18-2 


40 


24-6 


24-3 


PO 


30 -R 


30-4 






P.P. 



'70A 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
38° 39° 



Lg. Vers, 



32631 
32668 
32704 
32741 
32778 



32814 
32851 
32888 
32924 
32981 



32997 
33034 
33070 
33107 
33143 



33180 
33216 
33252 
3328? 

33325 



33381 
33398 
33434 
33470 
33507 



33543 
33579 
33815 
33652 
33888 

33724 
33760 
33796 
33833 
33869 



33905 
33941 
33977 
34013 
34049 



34085 
34121 
34157 
34193 
34229 



34265 
34301 
34337 
34373 

34408 



34444 
34480 
34516 
34152 
34587 



34623 
34659 
34695 
3473Q 
3476P. 

9. 34802 

Lor. Vers* 



n 

36 
36 
37 
36 

36 
38 
37 
36 
36 

36 
38 
36 
38 
36 

36 
38 

36 
36 
36 

36 

36 
36 
38 
36 

38 
38 
36 
36 
36 

36 
36 
38 
36 
38 

38 
38 
36 
36 
36 

36 
36 
38 
36 
36 

36 
36 
36 
36 
35 

38 
36 
35 
36 
35 

36 
35 
38 
35 
36 

35 



Log.Exs. 



42978 
43024 
43071 
43118 
43184 



43211 
43257 
43304 
43350 
43396 



43443 
43489 
43536 
43582 
43629 

43675 
43721 
43768 
43814 
43831 



43907 
43953 
43999 
44046 
44092 



44138 
44185 
44231 
44277 
44323 



44370 
44416 
44462 
44508 
44554 



44601 
44847 
44693 
44739 
44785 



44831 
44877 
44924 
44970 
45016 



45062 
45108 
45154 
45200 
45246 



45292 
45338 
45384 
45430 
45476 



45522 
45568 
45614 
45660 
45706 



D 



45752 



I> Log.Exs. 



46 
47 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
48 

46 
46 
48 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 



Lg. Vers. 



34802 
34837 
34873 
34909 
34944 



34980 
35016 
35051 
35087 
35122 



35158 
35193 
3522? 
35264 
35300 



35335 
35370 
35406 
35441 
35477 



35512 
35547 
35583 
35818 
35653 



35689 

85724 
35759 
35794 
35829 



35865 
35900 
35935 
35970 
36005 



16040 
36076 
36111 
36148 
36181 



36216 
36251 
36286 
36321 
36356 



36391 
36426 
38461 
36495 
36530 

36565 
36600 
36635 
36670 
36705 



36739 
36774 
36809 
36844 
3 6878 
36913 



1) Lg.Vers. I> 



n 

35 
36 
35 
35 

35 
36 

35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
35 
34 
35 
35 

34 
35 
34 
35 
34 

35 



Log.Exs. 



45752 
45797 
45843 
4588? 
45935 



45981 
46027 
46073 
46118 
46164 



46210 
46256 
46302 
46347 
46393 



46439 
46485 
46530 
46576 
46622 



46668 
48713 
46759 
46805 
46850 



46896 
46942 
46987 
47033 
41078 

47124 
47170 
47215 
47261 
47306 



47352 
47398 
47443 
4748? 
47534 



47580 
47625 
47671 
47716 
47782 



47807 

47852 

47898 

4794 

47989 



48034 
48080 
48125 
48170 
4821P 



4826] 
48306 
4835^ 
48397 
48442 



9 ■ 48488 
Log.Exs. 



n 

45 
46 
46 
46 

45 
46 
46 
45 
48 

46 
45 
46 
45 
46 

45 
46 
45 
46 
45 

46 
45 
45 
46 
45 

45 
46 
45 
45 
45 

46 
45 
45 
45 
45 

46 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 





1 

2 

3 

_4 

5 
8 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
2A 

25~ 
26 
27 
28 
2^ 

30 

31 
32 
33 
-34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
li 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
-59 
60 



P.P. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



47 



4 


7 


5 


5 


6 


2 


7 





7 


8 


15 


6 


23 


5 


31 


3 


39 


1 



46_ 

4-6 

5.4 

6-2 

7.0 

7.7 

15.5 

23.2 

31-0 

38.7 





46 


45 


6 


4-6: 4-5 


7 


5 


3 


5.3 


8 


6 


1 


6 


9 


6 


9 


6 8 


10 


7 


6 


7.6 


20 


15 


3 


15.1 


30 


23 


022-7 


40 


30 


6:30.3 


50 


38 


3137.9 



7 
8 
9 
10 
2Q 
SO 
40 
50 



45 

4.5 

5 

6 

6 

7 
15 
22 
30 
37 



6 

7 

8 

9 

10 

20 

30| 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



37 

37 



3.g 



36 

3.6 



35_ 

35 





35 


3l 




e 


3.5 


3-4 




7 


4.1 


4.0 




p 


4-6 


4.6 







5.2 


5.2 




lb 


5.8 


5.7 




20 


11.6 


11.5 




30 


17.5 


17.2 




40 


23.3 


23. 




50 


29.1I9R.7 


^ _ 



P.P. 



727 



S*S5&!L'te VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL "SECANTS 
40° 41° 



Lg. Vers. 



36913 
36948 
36982 
37017 
37052 



37088 
37121 
37156 
37190 
37225 



37259 
37294 
3732C 
37363 
37397 



37432 
37466 
37501 
37535 
37570 



37604 
37639 
37673 
37707 
37742 



37776 
37810 
37845 
37879 
37913 



37947 
37982 
38016 
38050 
38084 



38118 
38153 
38187 
38221 
38255 



38289 
883 2S 
38357 
38391 
38425 



38459 
38493 
38527 
38561 
38595 



3862P 
38663 
38697 
3873] 
38765 



38799 
38833 
38866 
38900 
38934 



60|9_389R8 
' iLg.Vers. 



D 

34 
34 
35 
34 

34 
35 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 

34 

34 
34 
34 
34 
34 

34 
34 
34; 
34 
34 

34 
34 

34: 

34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
33 
34 

34 
34 
33 
34 
33 

34 



Log.Exs. 



48488 
48533 
48578 
48624 
48669 



48714 
48759 
48805 
48850 
48895 



4894G 
48986 
49031 
49076 
49121 

49166 
49211 
49257 
49302 
49347 



49392 
49437 
49482 
49527 
49572 



49618 
49663 
49708 
49753 
49798 



49843 
49888 
49933 
49978 
50023 



50068 
50113 
50158 
50203 
50248 



50293 
50338 
50383 
50427 
50472 



50517 
50562 
50607 
50652 
50697 



50742 
50787 
5083] 
50876 
50921 



50966 
5101] 
51055 
51100 
51145 
51190 



D I Log.Exs. 



D 



45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
44 
45 

45 
45 
45 
44 
45 

45 
45 
44 
45 
45 

44 
45 
44 
45 
45 

44 



Lg. Vers. 



38968 
39002 
39035 
39069 
39103 



39137 
39170 
39204 
39238 
39271 



39305 
39339 
39372 
39406 
39439 



39473 
39507 
39540 
39574 
39607 



39641 
39674 
39708 
39741 
39774 



39808 
39841 
39875 
39908 
39941 



39975 
40008 
40041 
40075 
40108 



40141 
40175 
40208 
40241 
40274 



2) 



40307 
40341 
40371 
40407 
40440 



40473 
40506 
40540 
40573 
40606 



40639 
40672 
40705 
40738 
40771 



40804 
40837 
40870 
40903 
.49936 
409^9 



34 
33 
34 
33 

34 
33 
33 
34 
33 

33 
34 
33 
33 
33 

33 
34 
33 

33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

S3 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 

33 
33 

33 
33 
38 
33 
33 

33 



Log.Exs. 



i> ILg.Vers. XJJLog.Exs. J> 



51190 
51235 
51279 
51324 
51369 



51414 
51458 
51503 
51548 
51592 



51637 
51682 
51726 
51771 
51816 



51860 
51905 
51950 
51994 
52039 



52084 
52128 
52173 
52217 
5226 



52306 
52351 
52396 
52440 
52485 



52529 
52574 
52618 
52663 
52707 



52752 
52796 
52841 
52885 
52930 

52974 
53018 
53063 
53107 
53152 



53196 
53240 
53285 
53329 
53374 



53418 

53462 
E3507 
53551 
53595 



53640 
53684 
53728 
53773 
538] 7 



D 



45 
44 
45 
44 

45 
44 
45 
44 
44 

45 

44 
44 
45 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 



^i 24 



44 
45 
44 
44 
44 

44 
44 
44 
44 
44 

44 

44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 



25 
26 
27 
28 
29 



30 

31 
32 
33 
M. 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 



^f 54 



53RR] 



44 
44 
44 
44 
44 

44 



55 
56 
57 
58 
_5.9 
60 



P.P. 



6 


4 


7 


5 


8 


6 


9 


6 


10 


7 


20 


15 


3C 


22 


40 


30 


50 


37 



45 

5 
3 

8 
6 
1 
7 
3 
9 



46 

4.5 
5.§ 





44 


6 


4-4 


7 


5.2 


8 


5.9 


9 


6.7 


10 


7-4 


20 


14.8 


30 


22.2 


40 


29.6 


50 


37.1 



44 



5 


f 


5 


g 


fi 


6 


7 


3 


14 


6 


22 





29 


3 


36 


6 



6 
7 
8 
9 

10 
20 
30 
40 
50 



35 



3 


5 


4 


I 


4 


6 


5 


2 


5 


8 


11 


6 


17 


5 


23 


3 


29 


1 



31 

3.4 

4.0 

4-6 

5.2 

5.7 

11.5 

17-2 

23-0 

28.7 









34 33 


6 


34 


3.3 


7 


39 


39 


8 


4-5 


4.4 


9 


5-1 


5.0 


10 


5.^ 


5.6 


20 


11.3 


11. 1 


30 


17.0 


16. Z 


40 


22.6 


22.3 


50 


28.3 


27.9 



6 
7 
8 

9 
10 
20 
30 
40 
50 



33 

33 

3-8 

4.4 

4.9 

5.5 

11.0 

16.5 

22.0 

27.5 



P. P. 



728 



TABLE VIII —LOGARITHMIC VERSED SINES AND EXTERNAL SECANT^a, 



Lg. Vers. 



40969 
41001 
41034 
41067 
41100 



41133 
41166 
41199 
41231 
41264 



41297 
41330 
41362 
41395 
41428 



41481 
41493 
41526 
41559 
41591 



41624 
41657 
41689 
41722 
41754 



41787 
41819 
41852 
41885 
41917 



41950 
41982 
42014 
42047 
42079 



42112 
42144 
42177 
42209 
42241 



42274 
42306 
42338 
42371 
42403 



42435 
42467 
42500 
42532 
42564 



42596 
42629 
42661 
42693 
42725 



42757 
42789 
42822 
42854 
42886 



9-42918 



Lg. Vers. 



43"= 



43° 



D 

32 
33 
33 
33 
32 
33 
33 
32 
33 

32 
33 
32 
33 
32 

33 
32 
33 
32 
32 

32 
33 
32 
32 
32 

32 
32 
32 
33 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 



Log.Exs. 



53861 
53906 
53950 
53994 
54038 



54083 
54127 
54171 
54215 
54259 



54304 
54348 
54392 
54436 
54480 



54525 
54569 
54613 
54657 
54701 



54745 
54790 
54834 
54878 
54922 



54966 
55010 
55054 
55098 
55142 



55186 
55230 
55275 
55319 
55363 



55407 
55451 
55495 
55539 
55583 



55627 
55671 
55715 
55759 
55803 



55847 
55890 
55934 
55978 
56022 



56066 
56110 
56154 
56198 
56242 



56286 
56330 
56374 
56417 
56461 



56505 



I> Log.Exs. 



n 



44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 

44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
43 
44 
44 
44 

44 
44 
44 
43 
44 

44 
44 
44 
43 
44 

43 



Lg. Vers. 



42918 
42950 
42982 
43014 
43046 



43078 
43110 
43142 
43174 
43206 



43238 
43270 
43302 
43334 
43365 



43397 
43429 
43461 
43493 
43525 



43557 
43588 
43620 
43652 
43684 

43715 
43747 
43779 
43810 
43842 



43874 
43906 
43937 
43969 
44000 



44032 
44064 
44095 
44127 
44158 



44190 
44221 
44253 
44284 
44316 



44347 
44379 
44410 
44442 
44473 



44504 
44536 
44567 
44599 
44630 



44661 
44693 
44724 
44755 
44787 



44818 



I> Lg. Vers. 



D 



32 
32 
32 
32 

32 
32 
3l 
32 
32 

32 
32 
32 
32 
31 

32 
32 
32 
31 
32 

32 
31 
32 
31 
32 

3l 
32 
31 
31 
32 

3l 
32 
31 
31 
31 

32 
31 
31 
31 
31 

31 
31 

31 
31 
31 

31 
31 
31 
31 
31 

31 
3l 
31 
31 
31 

3l 
3l 
31 
31 
31 

31 



Log.Exs. 



56505 
56549 
56593 
56637 
56680 



56724 
56ro8 
56812 
56856 
56899 



56943 
56987 
57031 
57075 
57118 



57162 
57206 
57250 
57293 
57337 



57381 
57424 
57468 
57512 
57556 



57599 
57643 
57687 
57730 
57774 



57818 
57861 
57905 
57949 
57992 



58036 
58079 
58123 
58167 
58210 



58254 
58297 
58341 
58385 
58428 



58472 
58515 
58559 
58602 
58646 



58689 
58733 
58776 
58820 
58864 



58907 
58951 
58994 
59037 
59081 



59124 



I> Log.Exs. 
729 



D 

43 
44 
44 
43 

44 
44 
43 
44 
43 

44 
44 
43 
44 
43 

44 
43 
44 
43 
44 

43 
48 
44 
43 
44 

43 
43 
44 
43 
43 

44 
43 
43 
44 
43 

43 
43 
44 
43 

43 

43 

43 
44 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 



2> 



5 
6 
7 
8 
9^ 

10 

11 
12 
13 
14 



30 

21 
22 
23 
24 



30 

31 
32 
33 
31 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
j49 

50 

51 
52 
53 

55 
56 
57 
58 
59 

60 



P.P. 





41 


44 


6 


4.4 


4.4 


7 


5.2 


5.1 


8 


5-9 


5.8 


9 


6.7 


6.6 


10 


7.4 


7.3 


20 


14-8,14.6 


30 


22.2 22.0 


40 


29.6:29.3 


50 


37.1136.6 





43 


43 


6 


4.3 


4.3 


7 


5 


1 


5 





8 


5 


8 


5 


7 


9 


6 


5 


6 


4 


10 


7 


2 


7 


1 


20 


14 


5 


14 


3 


30 


21 


7 


21 


5 


40 


29 





28 


6 


50 


36 


2 


35 


8 





33 


33 


6 


3.3 


3.2 


7 


3 


8 


3-8 


8 


4 


4 


4.3 


9 


4 


9 


4.9 


10 


5 


5 


5.4 


20 


11 





10.8 


30 


16 


5 


16.2 


40 


22 





21.6 


50 


27 


5 


27.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



33 



3 


2 


3. 


3 


7 


3. 


4 


2 


4. 


4 


8 


4. 


5 


3 


5. 


10 


6 


10. 


16 





15. 


21 


3 


21. 


26 


6 


26. 



31 

1 

7 
2 
7 
2 
5 
7 

2 



6 

7 

8 

9 

10 

20 

30 

40 



31 

3.1 

3.8 

4.1 

4.6 

5-1 

10.3 

15.5 

20.6 



50125-8 



P. P. 



?fABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
44° 45° 



Lg, Vers. D 



35 9 

36 

37 

38 

39 



44818 
44849 
44880 
44912 
44943 



44974 
45005 
45036 
45068 
45099 



45130 
45161 
45192 
45223 
45254 



45285 
45316 
45348 
45379 
45410 



45441 
45472 
45503 
45534 
45565 



45595 
45626 
45657 
45688 
45719 



45750 
45781 
45812 
45843 
45873 



45904 
45935 
45966 
45997 
46027 



46058 
46089 
46120 
46150 
46181 



46212 
46242 
46273 
46304 
46334 



46365 
46396 
46426 
46457 
46487 



46518 
46549 
46579 
46610 
46R40 



46671 



31 
31 
3l 
31 

31 

31 
31 
3l 
31 

31 
31 
3l 
31 
31 

31 
31 
3l 
31 
31 

31 
31 
31 
31 
31 

30 
31 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 
30 
31 
30 

31 
30 
31 
30 
31 

30 
30 
31 
30 
30 

31 
30 
30 
30 
30 

31 
30 
30 
30 
30 

30 



Log.Exs. 



59124 
59168 
59211 
59255 
59298 



59342 
59385 
59429 
59472 
59515 



59559 
59602 
59646 
59689 
59732 



59776 
59819 
59863 
59906 
59949 



59993 
60036 
60079 
60123 
60166 



60209 
60253 
60296 
60339 
60383 



60426 
60469 
60512 
60556 
60599 



60642 
60685 
60729 
60772 
60815 

60858 
60902 
60945 
60988 
61031 



D 



61075 
61118 
61161 
61204 
61247 



61291 
61334 
61377 
61420 
61463 



61506 
61550 
61593 
61636 
61679 



61725! 



43 
43 
43 
43 

43 
43 
43 
43 

43 

43 
43 

43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 

43 

43 



Lg. Vers. 



46671 
46701 
46732 
46762 
46793 



46823 
46853 
46884 
46914 
46945 



46975 
47005 
47036 
47066 
47096 



47127 
47157 
47187 
47218 
47248 



47278 
47308 
47339 
47369 
47399 



47429 
47459 
47490 
4752C 
4755C 



4758C 
4761C 
47640 
4767C 
4770G 



47731 
47761 
47791 
47821 
47851 



47881 
47911 
47941 
47971 
48001 



48031 
48061 
4809G 
48120 
48150 



48180 
48210 
48240 
4827C 
48300 



48329 
48359 
48389 
48419 
484-59 



AMI?'. 



Lg. Vers. -O Log.Exs. I> Lg. Vers. -O Log.Exs, I> 



J) 

30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
29 
30 
30 

30 
30 
29 
30 
30 

29 
30 
30 
29 
30 

29 



Log.Exs. 



61722 
61765 
61808 
61852 
61895 



61938 
61981 
62024 
62067 
62110 



62153 
62196 
62239 
62282 
62326 



62369 
62412 
62455 
62498 
62541 



62584 
62627 
62670 
62713 
62756 



62799 
62842 
62885 
6292€ 
'62971 



63014 
63057 
63100 
63143 
63186 



63229 
63272 
63315 
63358 
63401 



63443 
63486 
63529 
63572 
63615 



63658 
63701 
63744 
63787 
63830 



63878 
63915 
63958 
64001 
64044 



64087 
64130 
64173 
64216 
64258 

6430T 



D 

43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

42 
43 
43 
43 
43 

43 
42 
43 
43 
43 

43 
42 
43 
43 
43 

42 
43 
43 
43 
42 

43 



5 
6 
7 
8 
. 9 

10 

11 
12 
13 
14 

15 
16 
17 
18 

11 
30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
31 
60 



P. P. 





43 


43 ' 


6 


4 £ 


4-3 


7 


5 


1 


50 


8 


5 


£ 


5-7 


9 


6 


5 


6-4 


10 


7 


2 


7-1 


20 


14 


R 


14-3 


30 


21 


7 


21.5 


40 


29 





28.6 


5C 


36 


2 


35.8 





43 


6 


4-2 


7 


4 


9 


8 


5 


6 


9 


6 


4 


10 


7 


1 


2C 


14 


1 


30 


21 


2 


4C 


28 


3 


50 


35 


4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



31_ 

31 



31 

31 





30 


30 


6 


3-0 


30 


7 


3 5 


3 


5 


8 


4-0 


4 





9 


4 


6 


4 


5 


10 


5 


1 


5 





20 


10 


1 


10 





30 


15 


2 


15 





4G 


20 


3 


20 





50 


25 


4 


25 








39 


6 


2-9 


7 


3 


4 


8 


3 


9 


9 


4 


4 


10 


4 


9 


20 


9 


8 


30 


14 


7 


40 


19 


6 


50 


24 


6 



P. p. 



730 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT3. 
46° 47° 



Ig, Vers. 



48478 
48508 
48538 
48568 
48597 



48627 
48657 
48686 
48716 
48746 



48775 
48805 
48835 
48864 
43894 



48923 
48953 
48983 
49012 
4904'^ 



49071 
49101 
49130 
49160 
49189 



49219 
49248 
49278 
49307 
49336 



49300 
49395 
49425 
49454 
49483 



49513 
49542 
49571 
49601 
49630 



49059 
49689 
49718 
49747 
49776 



49806 
49835 
49864 
49893 
49922 



49952 
49981 
50010 
50039 
50068 



50097 
50126 
50155 
50185 
50214 



50^43 



30 
29 
30 
29 

30 

29 
29 
30 
29 

29 
30 
29 
29 
29 

29 
30 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
2C 
29 

29 



Log.Exs, 



64301 
64344 
64387 
64430 
64473 



64515 

.64558 

.64601 

64644 

64687 



.64729 
64772 
64815 
64853 
64901 



. 64943 
•64986 
.65029 
.65072 
■65114 

.65157 

.65200 

.65243 

65285 

65328 



■65371 
■65414 
.65456 
■65499 
■65542 

65585 
■65627 
■65670 
■65713 

65755 



65798 
65841 
65884 
.65926 
65969 



66012 
66054 
66097 
66140 
66182 



66225 
66268 
66310 
66353 
66396 



63438 

66481 

.66523 

■66566 

■66609 



66651 
66694 
66737 
66779 
^66822 
•66864 



D 

43 
42 
43 
43 

42 
43 
43 
42 
43 

42 
43 
43 
42 
43 
42 
43 
42 
43 
42 

43 
42 
43 
42 
43 

42 
43 
42 
43 
42 

43 
42 
43 

42 
42 

43 

42 
43 
42 
42 

43 

42 
42 
43 
42 

42 
43 
42 
42 
43 

42 
42 
42 
43 
42 

42 
42 
43 
42 
42 

42 



Lg. Vers, 



Lg. Vers. I> Log.Exs.| ■'> Lg. Vers.l O Log.Exs. i> 



50243 
50272 
50301 
50330 
50359 



50388 
50417 
50446 
50475 
50504 



50533 
50562 
50591 
50619 
50648 



5067? 
50706 
50735 
50764 
50793 



Z) 



50821 
50850 
50879 
50908 
50937 



50965 
50994 
51023 
51052 
51080 



51109 
51138 
51167 
51195 
51224 



51253 
51281 
51310 
51338 
51367 



51396 
51424 
51453 
51481 
51510 



51539 
51567 
51596 
51624 
51653 



51681 
51710 
51738 
51767 
51795 



51823 
51852 
51880 
51909 
51937 



51965 



29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
28 
29 

29 
29 
28 
29 
29 

28 
29 
29 
28 
29 

28 
29 
28 
29 
28 

29 
28 
29 
28 
28 

29 
28 
28 
28 
29 

28 
28 
28 
28 
28 

29 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 



Log.Exs. -Z> 



66864 
66907 
66950 
66992 
67035 



67077 
67120 
67162 
67205 
67248 



67290 
67333 
67375 
67418 
67460 



67503 
67546 
67588 
67631 
67673 



67716 
67758 
67801 
67843 
67886 



67928 
67971 
68013 
68056 
68098 



68141 
68183 
68226 
68268 
68311 



68353 
68396 
68438 
68481 

68523 



68566 
68608 
68651 
68693 
68735 



68778 
68820 
68863 
68905 
68948 



68990 
69033 
69075 
69117 
69160 



69202 
69245 
69287 
69330 
69372 



6941-1 



42 
43 
42 
42 

42 
42 
42 
43 
42 

42 
42 
42 
42 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 





1 
2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 

14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
34 

35" 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49. 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59. 
HO 



P. P. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



43 



4 


3 


4. 


5 





4. 


5 


7 


5^ 


6 


4 


G. 


7 


1 


7. 


14 


3 


14 • 


21 


5 


21. 


28 


6 


28 • 


35 


8 


35. 



43_ 

2 
9 
6 
4 
1 
1 
2 
3 
4 



6 
7 
8 

9 
10 
20 
30 
40 
50 



43 

4.2 

4-9 

5.6 

6.3 

7.0 

14.0 

21.0 

28.0 

35.0 





30 


39 


6 


3.0 


2.9 


7 


3 


5 


3 


4 


8 


4 





3 


9 


9 


4 


5 


4 


4 


10 


5 





4 


9 


20 


10 





9 


8 


30 


15 





14 


7 


40 


20 





19 


6 


50 


25 





24 


6 





39 


31 


6 


2.9 


2. 


7 


3 


4 


3. 


8 


3 


8 


3. 


9 


4 


3 


4.- 


10 


4 


8 


4. 


20 


9 


6 


9. 


30 


14 


5 


14. 


40 


19 


3 


19. 


50 


24 


1 


23. 



• 8 

• 3 
8 
3 

• 7 
.5 

• 2 

■Q 
•7 



6 

7 

8 

9 

10 

20 

30 

40 

50 



38 

2^8 

3.2 

3.7 

4.2 

4.6 

9.3 

14.0 

18-6 

23.3 






P. P. 



731 



CABLE Vill— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



48' 



49° 



Lg. Vers. 



9.51965 
51994 
52022 
52050 
52079 



52107 
52135 
52164 
52192 
52220 



52249 
52277 
52305 
52333 
52362 



52390 
52418 
52446 
52474 
52503 



52531 
52559 
52587 
52615 
52643 



52671 
52699 
52727 
52756 
52784 



52812 
52840 
52868 
52896 
52924 



52952 
52980 
53008 
53036 
53064 



53092 
53i20 
53147 
53i75 
53203 



53231 
53259 
53287 
53315 
53343 



53370 
53398 
53426 
53454 
53482 



53505 
53537 
53566 
53593 
53620 



a.-53648 



Lg. Vers. 



I> 



28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
27 
28 
28 

28 
27 
28 
28 
28 

27 
28 
28 
27 
28 

27 
28 
27 
28 
27 
28 



Log.Exs. 



9.69414 
69457 
69499 
69542 
69584 



69626 
69669 
69711 
69753 
69796 



69838 
69881 
69923 
69965 
70008 



70050 
70092 
70135 
70177 
70220 



70262 
70304 
70347 
70389 
70431 



70474 
70516 
70558 
70601 
70643 



70685 
70728 
70770 
70812 
70854 



70897 
70939 
70981 
71024 
71066 



71108 
71151 
71193 
71235 
71^78 



71320 
71362 
71404 
71447 
71489 



71531 
71573 
71616 
71658 
71700 



71743 
71785 
71827 
71869 
71912 



71954 



Log.Exs. 



n 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



Lg. Vers. 



9-53648 
53676 
53704 
53731 
53759 



D 



53787 
53814 
53842 
53870 
53897 



53925 
53952 
53980 
54008 
54035 



54063 
54090 
54118 
54145 
54173 



54200 
54228 
5425b 
54283 
54310 



54338 
54365 
54393 
54420 
54448 



54475 
54502 
54530 
54557 
54585 



54612 
54639 
54667 
54694 
54721 



54748 
54776 
54803 
54830 
54858 



54885 
54912 
54939 
54967 
54JI94 

55021 
55048 
55075 
55103 
55130 



55157 
55184 
55211 
55238 
55265 



55292 



1> 



27 
28 
27 
27 

28 

27 
27 
28 
27 

27 
27 
28 
27 
27 

27 
27 
27 
27 
27 

27 

27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
22 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 



Lg. Vers.j ^ Log.Exs. 



Log.Exs. 



71954 
71996 
72038 
72081 
72123 



72165 
72207 
72250 
72292 
72334 



72376 
72419 
72461 
72503 
72545 



72587 
7263C 
72672 
7271| 
72756 



72799 
72841 
72883 
72925 
72967 



73010 
73052 
73094 
73136 
73178 



73221 
73263 
73305 
73347 
73389 



73431 
73474 
73516 
73558 
73600 

73642 
73685 
73727 
73769 
73811 



73853 
73895 
73938 
73980 
74022 



74064 
74106 
74148 
74191 
74233 

74275 
74317 
74359 
7440T 
74444 



74486 



2> 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



Z> 



10 

11 

12 

13 

14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
-29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49, 

So 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



P. P. 





43 


43 




6 


4-2 


4.2 




7 


4 


9 


4 


9 




8 


5 


6 


5 


6 




9 


6 


4 


6 


3 




10 


7 


1 


7 







20 


14 


1 


14 







30 


21 


2 


21 







40 


28 


3 


28 







50 


35 


4 


35 










21 


5 


38 


6 


2.8 


2.8 


7 


3 


3 


8 


2 


8 


3 


8 


3 


7 


9 


4 


3 


4 


2 


10 


4 


7 


4 


6 


20 


9 


5 


9 


3 


30 


14 


2 


:.4 


Q 


40 


19 





18 


§ 


50J 


23 


7J 


23 


3 





27 


37 


6 


2.7 


2.7 


7 


3 


2 


3 


1 


8 


3 


6 


3 


6 


9 


4 


1 


4 





10 


4 


6 


4 


5 


20 


9 


] 


9 





30 


13 


7 


13 


5 


40 


18 


3 


18 





50 


22 


9 


22 


5 



P. p. 



732 



f ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTa 
50" 51" 





1 
2 
3 

5 
6 
7 
8 

10 

11 

12 
13 
14 
15 
16 
17 
18 

ii 
30 

21 
22 
23 

2^ 

25 
26 
27 
28 
29 



Lg.Vers. 



30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



60 

51 
52 
53 
54 

55 
56 
57 
58 
59. 

60 
"7" 



55292 
55319 
55847 
55374 
55401 



55428 
55455 
55482 
55509 
55536 



55583 
55590 
55617 
55644 
55871 

55698 
55725 
55751 
55778 
55805 

55832 
55859 
55886 
55913 
55940 



55906 
55993 
56020 
56047 
56074 



56101 
56127 
56154 
56181 
58208 



56234 
56261 
56288 
56315 
56311 



56368 
56395 
56421 
56448 
58475 

56501 
56528 
56554 
56581 
58608 



56634 
56661 
56687 
56714 
56741 



56767 
58794 
56820 
58847 
56873 



9-58900 
Lg. Vers. 



27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
26 
27 
27 

27 
27 
26 
27 
27 

26 
27 
27 
26 
27 

27 
2g 
27 
26 
27 

26 
27 
26 
27 
26 

26 
27 
26 
26 
27 

26 
26 
26 
27 
26 

26 
26 
26 
26 
27 

26 
26 
26 
26 
28 

26 

1o 



Log.Exs. 



9.74486 
.74528 
.74570 
.74612 
•74654 



n 



9.74696 
.74739 
.74781 
.74823 
.74865 



9.74907 
•74949 
.74991 
.75033 
.75076 

9.75118 
.75160 
.75202 
.75244 
.7528Q 



9.75328 
.75370 
.75413 
.75455 
.75497 

9.75539 
.75581 
.75623 
.75665 
.75707 



9.75750 
.75792 
.75834 
.75876 
.75918 



9.75960 
•76002 
.76044 
.76086 

.761C?8 



9.76171 
.76213 
.76255 
.76297 
.76339 



9.76381 
.76423 
•76465 
•76507 

.76549 



9.76592 
.76634 
.76676 
.76718 
•76760 



9.76802 
.76341 
.768<'6 
.76928 
.76970 



9 ■ 77012 
Log.Exs. 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



Lg. Vers. 



58900 
.56926 

56953 
■56979 

57005 



1.57032 
.57058 
•57085 
.57111 
.57138 



.57164 
.57190 
.57217 
.57243 
j572_69 

. 57296 
.57322 
.57348 
.57375 
. 57401 

.57427 
.57454 
.57480 
•57506 
.57532 



•57559 
.57585 
.57611 
.57637 
.57664 



.57690 
.57716 
.57742 
.57768 
.57794 



57821 
.57847 
.57873 
.57899 
.579M 

.57951 
.57977 
.58003 
.58029 
■58055 

. 58082 
.58108 
.58134 
.58160 
.58186 



58212 
58238 
58264 
58290 
58316 



58342 
58367 
58393 
584i9 
58445 



9.58471 



I> Lg.Vers. 



D 

26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
28 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

22 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 

26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
25 
26 
26 
28 

26 



Log.Exs 



1.77012 
•77055 
•77097 
•77139 
•77181 



77223 
■77265 
■7730? 
■77349 

77391 



.77433 
.77475 
.77517 
.77580 
jlZZ602 

. 77644 
.77686 
•77728 
.77770 
.77812 

.77854 
.77896 
.77938 
.77980 
.78022 



.78064 
.78107 
.78149 
.78191 
.78233 



.78275 
.78317 
.78359 
.78401 
. 78443 



■78485 
■78527 
.78569 
.78611 
.7865R 



78696 
■78738 
■78780 
.78822 
lZ8864 

. 78906 
.78948 
.78995 
.79032 
l79074 

r7911g 
.79158 
.79200 
.79242 
■79285 



.79327 
•79369 
.79411 
■79453 
■79495 



9.79537 



^ Log.Exs. 
733 



n 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
L4 

15 
16 
17 
18 

Ji. 
30" 

21 
22 
23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 

34 



35 
36 
37 
38 

39 

40 

41 
42 
43 
ii 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
58 
57 
58 
59 



60 



P.P. 





43 42 


6 


42 4 2 


7 


4 9 4-9 


8 


5 6 5-8 


9 


6-4 6-3 


10 


7 i 7 


20 


14 1 14-0 


30 


21 221. 


40 


28-3 28 U 


50 


35 435-0 





2f 


37 


6 


2.7 


2-r 


7 


3 


2 


3-1 


8 


3 


6 


3-6 


9 


4 


1 


4-0 


10 


4 


8 


4-5 


20 


9 


1 


9-0 


30 


13 


7 


13 5 


40 


18 


3 


18 


50 


22- 


9 


22-5 



6 

7 
8 
9 

10 
20 
30 
40 
50 







36 36 


2-6 


2 6 


3 1 


30 


3 5 


3 4 


40 


3 9 


44 


4 3 


8 8 


8 6 


13-2 


13. 


17 6 


17 3 


22-1 


21-g 





3.n 


6 


2-5 


7 


3-0 


8 


3 4 


9 


3-8 


10 


4-2 


20 


8^5 


SO 


12^7 


40 


17-0 


50 


21.2 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
53° 63° 



Lg. Vers. 



58471 
58497 
58523 
58549 
58575 



58601 
58626 
58652 
58678 

58704 



58''30 
58755 
58781 
58807 
58833 



58859 
58884 
58910 
58936 
58962 



58987 
59013 
59039 
59064 
59090 



59116 
59141 
59167 
59193 
59218 



59244 
59270 
59295 
59321 
59346 



59372 
59397 
59423 
59449 
59474 



59500 
59525 
59551 
595'76 
59602 



59627 
59653 
59678 
59704 
59729 



59754 
59780 
59805 
59831 
59856 



59881 
59907 
59932 
59958 
59983 



9.6OOOR 



Lg> Versc 



D 



26 
25 
26 
26 

26 
25 
26 
26 

25 

'A\j 
25 
26 
26 
25 

26 
25 
26 
25 
26 

25 
25 
26 
25 
26 

25 
25 
26 
25 
25 

2o 
26 
25 
25 
25 

25 
25 
26 
25 
25 

21 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



Log.Exs. 



79537 
79579 
79621 
79663 
79705 



79747 
79789 
79831 
79874 
79916 



79958 
80000 
80042 
80084 
80126 



80168 
80210 
80252 
80294 
80S36 



80378 
80420 
80463 
80505 
80547 



80589 
80631 
80673 
80715 
80757 



80799 
80841 
80883 
80925 
80968 



81010 
81052 
81094 
81136 
81178 



81225 
81262 
81304 
81346 
81388 



81430 
81473 
81515 
81557 
81599 



81641 
81683 
81725 
81767 
81809 



81851 
81894 
81936 
81978 
82020 



82062 



D 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 

42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



Lg. Vers. 



60008 
60034 
60059 
60084 
60110 



60135 
60160 
60185 
60211 
60236 



60261 
60286 
60312 
60337 
60362 

60387 
60412 
60438 
60463 
60488 



60513 
60538 
60563 
60589 
60614 



60639 
60664 
60689 
60714 
60739 



60764 
60789 
6C814 
60839 

60864 



60889 
60914 
60939 
60964 
60989 

61014 
61039 
61064 
61089 
61114 



61139 
61164 
61189 
61214 
61239 



61264 
61289 
61313 
61338 
61363 



6138g 
61413 
61438 
61462 
6 1487 
61512 



i>|Log.Exs. l>|Lg. Versc 



D 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 

25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
24 
25 

25 
25 
24 
25 
25 

25 
24 
25 
24 
25 

25 
It 



Log.Exs. 



82062 
82104 
82146 
82188 
82230 



82272 
82315 
82357 
82399 
82441 



82483 
82525 
82567 
82609 
82651 



82694 
82736 
82778 
82820 
82862 



82904 
82946 
82988 
83031 
83073 

83115 
83157 
83199 
83241 
83283 



83325 
83368 
83410 
83452 
83494 



83536 
83578 
83620 
83663 
83705 



83747 
83789 
83831 
83873 
83916 



83958 
84000 
84042 
84084 
84.26 



84168 
84211 
84253 
84295 
84337 



84379 

84422 

.84464 

.84506 

. 84548 



9 ■ 845 90 

llogiExs. 



n 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

.42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



n 



o 

1 
2 
3 
4 

5 

6 

7 

8 

_9 

10 

11 
12 
13 
14 

1*1 
16 
17 
18 
19 

20 

21 
22 
23 

M. 
25 
26 
27 
28 
29 

30 

31 
32 
33 

35 
36 
137 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



P. P. 





43 


42 


6 


4.2 


4 2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.3 


10 


7.1 


70 


20 


14.1 


14.0 


30 


21.2 


21-0 


40 


28-3 


28.0 


50 


35.4 


35-0 





3G 


Sr 


6 


2.6 


2.S 


7 


3.0 


3.0 


8 


3.4 


3.4 


9 


3.9 


3. i 


10 


4.3 


4.2 


20 


8.6 


85 


80 


13.0 


12-7 


40 


17-3 


17.0 


60 


21-6 


21.2 



25 



6 


2 


5 


2. 


7 


2 


9 


2. 


8 


3 


3 


3. 


9 


3 


7 


3. 


10 


4 


1 


4. 


2J 


8 


3 


8. 


30 


12 


5 


12. 


40 


16 


6 


16. 


50 


20 


8 


20. 



8 
2 
7 
1 
I 
2 
3 
4 



P.P. 



734 



TABLE VIIX— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTE 



54° 



55 



flL° 



Lg. Vers. 



61512 
61537 
61562 
61586 
61611 



61638 
61661 
61685 
61710 
61735 



61760 
61784 
61809 
61834 
61858 



61833 
61908 
61932 
61957 
61982 



62006 
62031 
62055 
62080 
62105 



62129 
62154 
62178 
62203 
62227 



62252 
62276 
62301 
62325 
62350 



62374 
62399 
G2423 
82448 
62472 



62497 
62521 
62546 
62570 
62594 



6261? 
62643 
62668 
62692 
62716 



62741 
62765 
62789 
62814 
62838 



62882 
62887 
62911 
62935 
62960 



9.62984 



Lg. Vers. 



D 

24 
25 
24 
25 

24 
25 
24 
25 
24 

25 
24 
24 
25 
24 

25 
24 
24 
24 
25 

24 
24 
24 
25 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 



Log.Exs. 



84590 
8463 i 
84675 
84717 
84759 



84801 
84843 
84886 
84928 
84970 



85012 
35054 
85097 
85139 
85181 



85223 
85265 
85308 
85350 
85392 



85434 
85476 
85519 
85561 
85603 



85645 
85688 
85730 
85772 
85814 

85857 
8589? 
85941 
85983 
86026 



86068 

86110 
86152 
86195 
86237 



8627? 
86321 
86364 
86406 
88448 



86490 
86533 
86575 
86617 
86659 



86702 
86744 
86786 
86829 
86871 



86913 
86956 
86998 
87040 
87082 



87125 



D 

42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 

42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



Lg. Vers. 



-Z> Log.Exs. -O Lg. Vers. 



62984 
63008 
63032 
63057 
63081 

63105 
63129 
63154 
63178 
63202 



63226 
63250 
63274 
63209 
63323 



633-47 
633-71 
63395 
63419 
63443 



63468 
63492 
63516 
63540 
63564 



63588 
63612 
63636 
63660 
63684 



63708 
63732 
63756 
63780 
63804 



63828 
63852 
63876 
63900 
63924 



63948 
63972 
63996 
6401? 
64043 



64067 
64091 
64115 
64139 
64163 



64187 
64210 
64234 
64258 
64282 



64306 
64330 
64353 
64377 
64401 
64425 



D 



Log.Exs. 



24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 [ 

^^ Iq 

24 r 

24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
23 
24 

24 
24 
24 
23 
24 

24 
23 
24 
24 
23 

24 
24 
23 
24 
23 
24 



87125 
87167 
87209 
87252 
87294 



87336 
87379 
87421 
87463 
87506 



87548 
87590 
87633 
87675 
87717 



87760 
87802 
87844 
87887 
87929 

87971 
88014 
88056 
88099 

88141 



88183 
88226 
88268 
88310 
88353 



88395 
88438 
88480 
88522 
88565 



88607 
88650 
88692 
88734 
88777 



88819 
88862 
8890<! 
88947 
88989 



89031 
89074 
89116 
89159 
89201 

89244 
89286 
8932? 
89371 
89414 



89456 
89499 
89541 
89583 
89626 



89668 



7> 



42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 

42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 




1 
2 
3 

4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 

.11 
30 

21 
22 
23 
24 

25 
26 
27 
28 
11 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

60 



l> Log.Exs. I* 
735 



P.P. 





4^ 


43 


6 


4.2 


4.2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.3 


10 


7.1 


7.0 


20 


14.1 


14.0 


30 


21.2 


21.0 


40 


28.3 


28.0 


50 


35.4 


35-0 





25 


2i 


6 


2.5 


2.4 


7 


2.9 


2.8 


8 


3.3 


3.2 


9 


3.7 


3.7 


10 


4.1 


44 


20 


8-3 


8.1 


30 


12-5 


12.2 


40 


16.6 


16-3 


50 


20-8 


20.4 





24 


2 


3 


6 


2-4 


2 


3 


7 


2-8 


2 


7 


8 


3.2 


3 


I 


9 


3-6 


3 


5 


10 


4.0 


3 


9 


20 


8.0 


7 


& 


30 


12.0 


11 


7 


40 


16.0 


15 


6 


50 


20.0 


19 


6 



P.P. 



#ABLE VIII,— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT& 
56° 57° 



Lff. Vers, 



9.64425 
fi4448 
64472 
64496 
64520 



64543 
64567 
64591 
64614 
64638 



64662 
64685 
64709 
64733 
64756 



64780 
64804 
64827 
64851 
64875 



64898 
64922 
64945 
64969 
64992 



65016 
65040 
65063 
65087 
65110 



65134 
65157 
65181 
65204 
65228 



65251 
65275 
65298 
65321 
65345 



65368 
65392 
65415 
65439 
65462 



65485 
65509 
65532 
65556 
65579 



65602 
65626 
65649 
65672 
65686 
65719 
65742 
65765 
65789 
65812 



9.65835 



2> 



23 
24 
23 
24 

23 
24 
23 
23 
24 

23 
23 
24 
23 
23 

24 
23 
23 
23 
24 

23 
23 
23 
23 
23 

24 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 



Log.Exs. 



1-89668 
.89711 
.89753 
•89796 
•89838 



89881 

89923 

.89966 

.90008 

■90051 



.90094 
90136 
90179 
90221 
90264 



90306 
.90349 

90391 
•90434 
.90476 



■90519 
■90561 
.90604 
•90647 
■90689 



.90732 
.90774 
•90817 
.90860 
■90902 



•90945 
•90987 
.91030 
•91073 
.91115 



■91158 
•91200 
91243 
•91286 
•91328 



•91371 
•91414 
.91456 
•91499 
.91541 



•91584 
•91627 
•91669 
■91712 
■91755 



•91797 
•91840 
•91883 
•9192B 
•91968 



•92011 
•92054 
.92096 
.92139 
.92182 



9.92224 



9 



1> 

42 
42 
42 
42 

42 
42 
42 
42 
42 

43 
42 
42 
42 
42 

42 
42 
42 
42 
4? 

42 
42 
43 
42 
42 

42 
42 
42 
43 
42 

42 9 
42 ^ 
42 
43 
42 

42 
42 
42 
43 
42 

42 
43 
42 
42 
42 

43 
42 
42 
43 
42 

42 
43 
42 
43 
42 

42 
43 
42 
43 
42 

42 



Lg. Vers. 

9.65835 
•65859 
•65882 
•65905 
•65928 



9. 



65952 
65975 
65998 
66021 
66044 



■66068 
•66091 
•66114 
■66137 
66160 



66183 
•66207 
•66230 
•66253 
•66276 

■66299 
•66322 
•66345 
•66368 
•66391 



66415 
66438 
66461 
66484 
66507 



66530 

•66553 

.66576 

66599 

66622 



■66645 
•66668 
.66691 
■66714 
■66737 



66760 
.66783 
•66805 
■66828 
■66851 



66874 

•66897 

•66925 

66943 

66966 



• 66909 
■67012 
•67034 
•67057 
67080 



67103 

67126 

■67149 

■67171 

•67194 



9- 67217 



D 

23 
23 
23 
•23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 !9 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
22 
23 

23 
23 
22 
23 
23 

25 
23 
23 
22 
23 
22 



Log.Exs. 

92224 
92267 
92310 
92353 
92395 



9 



92438 
92481 
92524 
92566 
92609 



92652 
92695 
92737 
92780 
92823 

92866 
92909 
92951 
92994 
93037 



93080 
93123 
93165 
93208 
93251 

93294 
93337 
93380 

93422 
93465 



93508 
93551 
93594 
93637 
93680 



93722 
93765 
93808 
93851 
y3894 



93937 
93980 
94023 
94066 
94109 



94151 
94194 
94237 
94280 
94328 



94366 
94409 
94452 
94495 
94538 



94581 
94624 
94667 
94710 
94753 



94796 



I) 

43 
42 
43 
42 

43 
42 
43 

42 
43 

42 
43 
42 
43 
42 

43 
43 

42 
43 
42 

43 
43 
42 
43 
43 

42 
43 
43 
42 
43 

43 
42 
43 
43 
43 

42 
43 
43 
43 
42 

43 
43 
43 
43 
43 

42 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

30 

21 

22 
23 
24 

25 
26 
27 
28 
-2i 

30 

31 
32 
33 
_34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
j44 

45 
46 
47 
48 
49 



Lg. Vers. J> Lcg.Exs, Z>jLg. Vers. -O Log.Exs. Ji 

736 



50 

51 
52 
53 

55 
56 
57 
58 
59 



60 



p.p. 





43 


43 


6 


4^3 


4^2 


7 


5 





4 


9 


8 


5 


7 


5 


6 


9 


6 


4 


6 


4 


10 


7 


1 


7 


•1 


20 


14 


3 


14 


1 


3C 


21 


5 


21 


2 


4C 


28 


6 


28 


3 


50 


35 


8 


3[) 


4 





24 


2; 


fi 


2.4 


2 


7 


2 


• 8 


2 


8 


3 


2 


3. 


g 


3 


6 


3. 


10 


4 





3. 


20 


8 





7. 


30 


12 





11. 


4C 


16 





15. 


50 


20 





19 • 



6 

7 
8 
9 
10 
20 
3D 
40 
50 



23 



2 


3 


2 


2 


7 


2 


3 





3. 


3 


4 


3- 


3 


8 


3. 


7 


6 


7. 


11 


5 


11. 


15 


3 


15- 


19 


1 


18. 



22 

2 
6 

4 
7 
5 
2 

7 



P. P. 



1 

JTABLE VIII.-LOGARITHMIC VERSED SINES AND EXTERNAX. SECANTO. 

I 58° 59° 




s^ _£^ Log. Exs. D Lg. Vers. 



10 

11 

12 
13 
14 



■67331 
.67354 
.67376 
•67399 
.67422 



.67445 
■67467 
•67490 
■67513 
•67535 



15 
16 
17 
18 

19 

30 

21 
22 
23 
24 



25 
26 
27 
28 
29 



■67558 
■67581 
•67603 
•67626 
.67649 

■67671 
•67694 
•67717 
■67739 
•67762 



•67784 
•67807 
.67830 
■67852 
■67875 



30 

31 
32 
33 

3± 

35 9 

36 

37 

38 

39 

40 

41 

42 

43 

44 



•67897 
•67920 
•67942 
•67965 
•67987 



45 
46 
47 
48 
49 



•68010 
■68032 
.68055 
68077 
■68100 

"68122 
68145 
68167 
68190 
68212 



9 ■94796 
•94839 
•94882 
.94925 
•94968 
9.95011 
•95054 
.950971 
.951401 
•95183 

9^95226 
•95269 
.95313 
.95356 
.95399 

9.95442 
95485 
95528 
95571 
95614 

9.95657 
.9570.1 
.95744 
. 95787 
•95830 

9.95873 
.95916 
.95959 
.96002 
.96046 

9-96089 

.96132 

.96175 

.96218 

_ .96261 

^119.98305 
.96348 
.96391 
.96434 
.96478 



.68571 
.68593 
•68615 
.68637 
•68660 



.68682 
.68704 
.68727 
.68749 
.68771 



22 
22 
22 
22 



Log. Exs. 



9 



.68793 
.68816 
.68838 
.68860 
.68882 



97387 
97430 
97473 
97517 
97560 



.97603 
.97647 
-97690 
•97734 
■97777 



•68235 
.68257 
.68280 
.68302 
•68324 



50 19 

51 
52 
53 
54 



60 



68347 
68369 
683,92 
68414 
68436 




9.68571 



Lg. Versi 



!9. 96521 
.96564 
.96607 
.96650 
•96694 

19.96737 
•96780 
.96824 
.96867 
.98910 

9.96953 
96997 
97040 
no n .97083 
^£ -97127 

^§9.97170 
iil .97213 
iil .97257 
iil .97300 
'^f .97343 

2^ 9-97387 

D jLog. Exs. 



N- 9 



.68905 
.68927 
.68949 
.68971 
.68993 

.69016 
.69038 
.69060 
.69082 
.69104 



.69126 
.69149 
.69171 
.69193 
.69215 




.98038 
.98081 
.98125 
.98168 
.98211 



D 



.98255 
.98298 
.98342 
.98385 
.98429 



:f I 9 



.69237 
.69259 
.69281 
.69303 
.69325 



43 

43 



9. 



69347 
69369 
69392 
69414 
69436 



.69458 
.69480 
.69502 
.69524 
.69546 



.98472 

98516 

■98559 

98603 

98647 



.98690 
.98734 
.98777 
.98821 
■98864 



.98908 
.98952 
.98995 
.99039 
.99082" 



f^ 9 



69568 
69590 
69612 
69634 
6965 6 



f# 9 



43 

43 r 



69673 
69700 
6972_ 
69743 
69765 



.99126 
.99170 
.99213 
.99257 
.99300 



.99344 
•99388 
.99431 
.99475 
.99519 



D 



69787 
69809 
69831 
69853 
69875 



.69897 



Vers. 



.99562 
.99606 
.99650 
.99694 
.99737 



D 



.99781 
.99825 
,99868 
.99912 
.99956 



1.00000 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

4l 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 
44 
43 
43 
43 

43 
44 
43 
43 
43 

44 
43 
43 
44 
43 

43 
44 
43 
44 
43 

4^155 
f|<56 
^^ L 57 

— I 5a 

AQ i Jo 

^^[59 
44 60 



10 

11 

12 
13 
14 



15 
16 
17 
18 
19. 

20 

21 
22 
23 
24 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 
40 
41 
42 
43 
44 



45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



Log, Exs. D 



P.P. 



44 



6 


4.4 


7 


5.1 


8 


5-8 


9 


6.6 


10 


7-3 


20 


14-6 


30 


22^0 


40 29-3 
50 36.6 



43 

4-3 

1:1 

6-5 
7.2 

14.5 
21 
29 
36-9 



6 

7 
8 
9 

10 
20 
30 
40 
50 





33 


6 


2-3 


7 


2-7 


8 


3-0 


9 


3-4 


10 


3^8 


20 


7-6 


30 


11-5 


40 


15-3 


50 


19.1 



43 

4-3 

5-0 

5-2 

6.^ 

71 

14-3 

21.5 

28-6 

35. S 



SSL 

2^2 
2^6 
3-0 
3-4 
3-7 
7.§ 
11-2 



:? 



6 


22 

2-2 


1h 


7 


2^5 


2-9 


8 


2^9 


2-? 


» 


3^3 


3-2 


10 


3-6 


3.6 


20 


7-3 




30 


11-0 


10-7 


40 


14-6 


14-3 


50 


18.31 


17.9 



P.P. 



737 



«ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT8,i 
60° 61° 



Lg. Vers. 



69897 
69919 
69940 
69962 
69984 



70006 
70028 
70050 
700-72 
70093 



70115 
70137 
70159 
70181 
70202 



70224 
70246 
70268 
70289 
70311 



70333 
70355 
70376 
70398 
70420 



70441 
70463 
70485 
70507 
70528 



70550 
70572 
70593 
70615 
70636 



70658 
70680 
70701 
70723 
70745 



70766 
70788 
70809 
70831 
70852 



70874 
70896 
70917 
70939 
70960 



70982 
71003 
71025 
71046 
71068 



71089 
71111 
71132 
71154 
71175 



60 9-71197 
' JLg. Vers. 



D 



22 
21 
22 
22 

22 
21 
22 
22 
21 

22 
21 
22 
22 
21 

22 
2T 
22 
21 
22 

2T 
22 
2T 
22 
2T 

21 
22 
21 
22 
21 

2T 
22 
21 
21 
21 

22 
21 
21 
21 
22 

21 
21 
21 
21 
21 

22 
21 
21 
21 
21 

21 
21 

21 
21 
21 

2l 
21 
21 
21 
21 

2T 
~D 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



00000 
00044 
00087 
00131 
00175 



00219 
00262 
00306 
00350 
00394 



00438 
00482 
00525 
00569 
00613 



00657 
00701 
00745 
00789 
00833 



00876 
00920 
00964 
01008 
01052 



01096 
01140 
01184 
01228 
01272 



01316 

oisen 

01404 
01448 
01492 



01536 
01580 
01621 
01668 
01712 



01756 
01800 
01844 
01889 
01933 



01977 
02021 
02065 
02109 
02153 



02197 
02242 
02286 
02330 
02374 



02418 
02463 
02507 
02551 
02595 



02639 



Log. Exs. 



44 
43 
44 
43 

44 
43 
44 
44 
43 

44 
44 
43 
44 
44 

44 
43 
44 
44 
44 

43 

44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 



Lg. Vers. 



71197 
71218 
71239 
71261 
71282 



71304 
71325 
71346 
71368 
71389 



71411 
71432 
71453 
71475 
71496 



71517 
71539 
715 6S 
71581 
71603 

71624 
71645 
71667 
71688 
71709 



71730 
71752 
71773 
71794 
71815 



71837 
71858 
71879 
71900 
71922 



71943 
71964 
71985 
72006 
72028 



72049 
72070 
72091 
72112 
72133 



72154 
72176 
72197 
72218 
72239 



72260 
72281 
72302 
72323 
72344 



72365 
72386 
72408 
72429 
72450 
72471 



D iLg. Vers. 



D 

21 
21 
21 
21 

21 
21 

21 
21 
21 

21 
21 
21 
21 
21 

21 
24 
21 

21 
21 

2T 
21 
21 
21 
21 

21 
2T 
21 
21 
21 

2T 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
21 



Log, Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



02639 
02684 
02728 
02772 
02816 



02861 
02905 
02949 
02994 
03038 



03082 
03127 
03171 
03215 
03260 



03304 
03348 
03393 
03437 
03481 

03526 
03570 
03615 
03659 
03704 



03748 
03793 
03837 
03881 
03926 



03970 
04015 
04059 
04104 
04149 



04193 
04238 
04282 
04327 
04371 



04416 
04461 
04505 
04550 
04594 



04639 
04684 
04728 
04773 
04818 



04862 
04907 
04952 
04996 
05041 



05086 
05131 
05175 
05220 
05265 



05310 



Loff. Exs. 



D 

44 
44 
44 
44 

44 
44 
44 
44 
44 

4l 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
•44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
45 

44 
44 
44 
44 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
45 

44 
45 
44 
44 
45 

44 
45 
44 
45 
44 

45 



10 

11 

12 

13 

ii. 

15 

16 

17 

18 

ii. 
*^0 

21 
22 
23 
24 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

34 



35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

5_4_. 

55 

56 

57 

58 

59_ 

60 



P. P. 



6 
7 
8 

9 
10 
2C 
3C 
40 
50 



45 



4-5 


4. 


5.2 


5 


6-0 


5. 


6-7 


6. 


7.5 


7- 


15. C 


14. 


22-5 


22. 


30-0 


29. 


37.5 


37. 



44 

4 





44 


4i 


6 


4-4 


4. 


7 


5 


1 


5- 


8 


5 


t 


5. 


9 


6 


6 


6. 


10 


7 


3 


7. 


20 


14 


6 


14. 


SO 


22 





21. 


40 


29 


3 


29. 


50 


36 


6 


36. 





33 


31 


6 


2.2 


i'.l 


7 


2.5 


2.5 


8 


2.9 


2.8 


9 


33 


3.2 


10 


3.6 


3.e 


20 


7.3 


71 


30 


11.0 


10.7 


40 


14.6 


14.3 


50 


18.3 


17.9 



6 

7 

8 

9 

10 

20 
30 
40 
50 



31 

2.1 

2.4 

2. 3 

3 1 

3.5 

7.0 

10.5 

14.0 

17.5 



P. P. 



738 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
63° 63° 



Lg. Vers. 



72471 
72492 
72513 
72534 
72555 



72576 
72597 
72618 
72639 
72660 



72681 
72701 
72722 
72743 
72764 



72785 
72806 
72827 
72848 
72869 



72890 
72911 
72931 
72952 
72973 



72994 
73015 
73036 
73057 
73077 



73098 
73119 
73140 
73161 
73181 



73202 
73223 
73244 
73265 
73285 



73306 
73327 
73348 
73368 
73389 



73410 
73430 
73451 
73472 
73493 



73513 
73534 
73555 
73575 
73596 



73617 
73637 
73658 
73679 
73699 



73720 



D 

21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
20 
21 
21 
21 

2] 
21 
21 
20 
21 

21 
21 
20 
21 
21 

21 
20 
21 
21 
20 

21 
21 
20 
21 
20 

21 
21 
20 
21 
20 

2. 
20 
21 
20 
21 

2Q 
20 
21 
20 

21 

20 
20 
21 
~20 
20 

21 
20 
20 
21 
20 

20 



Log. Exs. 



10.05310 
.05354 
.05399 
•05444 
.05489 



10.05534 
•05579 
•05623 
.05668 
.05713 



10-05758 
•05803 
•05848 
•05893 
■05938 



10.05983 
.06028 
.06072 
.06117 
.06162 



10-06207 
.06252 
.06297 
.06342 
•06387 



10 •06432 
•06477 
.06522 
.06588 
•06613 



10.06658 
.06703 
.06748 
•06793 
•06838 



10-06883 
•06928 
.06974 
.07019 
.07064 

10.07109 
.07154 
.07200 
•07245 
■07290 



10-07335 
•07380 
.07426 
.07471 
.07516 



10-07562 
.07607 
.07652 
.07697 
•07743 



10.07788 
.07834 
•07879 
.07924 
•07970 



10-08015 

Lg.Vers.| l>[Log. Exs. 



JD 



44 
45 
45 
44 

45 
45 
44 
45 
45 

45 
44 
45 
45 
45 

45 
45 
44 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 



Lg, Vers. 



73720 
73740 
73761 
73782 
73802 



73823 
73843 
73864 
73884 
73905 



73926 
73946 
73967 
73987 
74008 



74028 
74049 
74069 
74090 
74110 



74131 
74151 
74172 
74192 
74213 



74233 
71254 
74274 
74294 
74315 



74335 
74356 
74376 
74396 
74417 



74437 
74458 
74478 
74498 
74519 



74539 
74559 
74580 
74600 
74620 



74641 
74661 
74681 
74702 
74722 



74742 
74762 
74783 
74803 
74823 



74844 
74864 
7488i 
74904 
74924 

74945 



I> jLg. Vers. 



I) 



20 
20 
21 
20 

20 
20 
20 
20 
21 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



08015 
08061 
08106 
08151 
08197 



08242 
08288 
08333 
08379 
08424 



08470 
08515 
08561 
08606 
08652 



08697 
08743 
08789 
08834 
08880 



08926 
08971 
09017 
09062 
09108 



09154 
09200 
09245 
09291 
09337 



09382 
09428 
09474 
09520 
09566 



09C11 
09657 
09703 
09749 
09795 



09841 
09886 
09932 
09978 
10024 



10070 
10116 
10162 
10208 
10254 



10300 
10346 
10392 
10438 
10484 



10530 
10576 
10622 
10668 
10714 



10-10760 
Log. Exs. 



D 



45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
46 
45 
45 
45 

46 
45 
45 
45 
46 

45 
46 
45 
45 
46 

45 
46 
46 
45 
46 

45 
46 
46 
45 
46 

46 
45 
46 
46 
46 

46 
46 
46 

45 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 



5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 



15 
16 
17 
18 
19 



20 

21 
22 
23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



60 



P. P. 





46 


46 


6 


4-6 


4.fi 


7 


5^4 


5.3 


8 


6^2 


6-1 


9 


7.0 


6.9 


10 


7^7 


7-6 


20 


15.5 


15.3 


30 


23-2 


23.0 


40 


31.0 


30.6 


50 


38-7 


38.3 



6 

7 
8 
9 
10 
20 
30 
40 
50 



45 



4.5 


4- 


5.3 


5- 


6.0 


6- 


6.8 


6- 


7.6 


7. 


15.1 


15. 


22.7 


22. 


30.3 


30- 


37.9 


37. 



45 

5 
2 

7 
5 

5 

5 



6 


4 


7 


5 


8 


5- 


9 


6. 


10 


7- 


20 


14^ 


30 


22 


40 


29 ■ 


50 


37^ 



45 

4 
2 
9 
7 
4 
8 
2 



31 



7 
8 
9 
10 
20 
30 
40 
50 



2 


1 


2- 


2 


4 


2 


2 


8 


2. 


3 


1 


3- 


3 


5 


3- 


7 





6. 


10 


5 


10- 


14 





13- 


17 


5 


17. 



30_ 


4 
7 
1 
4 
8 
2 
6 
1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



30 

2-0 

2 

2 

3 

3 

6 
10 
13 
16 



P. P. 



739 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
64° 65° 



Lg. Vers. 



9.74945 

74965 
74985 
75005 
75026 



9^ 



75046 
75066 
75086 
75106 
75126 



75147 
75167 
75187 
75207 
752271 



75247 
75267 
75287 
75308 
75328 



75348 
75368 
75388 
75408 
75428 



75448 
75468 
75488 
75508 
75528 



75548 
75568 
75588 
75608 
75628 



75648 
75668 
75688 
75708 
75728 



75748 
75768 
75788 
75808 
75828 



75848 
75868 
75888 
7590? 
759^8 



75947 
75967 
75987 
76007 
76027 



76047 
76067 
76087 
76106 
76126 



9.76146 



' Lg.Vers. 



n 

20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 

20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
19 

20 
20 
20 
20 
20 

19 
20 
20 
20 

19 

20 
20 
20 
19 
20 

20 



Log.Exs, 



10. 



10760 
}.0807 
10853 
10899 
10945 



10. 



10991 
11037 
11084 
11130 
11176 



10 



.11222 
11269 

.11315 
11361 
11407 



10. 



10 



11454 
11500 
11546 
11593 
11639 

11685 
11732 
11778 
11825 
11871 



10 



11917 
11964 
12010 
12057 
12103 



10. 



12150 
12196 
12243 
12289 
12336 



10 



12383 

12429 
12476 
12522 
12569 



10. 



12616 
12662 
12709 
12756 
12802 



10. 



10 



12849 
12896 
.12942 
12989 
13036 

'l3083 
.13130 
•13176 
.13223 
.13270 



10 



13^17 

13364 

.13411 

•13457 

•13504 



10.13551 



If Log.Exs, 



I) 



46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 

46 
46 
46 
46 

46 
46 
46 
46 
46 

47 
46 
46 
46 
46 

47 
46 
46 
47 
46 

46 
47 
46 
47 
46 

47 
47 
46 

47 
46 

47 
47 
47 
46 
47 

47 



Lg. Vers. 



76146 
76168 
76186 
76206 
76225 



76245 
76265 
76285 
76304 
76324 



76344 
76364 
76384 
76403 
76423 



76443 
76463 
76482 
76502 
76522 



76541 
76561 
76581 
76600 
76620 



76640 
76659 
76679 
76699 
76718 



76738 
76758 
76777 
76797 
76817 



76836 
76856 
76875 
76895 
76915 



76934 
76954 
76973 
76993 
77012 



77032 
77052 
77071 
77091 
77110 



77130 
77149 
77169 
77188 
77208 



77227 
77247 
77266 
77286 
77305 



77325 



I> Lg. Vers. 



Z> 



10 



10 



10 



10 



10 



1-9 10 

20 
20 
19 

20 
19 
20 
19 
20 

20 
19 
20 
19 
20 

19 
20 

19 
19 
20 

19 
20 
19 
19 
20 

19 
19 
20 
19 
19 

19 10 
19 
19 
20 

19 
19 
19 
20 
19 

19 
19 
19 
19 
19 

20 

19 
19 
19 
19 

19 
19 
19 
19 
19 

1§ 
19 
19 
19 
19 

19 



Log.Exs, 



10 



10 



10 



10 



10 



10 



13551 
13598 
13645 
13692 
13739 



13786 
13833 
13880 
13927 
13974 



14021 
14068 
14115 
14162 
14210 



14257 
14304 
14351 
14398 
14445 



14493 
14540 
14587 
14634 
14682 



14729 
14776 
14823 
14871 
14918 



14965 
15013 
15060 
15108 
15155 



15202 
15250 
15297 
15345 
15392 



15440 
15487 
15535 
15582 
15630 



15678 
15725 
15773 
15820 
15868 



15916 
15963 
16011 
16059 
16106 



16154 
16202 
16250 
16298 
16345 



16393 



T) 



47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
4f 
47 
47 
47 

47 
47 
47 
47 
47 

48 

41 
47 
47 
48 

47 
47 
48 
47 
47 

48 
47 
48 
48 
47 
48 



10 

11 
12 
13 
14 



I> Log. Exs. D 



15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
li 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P.P. 



6 

7 

8 

9 

10 

20 

3C 

4C 

50 



6 

7 

8 

9 

10 

2C 

30 

40 

50 



48 
48 



47 



4 


7 - 


5 


5 


6 


3 


7 


1 


7 


9 


15 


8 


23 


7 


31 


6 


39 


6 



47 



4 


7 


4. 


5 


5 


5. 


6 


2 


6 


7 





7. 


7 


8 


7. 


15 


6'15. 


23 


5 23- 


31 


3 31- 


39 


138- 



46 

6 
4 
2 

7 
5 
2 

7 



6 
7 
8 
9 
10 
20 



46 

4.6 



80 23 
40 30 
50 38 





30 


20 


61 20 


20 


7 


4 


2 


3 


8 


-; 


7 


2 


6 


9 


3 


1 


3 





10 


3 


4 


3 


3 


20 


6 


8 


6 


6 


30 


10 


2 


10 





40 


13 


6 


13 


3 


50 


17 


1 


16 


6 



6 
7 
8 
9 

10 
20 
30 
40 
50 



19 

9 
3 
6 
9 

2 
5 
7 



P.P. 



tABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL gECANTa. 
66° 67° 





1 

2 

3 

_4 

S 
6 
7 
8 
_9^ 

10 

11 

12 

13 

14 

15 
16 
17 
18 
19 

30 

21 

22 

23 

24 

25 
26 
27 
28 
29 

30 

31 
82 
33 
84 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

m 



Lg. Vers. 



9.77325 
.77344 
•77363 
.77383 

. -7 7402 

9-77422 
- 77441 
-77461 
.77480 
. 77499 



9-7751? 
.77538 
.77557 
-77577 
-77598 



9-77616 
-77635 
.77654 
-77674 
-.77093 



9-77712 
•77732 
-77751 
-77770 
-77790 



3. 7780? 

-77828 

-77847 

77867 

-77886 



3. 77905 
-77925 
- 77944 
-77963 
-779^2 



9-78002 
7S021 
78040 
78059 
7807§ 



3-78098 
.78117 
-78136 
.78155 
-78174 



9-78194 
-78213 
-78232 
-78251 
-78270 



9.78239 
•78309 
-78328 
-78347 
-78366 



9-78385 

-78404 

-78423 

.78442 

78462 



g- 78481 



Lg. Vers. 



1> 

19 

1? 
1? 
19 

1? 
1? 
19 
19 
19 

1? 
19 
19 

1? 
19 

19 
1? 

1? 
19 
19 

19 
19 
19 
19 
19 

19 
19 
1? 
19 
19 

is 

19 

1? 
19 
19 

19 
1? 
19 
19 
19 

19 

1? 
19 
19 
19 

1§ 
1? 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 



Log.Exs. 



10.16393 
.16441 
.16489 
.16537 
.16585 



10-16633 
.16680 
.16728 
.16776 
.16824 



10.16872 
•16920 
.16968 
.17016 
.17064 



10.17112 
.17160 
.17209 
.17257 
.17305 



10.17353 
.17401 
.17449 
.17498 
.17546 



10.17594 
.17642 
.17890 
.17739 
.17787 

10.17835 
.17834 
.17932 
.17930 

._iL802? 

10.18077 
.18126 
.18174 
.18222 
-18271 

10-18319 
.18388 
.18416 
.18465 
.18514 



I) 



10.18562. 
.18611 
.18659 
.18708 
.18757 



10.18805 
.18854 
.18903 
•18951 
.19000 



10.19049 
.19098 
•19146 
•19195 
. 19244 

10-19293 
Log.Exs. 



48 
47 
48 
48 

48 
47 
48 
48 
48 

•48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 

48 
48 
48 
48 
48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
43 
48 
4? 
48 

48 
43 
48 
48 
49 

48 
48 
49 
48 
49 

48 
49 
48 
49 
49 

4§ 



Lg. Vers, 



1-78481 
.78500 
.78519 
.78538 
.78557 



1-78576 
-78595 
.78614 
.78633 
-78612 

1-78671 
-78690 
-7870? 
.78728 
-78747 



1-78766 
.78785 
.78804 
.78823 
.78842 



1-78861 
.78880 
•78899 
•78918 
.78937 



•78956 
.78975 
.78994 
.79013 
.79032 



.79051 
•79069 
.79088 
.79107 
.79126 



79145 
■79164 
.79183 
■79202 

■79220 



•7923? 
.79258 
•79277 
-79296 
• 79315 

•79333 
•79352 
.79371 
79390 
-79409 



.79427 
.79446 
.79465 
•79484 
•79503 



-79521 
.79540 
.79559 
.79578 
■79596 



9.79615 



n 



19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
18 

19 
19 
19 
19 
19 

19 
18 
19 
19 
19 

19 
18 
19 
19 
18 
19 

1? 
18 
19 
19 

18 
19 
19 
18 
19 

18 
19 

1? 
18 
19 

18 
1? 
18 
19 
18 
19 



Log.Exs, 

10-19293 
•19342 
•19391 
•1943? 
-19488 



10-19537 

•19586 

•19635 

•19684 

19733 



10-19782 
.19831 
.19880 
•19929 
.19979 



10-20028 
•20077 
.20126 
•20175 
•20224 



10.20273 
•20323 
.20372 
•20421 
•20470 



10.20520 
•2056? 
.20618 
.20668 
.20717 



10.20767 
.20816 
.20365 
.20915 
.20964 

10.21014 
.21063 
•21113 
•2]162 
.21212 



10.21262 
.21311 
•21361 
.^1410 
.21460 



10-21510 
•21560 
.2160? 
.21659 
.21709 



10-2175? 
•21808 
.21858 
.21908 
.21958 



10-22008 
.22058 
.22108 
.22158 
.22208 



10 22258 



I> Lg.Vers. -» 



Log.Exs. 



D 

49 
49 
48 
49 

49 
49 
49 
49 
49 

49 
49 
49 
4? 
49 

49 
49 
49 
49 
49 

49 

49 
4? 
49 
49 
4? 
49 
4? 
49 
49 

49 
4? 
49 
4? 
49 

4? 
4? 
4? 
49 
50 

4? 
49 
49 
49 
50 

49 
50 
49 
50 
49 

50 
49 
50 
50 
49 

50 
50 
50 
50 
50 

50 





1 

2 
3 
4 

5 
6 
7 
S 
_9 

10 

11 

12 

13 

JL4 

15' 
16 
17 
18 

11 
30 

21 
22 
23 
24 

25 
26 
27 
28 

M. 

30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



55 
56 
57 
58 
59 

60 



P. P. 



ma^im 





50 


49 


6 


5-0 


4.9 


7 


5-8 


5 


H 


8 


6-6 


6 


6 


9 


7-5 


7 


4 


10 


8-3 


8 


2 


20 


16-6 


16 


5 


30 


25-0 


24 


7 


40 


33-3 


33 





50 


41.6 


41 


2 





49 


48 


6 


4-9 


4.§ 


7 


5-7 


5.§ 


8 


6-5 


6-4 


9 


7-3 


7-3 


10 


8-1 


8-1 


20 


16-3 


18. J, 


30 


24-5 


24-2 


40 


32-6 


32-3 


50 


40.8 


40.4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



48 

4-8 

5-6 

6-4 

7-2 

8-0 

16-0 

24-0 

32-0 

40-0 



47_ 
4.Z 





19 


11 


6 


1-9 


1- 


7 


2-3 


2- 


8 


2-6 


2 


9 


2-9 


2- 


10 


3-2 


3- 


20 


6-5 


6. 


30 


9-7 


9- 


40 


13-0 


12- 


50 


16-2 


15- 



5 „ 

6-3 

7.1 

7.9 

15-8 

23-7 

31-6 

39-8 



-9 

•'2 
5 

•i 
1 

• 3 
-5 

i 



6 

V 

8 

9 

10 

20 

80 

40 

50 



P. P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
68" 69° 



Lg. Vers. 



79615 
79634 
79653 
79671 
79690 



79709 
79727 
79746 
79765 
79783 

79802 
79821 
79839 
79858 
79877 

79895 
79914 
79933 
79951 
79970 



799fl8 
80007 
80026 
80044 
80063 



80081 
80100 
80119 
80137 
80156 



80174 
80193 
80211 
80230 
80248 



80267 
80286 
80304 
80323 
80341 



80360 
80378 
80397 
80415 
80434 



80452 
80470 
80489 
80507 
80528 

80544 
80563 
80587 
80600 
80618 

80636 
80655 
80673 
80692 
80710 



9-80728 



Lg. Vers 



18 
19 
18 
18 

19 
18 
19 
18 
18 

1? 
18 
18 
19 
18 

18 
18 
19 
18 
18 

18 
19 
18 
18 
18 

18 
19 
18 
18 
18 

18 
18 
18 
18 
18 

19 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 

18 
18 
18 
18 

18 
18 
18 
IK^ 
18 

18 
18 
18 
18 
18 

18 



Log.Exs, 



10 



22258 
22308 
22358 
22408 
22458 



10. 



22508 
22558 
22608 
22658 
22708 



10 



.22759 
.22809 
.22859 
.22909 
.22960 



10 



.23010 
.23060 
.23110 
.23161 
.23211 



10 



10 



10 



.23262 
.23312 
.23362 
.23413 
^23463 

.23514 
.23564 
.23615 
.23666 
.23716 

23767 
23817 
23868 
23919 
23969 



10 



.24020 
.24071 
.24122 
.24172 
.24223 



10. 



24274 
24325 
24376 
24427 
24478 



10 



.24529 
24580 
.24631 
.24682 
.24733 



10 



24784 
24835 
24886 
24937 
24988 



10 



10 



-•^5039 
.25090 
.25142 
.25193 
i25_2_4| 

•25295 

fxsT 



n 

50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
51 
50 
50 

51 
50 
51 
50 
51 

51 
50 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

51 
51 
51 
51 
51 

51 



Lg. Vers. 



•80728 
.80747 
.80765 
.80783 
.80802 



80820 
.80839 
.80857 
.80875 

80894 



.80912 
.80930 
.80949 
.8096Z 
•-809_8J 
.81003 
.81022 
.81040 
.81058 
; 81077 

81095 
.81113 
.81131 
.81150 
.81168 



.81186 
. 81204 
.81223 
.81241 
.81259 
.81277] 
.81295 
.81314 
.81332 
■81350 

.81368 
.81386 
.81405 
.81423 
.81441 



•81459 
.81477 
.81495 
.81513 
L8i532 

.81550 
.81568 
.81586 
.81604 
.81622 



.81640 
.81658 
.81676 
.81695 
.81713 



.81731 
•81749 
.81767 
.81785 
.81803 

.'81821 



1> jLg. Vers, 



D 



18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
"8 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 

la 

18 
18 



Log.Exs. 



10 



25295 
25347 
25398 
25449 
25501 



10 



25552 
25604 
25655 
25707 
25758 



10. 



25810 
25861 
25913 
25964 
2S016 



10. 



26067 
26119 
26171 
26222 
26274 



10. 



26326 
26378 
26429 
26481 
26533 



10. 



10. 



26585 
26637 
26689 
26741 
28793 

26845 
26897 
26949 
27001 
27053 



10. 



27105 
27157 
27209 
27261 
97314 



10 



27366 
27418 
27470 
27523 
27575 



10 



22627 
.27680 
.27732 
.27785 

27837 



10 



27890 
.27942 
27995 
28047 
28100 



10 



28152 
28205 
28258 
28310 
28363 



10.28416 



5l 
5l 
51 
51 

51 
51 
51 
51 
5l 

51 
51 
51 
51 
51 

5l 
52 
51 
51 

52 

5l 
52 
51 
52 
52 

52 
51 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
53 
52 
52 
53 
5§ 





1 
2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



30 

21 
22 
23 
24 



D Log.Exs.] I> 

742 



25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.45 
46 
47 
48 

J9 

50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P.P. 





53 


53 


6 


5-3 


52 


7 


6 


2 


6-1 


8 


7 





7.0 


9 


7 


9 


7.9 


10 


8 


8 


8.7 


20 


17 


6 


17.5 


30 


26 


5 


26.2 


40 


35 


3 


35. 


5C 


44 


1 


43.7 



6 

7 
8 

9 
10 
20 
30 
4G 
50 



6 
7 
8 

9 
10 
20 
80 
40 
50 



63 



5 


2 


5. 


6 





6. 


& 


9 


6. 


7 


8 


7. 


8 


6 


8 


17 


3 


17- 


26 





25- 


34 


C 


34 


43 


3 


42. 



51 



5 


1 


5^ 


5 


9 


5. 


6 


8 


6. 


7 


6 


7- 


8 


5 


8. 


17 





16. 


25 


5 


25 • 


34 





33 • 


42 


5 


42. 



51 

1 

8 
7 
6 
1 
7 
3 
9 

50 


9 
7 
6 
4 
8 
2 
6 
1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



60 

5.0 

5.8 

6.6 

7-5 

8.3 

16.S 

25^0 

33.5 

41.6 





19 


18 


6 


1.9 


1-8 


7 


2 


2 


2 


1 


8 


2 


5 


2 


4 


9 


2 


8 


2 


8 


10 


3 


1 


3 


1 


20 


6 


3 


6 


1 


30 


9 


5 


9 


2 


40 


12 


6 


12 


3 


50 


15 


8 


15 


4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



18 



• 8 
.1 
■ 4 

7 

• 


• 
.0 



15 



P.P. 



ITABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
70^ 71° 



/ 


Lg. Vers. 




1 

2 
3 

4 


9 


81821 
81839 
81857 
81875 
81893 


5 
6 
7 
8 

g 


9. 


81911 
81929 
81947 
81965 
81983 


10 

11 

12 
13 
14 


9 


82001 
82019 
82037 
82055 
82073 


15 
16 
17 
18 
19 


9 


82091 
82109 
82127 
82145 
82163 


20 

21 
22 
23 
24 


9 


82181 
82199 
82217 
82235 
82252 


25 
26 
27 
28 
29 


9. 


82270 
82288 
82306 
82324 
82342 


30 

31 
32 
33 
34 


9 


82360 
82378 
82396 
82413 
82431 


35 
36 
37 
38 
39. 

40 

41 
42 
43 
44 


8 


82449 
82467 
82485 
82503 
82520 

82538 
82556 
82574 
82592 
82609 


45 
46 
47 
48 
49 


9 


82627 
82645 

•82663 
82681 

.82693 


50 

51 
52 
53 
54 


9 


•82716 
.82734 
.82752 
.82769 
.82787 


55 
56 
57 
58 
59 


9 


82805 
.82823 
.82840 
.82858 
.82876 


60 


9 


82894 


/ 


U 


J. Vers. 



10. 



10 



10. 



10. 



18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
1? 
18 
18 
18 
18 
18 
18 
18 
17 

18 
18 
18 
18 
17 

18 
18 
18 
17 
18 

17 10' 
18 
18 
17 

18 
18 
17 
18 
17 
18 
18 
17 
18 
17 
18 
17 
18 
17 
18 

17 
18 
17 
18 
17 

18 



Log. Exs. 



28416 
28469 
28521 
28574 
28627 



28680 
28V33 
28786 
28839 
18892 

28945 
28998 
29051 
29104 
29157 

29210 
29263 
29316 
29370 
29423 



10. 



10- 



10. 



29476 
29529 
29583 
29636 
29689 

29743 
29796 
2985g 
29903 
29957 

30010 
30064 
30117 
30171 
30225 



30278 
30332 
30336 
30440 
3049^ 



10 



30547 
30601 
30655 
30709 
30763 



10. 



10. 



30817 
30871 
30925 
30979 
31033 



31087 
31141 
31195 
31249 
31303 



10 



31358 

31412 
31466 
31521 
31575 



10.31629 



-O Log. Exs. 



D 

53 
52 
53 
53 
52 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 

53 

53 
53 
53 
53 
53 

53 
53 
53 

53 

53 

53 
53 
53 
54 
53 

11 ^' 
53 
54 
53 

54 
53 
54 
54 

541 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 



Lg. Vers. 



82894 
.82911 
.829'?9 
.82947 

82964 



•82882 
•83000 
.83017 
83035 
:_83053 

.83070 
.83088 
.83106 
.83123 
•83141 

83159 
.83176 
.83194 
.83211 
.83228 



.83247 
.83264 
.83282 
.83299 
.83317 



83335 

83352 

.83370 

•83387 

.83405 



•83422 
• 83440 
•83458 
•83475 
•83493 



83510 
83528 
83545 
83563 
83580 

83598 
.83615 

83633 
•83650 
.83668 



83685 

83703 

.83720 

.83737 

.83755 



83772 
.83790 

83807 
.83825 
.83842 



83859 
.83877 
.83894 
.83912 
.83929 
^83946 
I> jLg. Vers. 



n 

17 
17 
18 
17 
18 

17 
17 
18 
17 

17 
18 
17 
IZ 
17 

18 
17 
IZ 
17 
18 

17 
17 
17 
17 
18 

17 
17 
IZ 
17 
17 

17 
17 
18 

17 
17 

17 

17 
17 
17 
17 

17 
17 
17 

17 
17 

17 
17 

17 
17 
17 

17 

17 
17 
17 
17 

17 

17 
17 
17 
17 
17 



Log, Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



31629 
31684 
31738 
31793 
31847 



31902 
31956 
32011 
32066 
32120 



32175 
32230 
32284 
32339 
32394 



32449 
32504 
32558 
32613 
32668 

32723 
32778 
32833 
32888 

32944 



32999 
33054 
33109 
33164 
33220 



33275 
33330 
33385 
33441 
33496 



33552 
33607 
33663 
33718 
33774 

33829 
33885 
33941 
33996 
34052 



34108 
34164 
34-220 
34275 
34331 



34387 
34443 
34499 
34555 
34611 



34867 
34723 
34780 
34836 
34892 



34948 



-?> j Log. Exsc 



n 

54 
54 
54 
54 

54 
54 
5i 
55 
54 

54 
55 
54 

ill 131 
04 1 14 



5 

6 

7 

8 

__9 

10 
11 
12 



55 
55 
54 
55 
55 

55 
55 
55 
55 

55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
56 

55 
55 



15 
16 
17 
18 
19 

30 

21 
22 
23 
24 



25 
26 
27 
28 
29_ 

30 

31 
32 
33 
34 

35 
36 
37 
38 

ii 

i40 
41 
42 
43 



5? 441 



55 
56 
56 
55 
56 



45 
46 
47 
48 
49 



?^850 
^5 51 
Rfi 52 
Rft 53 
56 I 54 



56 
56 
56 
56 
56 

56 



55 
56 
57 
58 
59 



60 



P.P. 



5(J 



6 


5.61 


7 


6 


6 


8 


7 


5 


9 


8 


5 


10 


9 


4 


20 


18 


8 


30 


28 


2 


40 


37 


6 


50 


47 


1 



50 
5.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



55 



5.5 


5. 


6.5 


6. 


7.4 


7. 


8.3 


8. 


9.2 


9. 


18.5 


18. 


27.7 


27. 


37.0 


36. 


46.2 


45 • 



6 

7 

8 

9 
18 
28.0 
37-3 
46.6 



55 

5 
4 
3 
2 
1 
3 
5 
6 
8 





51 


54 


6 


5.4 


5.4 


7 


6.3 


6 


3 


8 


7.2 


7 


2 


9 


8.2 


8 


1 


10 


9.1 


9 





20 


18-1 


18 





30 


27-2 


27 





40 


36^3 


36 





50 


45^4 


45 






6 

7 

8 

9 

10 

20 

30 

40 

50 



5.3 


5. 


6.2 


6- 


7.1 


7. 


8.0 


7. 


8.9 


8. 


17.8 


17. 


26.7 


26. 


35.6 


35- 


44.6 


44. 



53 

3 
2 

9 
8 
6 
5 
3 
I 



6 

7 
8 
9 
10 
20 
30 
40 
50 



53_ 

5-2 

6.1 

7.0 

7.9 

8-7 

17-5 

26-2 

35. 

43^7 





18 


17 , 


6 


18 


1-7| 


7 


2 


1 


2 





8 


2 


4 


2 


3 


9 


2 


7 


2 


6 


10 


3 





2 


9 


20 


6 





5 


8 


30 


9 





8 


7 


40 


12 





11 


6 


50 


15 





14 


R 



17 

1.7 



11. 3 



P. P. 



743 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 

72° 73° "" ' 



Lg. Vers, 



9.83946 
83964 
83981 
83999 
84016 



5 9 
61 

7 
8 

J_ 

10 

11 
12 
13 
i4 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

81 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 

52 

53 

54, 

55 

56 

57 

58 

59 

60 



84033 
84051 
84068 
84085 
841J3 

84120 
84137 
84155 
84172 
8418§ 

84207 
84224 
8424i 
84259 
84276 



84293 
84310 
84328 
84345 
84362 



84380 
84397 
84414 
84431 
84449 

84466 
84483 
84500 
84517 
84535 



84552 
84569 
84586 
84603 
84620 



84638 
84655 
84672 
84689 

84706 



84724 
84741 
8475P 
84775 
84792 



8480P 
84826 
84844 
8486] 
84878 



84895 
8491? 
84929 
84946 
84963 
84980 



jLg, Vers. 



J> 



17 

17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 

IZ 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 

17 

17 

17 

IZ 
17 

IZ 
17 

IZ 
17 
17 
17 
17 

17 

17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 

IZ 
17 
17 
17 
17 



Log.Exs, 



10.34948 
35005 
35061 
35117 
35174 



10 



10 



10 



10 



10 



10 



llO 



10 



10 



10 



10 



]0 



35230 
35286 
35343 
35399 
35456 

35513 
35569 
35626 
35683 
35739 



35796 
35853 
35910 
35987 
36023 

36080 
36137 
36194 
36251 
36308 



36366 
36423 
36480 
36537 
36594 



36652 
36709 
36766 
36824 
36881 



36938 
36996 
37054 
37111 
37169 



37226 
37284 
37342 
37399 
37457 



37515 
37573 
37631 
37689 
37747 



37805 
37863 
37921 
37979 
3 8037 

38095 
38153 
38212 
38270 
38328 



38387 



Log.Exs. 



D 

56 
56 
56 
56 

56 
56 

56 
56 
57 
56 
56 
56 i 

57 
56 

57 
56 
57 
57 
56 

57 
57 
57 
57 
57 

57 
57 

5Z 
57 
57 

57 
57 
5? 
57 
57 

57 
57 
58 
57 
57 

57 
57 
58 
57 
58 

58 
57 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 



Lg. Vers 



9. 84980 
84997 
85014 
85031 
85049 



85066 
85083 
85100 
85117 
85134 

85151 
85168 
85185 
85202 
85219 



85236 
85253 
85270 
85287 
85304 



85321 
85338 
85355 
85372 
85389 



85405 
85422 
85439 
85456 
85473 



85490 
85507 
85524 
85541 
85558 



85575 
85592 
85608 
85625 
85642 



85659 
85676 
85693 

85710 
S5726 



85743 
85760 
85777 
85794 
85811 



8582_ 
85844 
85861 
85878 
85895 

85911 
85928 
85945 
85962 
85979 



ti.g'iggR 



'-* JLg> Vers. 



D 



Log. Exs. 



10 



10 



10 



10 



10 



10 



17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 

17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

16 
17 
17 
17 
17 

17 
17 
16 
17 
17 

17 
17 
18 
17 
17 

i^iio 

16 UU 

17 
17 
16 

17 

17 
16 
17 
17 

16 
17 
17 
16 
17 

16 
17 
17 
16 
17 

16 



10 



10 



10 



10 



10 



10 



38387 
38445 
38504 
38562 
38621 



38679 
38738 
38796 
38855 
38914 



38973 
39031 
39090 
39149 
39208 



39267 
39326 
39385 
39444 
39503 

39562 
39621 
39681 
39740 
39799 

39859 
39918 
39977 
4CC37 
4CC96 



40156 
40216 
40275 
4C335 
40395 



40454 
40514 
40574 
40634 
40694 
40754 
40814 
40874 
40934 
40994 



41054 
41114 
41174 
41235 
41295 



41355 
41416 
41476 
41537 
41597 



41658 
41719 
41779 
41840 
41901 



-O,, 



41962 
I Exs. 



n 



58 
58 
58 
58 

58 
58 
58 
59 
58 

59 
58 
59 
59 
58 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
60 
59 
59 
60 

59 
60 
59 
60 

60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 

60 

60 
61 
60 
60 
61 

6] 

Id 



10 

11 

12 
13 
14 



20 

21 
22 
23 
24 



25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P.P. 





61 


6 


61 


7 


7.1 


8 


81 


9 


9.1 


10 


10. 1 


20 


20.3 


30 


30.5 


40 


40-6 


50 


50.8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 
30 
40 
50 



60 

6.0 

7.0 

8.0 

9.0 

10.0 

20.0 

30.0 

40-0 

50-0 

59 



5 


9 


5. 


6 


9 


6. 


7 


8 


7. 


8 


8 


8. 


9 


8 


9. 


19 


6 


19. 


29 


5 


29. 


39 


3 


39. 


49 


1 


48. 



60_ 

60 

7.0 

8.0 

91 

10.1 

20. i 

30.2 

40. 3 

59 

5.9 

6.9 

7.9 

8.9 

9.9 

19-8 

29.7 

39. S 

49-6 

58 
8 





58 


5 


7 


6 


5.8 


5.7 


7 


6 


7 


6 


7 


8 


7 


7 


7 


6 


9 


8 


7 


8 


6 


10 


9 


6 


9 


5 


20 


19 


3 


19 


1 


30 


29 


C 


28 


7 


40 


38 


6 


38 


3 


50 


48 


3 


47 


9 



6 
7 
9 
9 
10 



6 
7 
8 
9 

20 19 
8028 
40138 
5047 



57 
57 



56_ 

5.6 





17 


17 


16 


6 


17 


1.7 


1.6 


7 


2 





2.0 


1 


9 


8 


2 


3 


2.2 


2 


2 


C 


2 


6 


2-5 


2 


5 


10 


2 


9 


2.8 


2 


7 


20 


5 


8 


5.6 


5 


5 


30 


8 


y 


8 5 


8 


2 


40 


11 


6 


11-3 


n 





BO 


14 


6 


]4.1 


-.3 


7 






P 


. P. 







744 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTa 



74" 



75° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 

12 
13 
11 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



Lg. Vers, 



.85995 

.86012 

.86029 

86046 

86062 



■86079 
•86096 
.86113 
•86129 
J6J46 

•86163 
.86179 
.86196 
.86213 
.86230 



.86246 
.86263 
.86280 
.86296 
.86313 



9.86330 
86346 
86363 
86380 
86396 



9-86413 
.86430 
.86446 
.86463 
•86479 



9-86496 
•86513 
•86529 
•86546 
•86562 



9. 86579 
•86596 

• 86612 

• 86629 
-86645 



9-86662 
•88678 
.86695 
•86712 
•86728 



6019 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



9-86745 
-86761 
.86778 
.86794 
.86811 



1.86827 

86844 

86860 

.86877 

86893 



9.86910 
.86926 
.86943 
.86959 
-86976 



9 - 8699-:? 



Lg. Vers. 



JD 



17 
16 
17 
16 

17 

16 

17 
16 
17 

16 
16 
17 
16 
17 

16 
16 
17 
16 
16 

17 
16 
IB 
17 
16 

16 
17 
16 
16 
16 

17 
16 
16 
16 
16 

17 
16 
16 
16 
16 

16 
16 
17 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 



Log. Exs. 



10. 



10. 



41962 
42022 
42083 
42144 
42205 

42266 
42327 
42388 
42450 
42511 



10 



42572 
42633 
42695 
42756 
42817 



10. 



42879 
42940 
43002 
43083 
43125 



10. 



43187 
43249 
43310 
43372 
43434 



10. 



43496 
43558 
43620 
43682 
43744 



10. 



43806 
43868 
43931 
43993 
44055 



10 



.44118 
.44180 
. 44242 
.44305 
.44368 



10. 



44430 
44493 
44556 
44618 
44681 



10. 



44744 
44807 
44870 
44933 
44996 



10. 



45059 
45122 
45185 
45248 
45312 



10. 



45375 
45439 
45502 
45565 
45629 



10-45693 



D Log. Exs. 



J) 

60 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
6l 
62 

61 
62 
61 
62 
61 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
63 

62 
62 
63 
62 
63 
62 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 

64 



Lg, Vers 



86992 
87009 
87025 
87042 
87058 



87074 
87091 
87107 
87124 
87140 



87157 
87173 
87189 
87206 
87222 



87239 
87255 
87271 
87288 
87304 

87320 
87337 
87353 
87370 
87386 



87402 
87419 
87435 
87451 
87468 



87484 
87500 
87516 
87533 
87549 



87565 
87582 
87598 
87614 
87631 

87647 
87653 
87679 
87696 

87712 



87728 
.87744 
.87761 
.87777 
.87793 



87809 
.87825 
.87842 
-87858 
.87874 



9- 



87890 
87906 
87923 
87939 
87955 



9-8797] 



-O Lg. Vers 



Z) 



16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 

16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
.16 
16 

16 

16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



45693 
45756 
45820 
45884 
45947 



46011 
46075 
46139 
46203 
46267 



46331 
46395 
46460 
46524 
46588 



46652 
46717 
46781 
46846 
46910 



46975 
47040 
47104 
47169 
47234 



47299 
47364 
47429 
47494 
47559 



47624 
47689 
47754 
47820 
47885 



47950 
48016 
48081 
48147 
48213 



48278 
48344 
48410 
48476 
48542 



48607 
48674 
48740 
48806 
48872 



48938 
49004 
49071 
49137 
49204 



49270 
49337 
49403 
49470 
49537 



49604 



D jLog. Exs. 



z> 



63 
63 
64 
63 
64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
65 
64 
65 
64 

65 
65 
65 
65 
65 

65 
65 
65 
65 
65 

65 
65 
65 
65 
66 

65 
65 
66 
66 
66 

65 
66 
66 
66 
66 

66 
66 
66 
66 
66 

66 
66 
66 
67 
66 

67 





1 

2 
3 

_4 

5 
6 
7 
8 
__9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
-29 

30 

31 
32 
33 
li 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 

48 

19 

50 

51 

52 

53 

_51 

55 

56 

57 

58 

^ 

60 



P.P. 





67 


66 


6 


6-7 


6.6 


7 


7.8 


7.7 


8 


8.9 


8.8 


9 


10-0 


10.0 


10 


11.1 


11.1 


20 


22.3 


22.1 


30 


33.5 


33-2 


40 


44.6 


44.3 


50 


55.8 


55-4 





65 


65 


6 


6-5 


6-5 


7 


7 


6 


7.6 


8 


8 


7 


8.6 


9 


9 


8 


9.7 


10 


10 


9 


10.8 


20 


21 


8 


21.6 


30 


32 


7 


32.5 


40 


43 


6 


43.3 


50 


54 


6 


54.1 





64 


63 


6 


6-4 


6.3 


7 


7.4 


7-4 


8 


8-5 


8-4 


9 


9.6 


9.5 


10 


10.6 


10.6 


20 


21.3 


21.1 


30 


32.0 


31-7 


40 


42.6 


42-3 


50 


53.3 


52.9 





62 


62 


6 


6.2 


6-21 


7 


7 


3 


7 


2 


8 


8 


3 


8 


2 


9 


9 


4 


9 


3 


10 


10 


4 


10 


3 


20 


20 


8 


20 


6 


30 


31 


2 


31 





40 


41 


6 


41 


3 


50 


52 


1 


51 


6 



66 

6.6 

7.7 

8.8 

9.9 

11-0 

22.0 

330 

44^0 

55-0 

iB5 

6.4 

7.5 

8.6 

9.7 

10.7 

21.5 

32.2 

43.0 

53.7 

63 

6.3 

7.3 

8.4 

9.4 

10.5 

21-0 

31.5 

42 

52.5 

61 

6.1 

7.2 



8. 

9. 
10. 
20. 
30. 
41.0 
51.2 





61 


6 


61 


7 


7^1 


8 


8^1 


9 


9.1 


10 


10.1 


20 


20.3 


30 


30.5 


40 


40.6 


50 


50.8 



60_ 

6.0 
7.0 
8.0 
9.1 

10.1 
20.1 
30.2 
40.3 
50.4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



17 

1.7 
2.0 
2.2 
2.5 
2-8 
5^6 
8^5 
11^3 
14-1 



16_ 

1-6 
1-9 
2-2 
2-5 
2-7 
5.5 
8.2 
11.0 
13-7 



16 

1.6 
1-8 
2.1 
2.4 
2.§ 
5.3 
8.0 
10. g 
13.3 



P. P. 



745 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



76° 



77° 



Lg. Vers. 



87971 
87987 
88003 
88020 
88036 



88052 
88068 
88084 
88100 
88116 



88133 
88149 
88165 
88181 
88197 

88213 
88229 
88245 
88261 
88277 



88294 
88310 
88326 
88342 
88358 



88374 
88390 
88406 
88422 
88438 



88454 
88470 
83486 
88502 
88518 



88534 
88550 
8856G 
88582 
88598 



88614 
88630 
88646 
88662 
88678 



88694 
88710 
88726 
88742 
88758 



88774 
88790 
88805 
88821 
88837 



88853 
88869 
88885 
88901 
_ 88917 
9 88933 



n 

16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
15 
16 

16 
16 
16 
16 
16 

16 
16 
15 
16 
16 

16 
16 
15 
18 
16 

16 



Log.Exs, 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



49604 
49670 
49737 
49804 
49871 



49939 
50006 
50073 
50140 
50208 



50275 
50342 
50410 
50477 
50545 



50613 
50681 
50748 
50816 
50884 



50952 
51020 
51088 
51157 
51225 



51293 
51361 
51430 
51498 
51567 



51636 
51704 
51773 
51842 
51911 



51980 
52049 
52118 
52187 
52256 



52325 
52394 
52464 
52533 
52603 



52672 
52742 
52812 
52881 
52951 



53021 
53091 
53161 
53231 
53301 



53372 
53442 
53512 
53583 
53653 

5 3 724 



' Lg. Vers. D Log.Exs. ?>JLg. Vers. 



J) 

66 
67 
67 
67 

67 
67 
67 
67 
67 

67 
67 
67 
67 
68 

67 
68 
67 
68 
68 

68 
68 
68 
68 
68 

68 
68 
68 
68 
68 

69 
68 
68 
69 
69 

69 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
70 
69 
63 
70 

70 
70 
70 
70 
70 

70 
70 
70 
70 
70 

70 



9 



Lg. Vers, 



8893? 
88949 
88964 
88980 
88996 



89012 
89028 
89044 
89060 
89075 



89091 
89107 
89123 
89139 
89155 



89170 
89186 
89202 
89218 
89234 



89249 
89265 
89281 
89297 
89312 



89328 
89344 
89360 
89376 
89391 



89407 
89423 
89438 
89454 
89470 



89486 
89501 
89517 
89533 
89548 

89564 
89580 
89596 
89611 
89627 
89643 
89658 
89674 
89690 
89705 



89721 
89737 
89752 
89768 
89783 



89799 
89815 
89830 
89846 
89862 



9. 89877 



I> 



16 
15 
16 
16 

16 
15 
16 
16 
15 

16 
16 
15 
16 
16 

15 
16 
15 
16 
16 

15 
16 
15 
16 
15 

16 
15 
16 
16 
15 

15 
16 
15 
16 
15 

16 
15 
16 
15 
15 

16 
15 
16 
15 
15 

16 
15 
15 
16 
15 

15 
16 
15 
15 
15 

16 
15 
15 
15 
16 
15 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



53724 
53794 
53865 
53936 
54007 



54078 
54149 
54220 
54291 
54362 



54433 
54505 
54576 
54647 
54719 



54791 
54862 
54934 
55006 
55078 



55150 
55222 
55294 
55366 
55438 



55511 
55583 
55655 
55728 
55801 



55873 
55946 
56019 
56092 
56165 

56238 
56311 
56384 
56457 
56531 

56604 
56678 
56751 
56825 
56899 



56973 
57047 
57120 
57195 
5.7269 

57343 
57417 
57491 
57566 
57640 

57715 
57790 
57864 
57939 
58014 



10.58089 



Log.Exs. 



D 



70 
71 
70 
71 

71 
71 
71 
71 
71 

71 
71 
71 
71 
72 

7l 
71 
72 
7l 

72 

72 
72 
72 
72 
72 

72 
72 
72 
73 
72 

72 
73 
72 
73 
73 

73 
73 
73 
73 
73 

73 
73 
73 
74 
73 

74 
74 
73 
74 
74 

74 
74 
74 
74 
74 

75 
74 
74 
75 
75 
75 





1 
2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
19 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 

-34 
55 
56 
57 
58 

_59. 

60 



P.P. 





75 


74 


6 


7.5 


7.4 


7 


8.7 


8.6 


8 


10.0 


9.8 


9 


11.2 


11.1 


10 


12.5 


12.3 


20 


25.0 


24.6 


30 


37.5 


37.0 


40 


50.0 


49.3 


50 


62.5 


61.6 



73 

7.3 





73 


71 


6 


7.2 


7.1 


7 


8.4 


83 


8 


9.6 


9.4 


9 


10-8 


10.6 


10 


12.0 


11.8 


20 


24-0 


23.6 


30 


36.0 


35-5 


40 


48-0 


47.3 


50 


60.0 


59.1 





69 


68 


67 


6 


8-9 


6.8 


6. 


7 


8 





7 


9 


7. 


81 9 


2 


9 





8. 


9 10 


3 


10 


2 


10. 


10 11 


5 


11 


3 


11. 


20 23 





22 


6 


22. 


30 34 


5 


34 





33. 


40 46 





45 


3 


44. 


50 


57 


5 


56 


6 


55. 



85 
9.7 
10.9 
12.1 
24-3 
36-5 
48.6 
60-8 



70 

7.0 
8.2 
9-4 
10.6 
11.7 
23.3 
3 5. 2 
47.0 
58-7 





66 





6 


6-6 


o.b 


7 


7 


7 


0.0 


8 


8 


8 


0.0 


9 


9 


9 


0.1 


10 


11 





01 


20 


22 





0.1 


30 


33 





0.2 


40 


44 





0.3 


50 


55 





0.4 





16 


16 


1 


6 


1.6 


1-6 


1. 


7 


1-9 


18 


1- 


8 


2-2 


2.1 


2. 


9 


2.5 


2-4 


2. 


10 


2.7 


2 6 


2. 


20 


5.5 


5.3 


5- 


30 


8.2 


8 


7. 


40 


11-0 


10.6 


10. 


50 


13.7 


13-3 


12. 



P. p. 



746 



tABLE Vin— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT^. 

78° ■ 79° 



Ljr, Vers. 



89877 
89893 
39908 
89924 
89939 



89955 
89971 
89986 
90002 
90017 



90033 
90048 
90064 
90080 
90095 



90111 
90126 
90142 
90157 
90173 



90188 
90204 
90219 
90235 
90250 



90266 
90281 
90297 
90312 
90328 



90343 
90359 
90374 
90389 
9 0405 

90420 
90436 
90451 
90467 
90482 



90497 
90513 
90528 
90544 
90559 



90574 
90590 
90605 
90621 
90636 



90651 
906&7 
90682 
90697 
90713 



90728 
90744 
90759 
90774 
90790 



90805 
Lg. Vers. 



1> 



15 
15 
15 
15 

16 

15 
15 
15 
15 

15 
15 
15 
16 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 

~Z7 



Log.Exs, 



10.58089 
•58164 
■58239 
•58315 
•58390 



10. 58465 
•58541 
•58616 
•58692 
•58768 



10.58844 
•58920 
•58995 
.59072 
•59148 



10-59224 
•59300 
.'59377 
•59453 
•59530 



10. 59606 
•59883 
-59760 
•59837 
.59914 



10-59991 
-60068 
•60145 
•60223 
•60300 



10.60378 
.60455 
.60533 
-60811 
-60688 



10-60766 

- 60844 

.60923 

61001 

61079 

10.61158 
61236 
61315 
61393 
6.1472 



10-61551 
•61630 
-61709 
.61788 
•61867 



10^61947 
•62026 
•62105 
•62185 
•62265 



10-62345 
-62424 
.62504 
.62585 
.62665 



J> 



10.62745 
Log.Exs. 



75 
75 
75 
75 

75 
75 
75 
76 
75 

76 
76 
75 
75 
76 

76 
76 
76 
76 
76 

76 
77 
.76 
77 
77 

77 
77 
77 
77 
77 

77 

77 
77 
78 
77 

78 
78 
78 
78 
78 

78 
78 
78 
78 
79 

78 
79 
79 
79 
79 

79 
79 
79 
80 
79 

80 
79 
80 
80 
80 

80 



Lg. Vers. 



90805 
90820 
90835 
90851 
9^0866 

90881 
90897 
90912 
90927 
909JL3 

90958 
90973 
90988 
91004 
91019 



91034 
91049 
91035 
91080 
91095 



91110 
91126 
91141 
91156 
91171 



91187 
91202 
91217 
91232 
91247 



91263 
91278 
91293 
91308 
91323 



91338 
91354 
91369 
91384 
91399 

91414 
91429 
91445 
91460 
91475 



91490 
91505 
91520 
91535 
91550 



91565 
91581 
91596 
91611 
9 1626 

91641 
91656 
91671 
91686 
91701 



91716 



-Z> jig. Vers. 



15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 

15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
15 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



62745 
62825 
62906 
62986 
63067 



63148 
63229 
63310 
63391 
63472 



63553 
63634 
63716 
63797 
63879 



63961 
64043 
64125 
64207 
64289 



64371 
64453 
64536 
64818 
64701 



64784 
64867 
64950 
65033 
65116 



65199 
65283 
65366 
65450 
65534 



65617 
65701 
65785 
65870 
65954 



66038 
66123 
66207 
66292 
66377 



66462 
66547 
66632 
66717 
66803 



66888 
66974 
67059 
67145 
67231 



67317 
67403 
67490 
67576 
67663 

67749 



-Z> Log.Exs. 



D 



80 
80 
80 
81 

80 
81 
81 
81 
81 

8l 
81 
81 
81 
81 

82 
82 
82 
82 
82 

82 
82 
82 
82 
83 
82 
83 
83 
83 
83 

83 
83 
83 
83 
84 

83 
84 
84 
84 
84 

84 
84 
84 
84 
85 

85 
85 
85 
85 
85 

85 
85 
85 
86 
86 

86 
86 
86 
86 
86 

86 

"z> 



10 

11 

12 
13 
14 



30 

21 
22 
23 
2j4 

25 
26 
27 
28 
29 



30 

31 
32 
33 

34 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 

54 



55 
56 
57 
58 
59 

60 



P. P. 





86 


85 


84 


6 


8.6 


8.5 


8. 


7 


10 





9 


9 


9. 


8 


11 


4 


11 


3 


11. 


9 


12 


9 


12 


7 


12. 


10 


14 


3 


14 


^ 


14. 


20 


28 


6 


28 


3 


28. 


30 


43 





42 


5 


42. 


40 


57 


3 


56 


6 


56. 


50 


71 


6 


70 


8 


70. 





83 


83 


83 


6 


8.3 


8.2 


8. 


7 


9 


7 


9 


5 


9. 


8 


11 





10 


9 


10. 


9 


12 


4 


12 


3 


12. 


10 


13 


8 


13 


6 


13. 


20 


27 


6 


27 


3 


27. 


30 


41 


5 


41 





40. 


40 


55 


3 


54 


6 


54. 


50 


69 


1 


68 


3 


67- 



6 

7 

8 

9 

10 

20 

30 

40 

50 



80 79 



8 


o; 7 


9 


7. 


9 


3| 9 


2 


9^ 


10 


6 10 


5 


10. 


12 


11 


8 


11. 


13 


3 13 


1 


13. 


26 


6 26 


3 


26- 


40 


39 


5 


39. 


53 


3 52 


6 


52. 


66 


6 65 


8 


65. 





77 


76 


7^ 


6 


7-7 


7.6 


7. 


7 


9 





8 


8 


8. 


8 


10 


2 


10 


1 


10. 


9 


11 


5 


11 


4 


11. 


10 


12 


8 


12 


6 


12- 


20 


25 


6 


25 


3 


25. 


30 


38 


5 


38 





37. 


40 


51 


3i50 


6 


50. 


50 


64 


1 


63 


3 


62. 



78 
8 
1 
4 
7 








6 

7 

8 

9 

10 

20 

30 

40 

50 







0.0 

















1 





1 





1 





2 





3 





.4 



6 


1 


7 


1 


8 


2 


9 


2 


10 


2 


20 


5 


30 


8 


40 


10 


50 


13 



16 

6 



15_ 

1.5 



15 

1.5 
1.7 
2.0 
2.2 
2.5 
5.0 
7.5 
10.0 
12.5 



P, P. 



TABLE Vlir.— 1.0GAIHTHMIC VERSED SINES AND EXTERNAL SECANTS, 
80° 81° • 





1 
2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
3i 
35 
36 
37 
38 
39 



Lg. Vers. 



9-91716 
91731 
91746 
91761 
91776 



40 

41 
42 
43 
44 

45 

46 

47 

48 

49_ 

50 

51 

52 

53 

54 

55 
56 
57 
58 
59 
60 



91791 
91807 
91822 
91837 
91852 



91867 
91882 
91897 
91912 
91927 

91942 
91957 
91972 
91987 
92002 



92016 
92031 
92046 
02061 
92076 



92091 
92106 
92121 
92136 
92151 



92166 
92181 
92196 
92211 
92226 

92240 
92255 
92270 
92285 
92300 



92315 
92330 
92345 
92360 
92374 

92389 
92404 
92419 
92434 
9 2449 

92463 
92478 
9.2493 
92508 
92523 



92538 
92552 
92567 
92582 
92597 
9-92612 



Lg. Vers. 



1> 



15 
15 
15 
15 
5 
lo 
15 
15 
15 
15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
14 
15 
15 
15 
15 

15 
15 
15 
15 
14 

15 
15 
15 
15 
15 

14 
15 
15 
15 
15 

li 
15 
15 
15 
14 

15 

15 

14 
15 
15 

14 
15 
15 
14 
15 

15 
14 
15 
15 
14 

15 



J> 



Log.Exs. 



10-67749 
.67836 
•67923 
.68010 
•68097 



10-68184 
•68272 
-68353 
•68447 

_^8534 

10-68622 
.6871Q 
.68798 
.68886 
.68975 



10.69063 
•69152 
.69240 
•69329 
.69418 



10-69507 
.69596 
.69686 
•69775 
.69865 



10-69955 
. 70044 
.70134 
.70224 
.70315 



10.70405 
.70495 
.70586 
.70677 
.70768 



10-70859 
.70950 
.71041 
.71133 
-71224 



10^71316 
.71408 
.71500 
.71592 
.71684 



10.71776 
.71869 
.71961 
.72054 
•72147 



10. 



72240 
72333 
72427 
72520 
72614 



10.72707 
.72801 
. 72895 
.72990 
.73084 



10-73178 



2> 

86 
87 
87 
87 

87 
87 
87 
87 
87 

08 
88 
88 
88 
88 

88 
88 
88 
89 
89 

89 q 
89 ^ 
89 
89 
89 

90 
89 
90 
90 
90 

90 
90 

91 
90 

91 

91 

91 
91 
91 
91 

91 
92 
92 
92 
92 

92 
92 
92 
93 
92 

93 
93 
93 
93 
93 

93 

94 
94 
94 
94 

94 



Lg. Vers. 



92612 
92626 
92641 
92656 
92671 



92686 
9270g 
S2715 
92730 
92745 

92759 
92774 
92788 
92804 
92818 

92833 
92848 
92862 
92877 
92892 



92907 
92921 
92936 
92951 
92935 



92980 
92995 
93009 
93024 
93039 

93053 
93068 
93083 
93097 
93112 



93127 
93141 
93156 
93171 
93185 

93200 
93214 
93229 
93244 
93258 



93273 
93287 
93302 
93317 
93331 



93346 
93360 
93375 
93389 
93404 



93^19 
93433 
93448 
93462 
93477 

93491 



Log.Exs. ^ Lg. Vers. 



2> 



Log. Exs. 



I 10. 



14 
14 
15 
14 
14 

14 

14 
14 
14 
15 

14 
14 
14 
14 
14 

14 



14 
15 
14 
15 

15 
14 
15 
14 
15 

14 
15 
14 
15 
14 

15 
14 
14 
15 
14 
15 
14 
14 
15 
14 

15 
14 
14 
15 
14 

14 
15 
14 
14 
15 

14 
14 
14 
15 
14 

14 
14 

15 
-4 
14 



10 



73178 
73273 
73368 
73463 
73558 



n 



10 



73653 

73748 
73844 
73940 
74035 



10. 



74131 
74227 
74324 
74420 
74517 



10 



10 



10. 



74613 
74710 
74807 
74905 
75002 

75099 
75197 
75295 
75393 
7 5491 

75589 
75688 
75786 
75885 
75984 



10- 



76083 
76182 
76282 
76382 
76481 



10. 



10 



76581 
76681 
76782 
76882 
76983 

77083 
77184 
77286 
77387 
77488 

77590 
77692 
77794 
77896 
77998 



10. 



I 10 



78101 
78203 
78306 
78409 
78513 

78616 
78720 
78823 
78927 
79031 



10-79136 



^ Logo Exs. 
748 



95 
94 
95 
95 

95 
95 
95 
96 
95 

96 
96 
96 
96 
96 

86 
97 
97 
97 
97 

97 
98 
97 
98 
98 

98 
98 
98 
99 
99 

99 
99 
99 
100 
99 

100 
100 
100 
lOO 
100 

100 
101 
lOl 
101 
101 

101 

102 
102 
102 
102 

102 
102 
103 
103 
103 

103 
104 
103 
104 
104 
104 





1 
2 
3 
± 
5 
6 
7 
8 
9 

10 

11 
12 
13 
14 



15 
16 
17 
18 
19 

*^0 

21 

22 

23 

_24 

25 
26 
27 
28 
_29 

30 

31 
32 
33 
11 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



55 
56 
57 
58 
59 



60 



P. P. 





90 


80 


6 


9.0 


8.0 


7 


10-5 


9.3 


8 


12.0 


10.6 


9 


13-5 


12-0 


10 


15-0 


13-3 


20 


30-0 


26.6 


30 


45-0 


40.0 


40 


80-0 


53.3 


50 


75-0 


66.6 





9 


8 


6 


0.9 


0.8 


7 


1.0 


0.9 


8 


1-2 


1-0 


9 


1-3 


1.2 


10 


1.5 


1.3 


20 


3.0 


2.6 


30 


4-5 


4.0 


40 


6-0 


5.3 


50 


7.5 


6.^ 





K 


7 


6 


6 


0.7 


0.6 


7 





8 


0.7 


8 








0.8 


9 


1 





0.9 


10 


1 


1 


1.0 


20 


2 


3 


2.0, 


30 


3 


5 


3.Q 


40 


4 


6 


4.0 


50 


5 


8 


5.0 



7 
8 
9 
10 
20 
30 
40 
50 






5 





4 





6 





4 





6 





5 





7 





6 





8 





6 


1 


6 


1 


3 


2 


5 


2 





3 


3 


2 


6 


4 


1 


3 


3 





15 


6 


1.5 


7 


1.8 


8 


2.0 


9 


2.3 


10 


2.6 


20 


5.1 


30 


7.7 


40 


10-3 


50 


12^9 



6 

7 

8 

9 

10 

20 

80 

40 

50 



15 

1-5 
1.7 
2^0 
2^2 
2^5 
5^0 
7^5 
10^0 
12.9 

1-i 
1.7 



12.1 



P.P. 



f ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

83" 83° 



Lg.Vers. 



93191 
93506 
93520 
93535 
93549 



93564 
93578 
93593 
93607 
93622 



93636 
93651 
93665 
93680 
93894 



93709 
93723 
93738 
93752 
93767 
93781 
93796 
93810 
93824 
93839 



93853 
93868 
93882 
93897 
93911 



93925 
93940 
93954 
93969 
93983 



93997 
94012 
94026 
94041 
94055 

94089 
94084 
94098 
94112 
94127 



94141 
94155 
94170 
94184 
94198 



94213 
94227 
94241 
94256 
94270 



94284 
94299 
94313 
94327 
94341 



9-94356 



Lg. Vers. 



n 

14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 

14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
14 



Log. Exs. 



10.79136 
79240 
79345 
79450 
79555 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



1} 



79660 
79766 
79871 
79977 
80083 



80189 
80296 
80402 
80509 
80616 



80723 
80831 
80938 

81046 
81154 



81262 
81371 
81479 
81583 
81697 



81806 
8191g 
82025 
82135 
82245 



82356 
82466 
82577 
82688 
82799 



82910 
83022 
83133 
83245 
83358 

83470 
83583 
83695 
83809 
83922 



84035 
84149 
84283 
84377 
84492 



84607 
84721 
84837 
84952 
85068 

85183 
85299 
85416 
85532 
85649 



10-85766 
Log. Exs. 



D 

104 
105 
1.04 
lOf? 

105 
105 
105 
106 
106 

106 
106 
106 
107 
107 

107 
107 
107 
108 
108 

108 
108 
108 
109 
109 
109 
109 
109 
110 
110 

110 
110 
110 
111 
111 

111 
111 
111 
112 
112 

112 
112 
112 
113 
113 

113 
114 
114 
114 
114 

115 

114 
115 
115 
116 

115 
116 
116 
116 
117 

117 



Lg. Vers. 



94356 
94370 
94384 
94398 
94413 

94427 
94441 
94456 
94470 
94484 



94498 
94512 
94527 
94541 
94555 



94569 
94584 
94598 
94612 
94626 



94640 
94655 
94669 
94683 
94697 



94711 
9.4726 
94740 
94754 
94768 



94782 
94796 
94810 
94825 
94839 



94853 
94867 
94881 
94895 
94909 



94923 
94938 
94952 
94966 
94980 



94994 
95008 
95022 
95036 
95050 



95064 
95078 
95093 
95107 
95121 



95135 
95149 
95163 
95177 
95191 



95205 



I> Lg.Vers. 



n 

14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 

14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 



Log, Exs. 



10. 



85766 
85884 
86001 
86119 
86237 



10. 



86355 
86474 
86592 
86711 
86831 



10. 



86950 
87070 
87190 
87310 
87431 



10. 



87552 
87673 
87794 
87916 
88038 



10. 



88160 
88282 
88405 
88528 
88651 



10 



88775 
88898 
89022 
89147 
89271 



10 



.89896 
.89521 
.89647 
.89773 
•89899 



10 



90025 
90152 
90279 
90406 
90533 



10 



90661 
90789 
90917 
91046 
91175 



10. 



91304 
91434 
91564 
91694 
91825 



10 



10 



91956 
92087 
92218 
92350 
92482 

92614 
92747 
92880 
93014 
93147 



93^81 



:• Exs. 



2> 

117 
117 
117 
118 

118 

lis 

118 
119 
119 

119 
120 
120 
120 
120 

121 
121 
121 
121 
122 

122 
122 
122 
123 
123 

124 
123 
124 
124 
124 

125 
125 
125 
126 
126 

126 
126 
127 
127 
127 

128 
127 
128 
129 
129 

129 
130 
129 
130 
130 

131 
131 
131 
131 
132 

132 
133 
133 
133 
133 

134 





1 

2 
3 

4 

T 
6 
7 
8 
9 



10 

11 
12 
13 
14 



15 
16 
17 
18 
19 

30 

21 
22 
23 
24 



30 

31 
32 
33 
34 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



60 



P.P. 





130 


6 


13-0 


7 


15.1 


8 


17.3 


9 


19.5 


10 


21.6 


20 


43-3 


30 


65-0 


40 


86.6 


50 


108.3 



110 



6jll 





7,1 12 


8 


8 14 


6 


9 


16 


5 


10 


18 


3 


20 


36 


6 


30 


55 





40 


73 


3 


50 


91 


6 



130 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



lOQ 

10.0 
11-6 
13.3 
15.0 
16.6 
33-3 
50.0 
66-6 
83-3 





3 


3 


6U.3 


0-2 


7 


0.3 


0.2 


8 


04 


0.2 


9 


".4 


0-3 


10 


0.5 


0.3 


20 


1.0 


0.6 


30 


1.5 


1.0 


40 


2.0 


1-3 


50 


2-5 


1.6 



6 

7 
8 
9 
10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



15 

1.4 



1.7 
1-9 
2.2 
2.4 
4.8 
7.2 
9.6 
12.1 



0„ 

O.Q 
O.Q 
0.0 
0.1 
0.1 
0.1 
0.2 
0.3 
0.^ 



14 

1.4 
1.6 
1-8 
2.1 
2-3 
4.6 
7.0 
9.3 
11.6 



P.P. 



749 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT^ 



sr 



85° 



o 

1 

2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
33 
39_ 

40 

41 
42 
43 
44 

45 

46 

47 

48 

M. 

50 

51 

52 

53 

54 

55 9 

56 

57 

58 

59 

60 



Lg. Vers, 



95205 
95219 
95233 
95247 
95261 



95275 
95289 
95303 
95317 
95331 



95345 
95359 
95373 
95387 
95401 



95415 
95429 
95443 
95457 
95471 



95485 
95499 
95513 
95527 
95540 



95554 
95568 
95582 
95596 
95610 



95624 
95638 
95652 
95666 
95680 



95693 
95707 
95721 
95730 
95749 



95763 
95777 
95791 
95804 
95818 



95832 
95846 
95860 
95874 
95888 



95901 
95915 
95929 
95943 
95957 



95970 
95984 
95998 
96012 
96026 



9-96039 



Lg. Vers. 



2> 

14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
13 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
13 

14 
14 
14 
14 
14 

14 
13 
14 
14 
14 

13 
14 
14 
14 
14 

13 
14 
14 
13 
14 

14 
14 
13 
14 
14 

13 
14 
13 
14 
14 

13 
14 
14 
13 
14 

13 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



11 



11 



11 



93281 
93416 
93551 
93686 
93821 



93957 
94093 
94229 
94366 
94503 



94641 
94778 
94917 
95055 
95194 



95333 
95473 
95613 
95753 
95894 



96035 
96176 
96318 
96461 
96603 



96746 
96889 
97033 
97177 
97322 



97467 
97612 
97758 
97904 
98050 



98197 
98345 
98492 
9B640 
98789 



98938 
99087 
99237 
99387 
99538 



99689 
99841 
99993 
00145 
00298 



J> 



Lg. Vers. 



00451 
00605 
00759 
00914 
01069 



01225 
01381 
01537 
01694 
01852 



02010 



134 
135 
135 
135 

135 
136 
136 
137 
137 

137 
137 
138 
138 
139 

139 
139 
140 
140 
140 

14l 
141 
142 
142 
142 

143 
143 
144 
144 
144 

145 
145 
145 
146 
146 

147 
147 
147 
148 
149 

149 
149 
150 
150 
151 

151 
15l 
152 
152 
153 

153 
154 
154 
155 
155 

155 
156 
156 
157 
157 
158 



i> Log.Exs. D Lg. Vers. 



96039 
96053 
96067 
96081 
96095 



96108 
96122 
96136 
96150 
96163 



96177 
96191 
96205 
96218 
96232 



96246 
96259 
96273 
96287 
96301 



96314 
96328 
96342 
96355 
96369 



96383 
96397 
96410 
96424 
96438 



96451 
96465 
96479 
96492 
96506 



96519 
96533 
96547 
96560 
96574 



96588 
96601 
96615 
96629 
96642 



96656 
96669 
96683 
96697 
96710 



96724 
96737 
96751 
96764 
96778 



96792 
96805 
96819 
96832 
96846 



96859 



14 
13 
14 
14 

13 
14 
13 
14 
13 

14 
13 
14 
13 
14 

13 
13 
14 
13 
14 

13 
14 
13 
13 
14 

13 
14 
13 
13 
14 

13 
13 
14 
13 
13 

13 
14 
13 
13 
14 

1§ 
13 
13 
14 
13 

13 
13 
13 
14 
13 

15 
13 
13 
13 
14 

13 
13 
13 
13 
13 

13 



Log.Exs. -Z> 



11 



11 



11 



11 



11 



11 



11 



D 



11 



11 



11 



11 



11 



11 



02010 
02168 
02327 
02487 
02646 



02807 
02968 
03129 
03291 
03453 



03616 
03780 
03944 
04108 
04273 



04438 
04604 
04771 
04938 
05106 



05274 
05443 
05612 
05782 
05952 



06123 
0&295 
06467 
06640 
06813 



06987 
07161 
07336 
07512 
07688 



07865 
08043 
08221 
08400 
08579 



08759 
08940 
09121 
09303 
09486 



09669 
09853 
10038 
10223 
10409 



10595 
10783 
10971 
11160 
11349 



11538 
11730 
11922 
12114 
12307 



12501 



158 
159 
159 
159 

160 
161 
161 
161 
162 

163 
163 
164 
164 
165 

165 
166 
167 
167 
167 

168 
169 
169 
169 
170 

171 
171 
172 
173 
173 

174 
174 
175 
176 
176 

177 
177 
178 
179 
179 

180 

180 
181 
182 
182 

183 
184 
185 
185 
186 

186 
187 
188 
189 
189 

190 
191 
191 
192 
193 

193 



Log.Exs. X> 



10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_59 
60 



P. P. 





190 


6 


19.0 


7 


22.1 


8 


25-3 


9 


28-5 


10 


31-6 


20 


63.3 


SO 


95-0 


40 


126.6 


50 


158.3 



180 

18- 
21.0 
24-0 
27.0 
30.0 
60.0 
90.0 
120.0 
150.0 





170 


160 


6 


17.0 


16. i 


7 


19 


8 


18 


6 


8 


22 


6 


21 


S 


9 


25 


5 


24 





10 


28 


3 


26 


6 


20 


56 


6 


53 


3 


30 


85 





80 





40 


113 


3 


106 


6 


50 


141 


6 


133 


3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



150 

15.0 
17 



20 
22 
25 
50 
75 
100 
125 



140 

14.0 
16 



18 
21 
23 
46 
70 
93 
116 





130 


9 


8 


6 


13-01 


0-9 


0. 


7 


15 


1 


1 





0. 


8 


17 


3 


1 


2 


1. 


9 


19 


5 


1 


3 


1. 


10 


21 


6 


1 


5 


1. 


20 


43 


3 


3 





2. 


30 


65 





4 


5 


4- 


40 


86 


6 


6 





5. 


50 


108 


S 


7 


5 


6. 



8 
.9 
.S 
•2 
•3 

6 
-0 
-3 
.6 





7 


6 


5 


6 


0.7 


0-6 


0-5 


7 


0-8 





7 





6 


8 


0-9 





8 





6 


9 


1-0 





9 





7 


10 


1-1 


1 








8 


20 


2-3 


2 





1 


6 


30 


3.5 


3 





2 


5 


40 


4-6 


4 





3 


3 


50 


5.8 


5 





4 


1 





14 


14. 


6 


1-4 


1-41 


7 


1 


7 


1 


6 


8 


1 


9 


1 


8 


9 


2 


2 


2 


1 


10 


2 


4 


2 


3 


20 


4 


8 


4 


6 


30 


7 


2 


7 





40 


9 


6 


9 


3 


50 


12 


1 


11 


6 



13 

1.3 



4.5 

6.7 

9-0 

11.2 



P.P. 



750 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

86° 87° 



Lg. Vers. 



10 

11 

12 
13 
14 



ao 

21 
22 
23 
24 



25 

26 

27 

28 

29. 

30 

31 

32 

33 

34 



9. 



40 

41 
42 
43 
44 



50 

51 
52 
53 
54 



60 



9-96859 
-96837 
-96887 
-96900 
-96914 



D 



1-96927 
-96941 
-96954 
-96968 
-96981 



13 
14 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

-97197J }| 
• 97210! i4 



-96995 
-97008 
-97022 
-97035 
-97049 



-97062 
-97076 
-97089 
-97103 
■97116 



-97130 
-97143 
-97157 
-97170 
-97183 



-97224 
-97237 
.97251 



97264 
97277 
97291 
97304 
97318 



-97331 
-97345 
-97358 
-97371 

-973S5. 



-97398" 
-97412 
-97425 
-97438 
-9745? 



9- 



97532 
97545 
97559 
97572 
97585 



-97599 
-97612 
•97625 
-97639 
-97652 



9-97665 



Log. Exs. 



11-12501 
.12696 
.12891 
.13087 
.13284 



13 
13 
13 

13 
13 
13 
13 

13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 



11.17609 
.17824 
.18041 
.18259 
.1847 7 



11.18697 
.18917 
.19138 
.19361 
.19584 



-97465 

-97478 

-97492 

-975051 tl 

■ 97519! ^'^ 
13 
13 
13 
-13 
13 

13 
13 
13 
13 
13 

13 



Lg. Vers, D 



11-13482 
.13680 
.13879 
.14079 
■14280 

11.14482 
.14684 
.14887 
.15092 
.15297 



11.15502 
.15709 
.15917 
.16125 
■16^^4 



11.16544 
.16755 
.16967 
.17180 
.17394 



11-19809 
.20034 
.20261 
.20489 
-207T7 



11.20947 
.21178 
.21410 
.21643 
.21877 



11.22112 
.22349 
.22583 
.22825 
-23065 



11-23303 
.23548 
.23792 
.24037 
-24283 



11-24530 
-24778 
-25028 
.25279 
-25531 



11-25785 



Log. Exs. 



195 
195 
196 
196 

198 
198 
199 
200 
201 

201 
202 
203 
204 
205 

205 
206 
208 
208 
209 

210 
211 
212 
213 
214 
214 
215 
216 
218 
218 

219 
220 
221 
222 
223 

224 
225 
227 
227 
228 

230 
230 
232 
233 
234 

235 
236 
237 
239 
239 

241 
242 
243 
24! 
246 

247 
248 
250 
251 
252 

254 



Lg. Vers, 



9-97665 
97679 
97692 
97705 
97718 



97732 
97745 
97758 
97772 
97785 



97798 
97811 
97825 
97838 
97851 



97864 
97978 
97891 
97904 
97917 



97931 
97944 
97957 
97970 
97984 



97997 
98010 
98023 
98036 
98050 



98063 
98076 
98089 
98102 
98116 



98129 
98142 
98155 
98168 
98181 

98195 
98208 
98221 
98234 
98247 



98260 
98273 
98287 
98300 
98313 



98326 
98339 
98352 
98365 
98378 



98392 
98405 
98418 
98431 
98444 



D\.g 



98457 
, Vers. 



D 

13 
13 
13 
13 

13 
13 
13 
13 

13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
15 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 



Log. Exs. 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



D 



11 



11 



25785 
26040 
26297 
26554 
26814 



27074 
27336 
27599 
27864 
28131 



28398 
28668 
28938 
29211 
29485 



29760 
30037 
30316 
30596 
30878 

31162 
31447 
31734 
32023 
32313 



32606 
32900 
33196 
33494 
33793 



34095 
34398 
34704 
35011 
35321 



35632 
35946 
36261 
36579 
36899 

3722T 
37546 
37872 
38201 
38532 



38866 
39201 
39540 
39880 
40224 



40569 
40918 
41269 
41622 
41979 



42338 
42699 
43064 
43431 1 
43802 



44T75 



Log. Exs. 



255 
25? 
257 
259 

260 
262 
263 
265 
266 

267 
269 
270 
272 
274 

275 
277 
278 
28G 
282 

283 

28 

287 

288 

290 

292 
294 
296 
298 
299 

301 
303 
305 
30_ 
309 

311 

313 
315 
318 
320 

322 
324 
326 
328 
331 

333 
335 
338 
340 
343 

341 
348 
351 
353 
356 

359 
361 
364 
367 
370 

373 



10 

11 
12 
13 
14 
15 
16 
17 
18 
]9 



20 

21 
22 
23 
24 



30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 



45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 



D 



55 
56 
57 
58 
59 
60 



P. P. 





•^50 


6 


25-0 


7 


29-1 


8 


33-3 


9 


37-5 


10 


41-6 


20 


83-3 


30 


125.0 


40 


166-6 


50 


208.3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



330 

23.0 

26.8 

30-6 

34-5 

38-3 

76-6 

115-0 

153-3 

191.6 

310 

21-0 

24-5 

28-0 

51.5 

35.0 

70-0 

105-0 

140-0 

1750 





19 





4 


6 


19-0 


0.4 


7 


22 


1 


0.4 


8 


25 


3 


0-j 


9 


28 


5 


0-6 


10 


31 


6 


0.6 


20 


63 


3 


1.3 


30 


95 





2.0 


40 


126 


6 


2.6 


50 


158 


3 


3.3 





3 


1 


6 


0-2 


0.1 


7 





2 


0.1 


8 





2 


0-1 


9 





3 


0-1 


10 





3 


0-1 


20 





6 


0-3 


30 


1 





0-5 


40 


1 


3 


0.6 


50 


1 


6 


0-8 



14 


13 


1.4 


1.3| 


1 


6 


1 


6 


1 


8 


1 


8 


2 


1 


2 





2 


3 


2 


2 


4 


6 


4 


5 


7 





6 


7 


9 


3 


9 





11 


6 


11 


2 



340 

24.0 

28.0 

32.0 

36.0 

40.0 

80-0 

120.0 

160.0. 

200.0 

330 

22.0 

25-6 

29-3 

33.0 

36.6 

73.3 

110:0 

146.6 

183.3 

300 

20.0 

23.3 

26.6 

30.0 

33.3 

66.6 

100.0 

133.3 

166.6 



3 

0.3 
0.3 
0.4 
0.4 
0-5 
1.0 
1-5 
2-0 
2-5 

O 

0.0 

0.0 . 

0.0 

0-1 

0.1 

0.1 

0-2 

0.3 

0.4 

13 

1.3 
1.5 
1-7 
1-9 
2.1 
4.3 
6.5 
8-6 
10-8 



P. P. 



751 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 



88° 



89° 



1 
2 
3 

5 
6 
7 
8 
^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
58 
57 
58 
59 

eo 



Lg. Vers. 



9.98457 



.98470 
.98481 
•98496 
•98509 



D I Log.Exs. 



9.98522 
•98535 
•98548 
•98562 

^98575 

9 • 98588 
98601 
98614 
98627 
98640 



9^98653 
• 98666 
•98679 
•98692 
.98705 



.98718 

•98731 

.98744 

98757 

98770 



98783 
98796 
98809 
98822 
98835 



■98848 
98861 
98874 
98887 

.98900 

9-98913 
•98925 
.98931 
•98951 
.98964 



9-98977 

.98990 

•99003 

•99016 

99C29 



9-99042 
•99055 
•99068 
•99081 
-99093 



9-99106 
•99119 
•99132 
•99145 
-99158 



9-39171 
-99184 
-99197 
-9920P 
•99222 



9. 9993 R 



Lg. Vers. 



13 
13 

13 
13 

13 

13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 

13 

13 
13 
13 
13 
13 

13 
12 
13 
13 
13 

13 
13 
13 
13 
12 

13 
13 
13 
13 
12 

13 
13 
13 
13 
12 

13 
13 
13 
12 
13 
13 



11 



•44175 
•44551 
•44931 
•45313 
•45699 



11 



•46088 
.46480 
•46876 
.47275 
•47677 



11 



•48083 
•48493 
•48906 
•49323 
•49743 



11 



11 



50168 
50597 
51029 
51466 
51906 

52351 
52801 
53255 
53713 
54176 



JD 



11 



54643 
55115 
55593 
56076 
5B563 



11 



57056 
57554 
58058 
58567 
59082 



11 



59602 
60129 
60662 
61202 
61747 



11. 



62300 
62859 
63425 
63998 
64579 



11. 



65167 
65762 
66366 
66978 
67598 



11. 



68227 
68865 
69511 
70168 
70834 



11^ 



71509 
72196 
72892 
73600 
74319 



n .75050 



376 
379 
382 
386 

389 

392 
395 
399 
402 

406 

409 
413 
417 
420 

425 
428 
432 
436 
440 

445 
449 
454 
458 
463 

467 
472 
477 
482 
487 

492 
498 
504 
509 
5l5 
520 
527 
533 
539 
545 

552 
559 
566 
573 
581 

588 
595 
604 
6ll 
620 

628 
638 
646 
656 
666 

675 
686 
696 
707 
719 

730 



Lg. Vers. 



9 



99235 
99248 
99261 
99274 
99287 



.99299 

.99312 

.93325 

99338 

99351 



•99363 
•99376 
•99389 
.99402 
99415 



99428 
99440 
99453 
99466 
99479 



n 



.99491 
.99504 
.99517 
.99530 
99543 



99555 

.99568 

.99581 

99594 

99606 



99619 
99632 
99645 
9965? 
99670 



99683 
99695 
.99708 
99721 
99734 

•99746 

.99759 

.99772 

99784 

99797 



9- 



99810 
99823 
99835 
99848 
99861 



.99873 
.99886 
.99899 
.99911 
.99924 



.99937 
.99949 
.99962 
.99974 
•99987 



lO-ooono 



12 
13 
13 
13 
12 
13 
13 
12 
13 
12 
13 
13 
12 
13 

13 
12 
13 
12 
13 
12 
13 
13 
12 
13 
12 
13 
12 
13 
12 

13 
12 
13 
12 
13 

12 
12 
13 
12 
13 

12 
13 
12 
12 
13 

12 
13 
12 
12 
13 

12 
12 
13 
12 
12 

13 

12 
12 
12 
13 
12 



Log.Exs. 



11-75050 
•75792 
•76547 
•77316 
•78097 

11.78892 
•79702 
•80527 
•81367 

J 2 223 

11 •83095 
•83986 
•84894 
•85821 
•86768 



11^87735 
•88724 
•89735 
•90769 
•91829 



11-92914 
•94026 
•95167 
•96338 
•97541 



11^98777 

12^0004§ 

•01358 

•02707 

•04098 



12-05535 
•07020 
•08557 
•10149 
•11801 



12-13517 
•15302 
•17163 
•19106 
-21139 



12-23271 
.25511 
•27872 
•30367 
-33013 



i> Log. Exs. J 1> Lg. Vers, j J> J Log. Exs. J 1> 



12-35828 
•38837 
•42068 
.45557 
•49349 



12 •53501 
•58089 
•63217 
•69029 
•75736 



12-83687 
•93371 

13-05877 
-23499 
-53615 



742 
755 
76§ 
781 

795 
809 
825 
840 
856 

872 
890 
908 
927 
947 

967 

989 

1009 

1034 

1059 

1085 
1112 
1140 
1171 
1203 

1236 
1271 
1309 
1349 
1391 

1436 
1485 
1537 
1592 
1652 

1716 
1785 
1861 
1943 
2033 

2131 
2240 
2361 
2495 
2645 

2815 
3009 
3231 
3489 
3791 

4152 
4588 
5127 
5812 



10 

11 
12 
13 
ii 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
34 



35 
36 
37 
38 
39. 

40 

41 
42 
43 
44 



6707 54 



50 

51 
52 
53 



Infinity 



7931 

9704 

12506 

17621 

30116 



55 
56 
57 
58 
59 



eo 



p. p. 





13 


6 


1^3| 


7 


1 


6 


8 


1 


8 


9 


2 





10 


2 


2 


20 


4 


5 


30 


6 


7 


40 


9 





50 


11 


2 



13 

1^3 
1.5 
1.7 

1-9 
2-1 
4.3 
6^5 
8^6 
10.8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



13 

1-2 



P.P. 



752 



i'ABLE IX.~ NATURAL SINES. COSINES, TANGENTS. AND COTANGENm 
0° l* 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos- 


Tan. 


Cot. 


r 




1 

2 
3 

4 


•00000 
•00029 
•00058 
.00087 
.00116 


One 
One 
One 
One 
One 


.00000 
.00029 
.00058 
.00087 
.00116 


Infinite 
3437.75 
1718.87 
1145.92 
859 • 436 

687.549 
572.957 
491.106 
429.718 
381.971 


.01745 
.01774 
.01803 
.01832 
.01862 

.01891 
.01920 
.01949 
.01978 
.02007 


-99985 
•99984 
-99984 
•99983 
•99983 

•99982 
.99982 
.99981 
.99980 
.99980 

.99979 
.99979 
.99978 
.99977 
.99977 

=99976 
.99976 
.99975 
.99974 
.99974 


.01746 
.01775 
.01804 
.01833 
.01862 

-01891 
.01920 
.01949 
.01978 
.02007 


57.2900 
56.3506 
55.4415 
54.5613 
53.7086 

52.8821 
52-0807 
51.3032 
50-5485 
49.8157 


60 

59 
58 
57 
56 


5 
6 
7 
8 

9 


.00145 
.00175 
.00204 
.00233 
.00282 


One 
One 
One 
One 
One 


.00145 
.00175 
.00204 
.00233 
.00262 


55 
54 
53 
52 
51 


10 

11 

12 
13 
14 


.00291 
.00320 
.00349 
.00378 
.00407 


One 
.99999 
.99999 
.99999 
.99999 

.99999 
.99999 
.99999 
.99999 
.99998 


.00291 
.00320 
.00349 
.00378 
.00407 


343.774 
312.521 
286.478 
264.441 
245.552 


.02036 
•02065 
.02094 
-02123 
.02152 


.02036 
.02066 
.02095 
•02124 
.02153 


49-1039 
48.4121 
47.7395 
47.0853 
46.4489 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.00436 
.00465 
.00495 
.00524 
.00553 


.00436 
.00465 
.00495 
.00524 
.00553 


229.182 
214.858 
202.219 
190.984 
180.932 


.02181 
-02211 
-02240 
.02269 
•02298 


.02182 
.02211 
.02240 
.02269 
.02298 


45.8294 
45.2261 
44.6386 
44-0661 
43-5081 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.00582 
.00611 
.00640 
.00669 
•00698 


.99998 
.99998 
.99998 
.99998 
.99998 

.99997 
.99997 
.99997 
.99997 
.99998 

.99996 
•99993 
.99996 
.99995 
.99995 


•00582 
.00611 
.00640 
.00669 
.00698 


171-885 
163.700 
156.259 
149.465 
143.237 


•02327 
•02356 
•02385 
•02414 
.02443 


.99973 
.99972 
-99972 
-99971 
-99970 

-99969 
-99969 
-99968 
.99967 
.99966 

.99966 
.99965 
.99964 
.99963 
.99963 


.02328 
.02357 
.02386 
-02415 
.02444 


42-9641 
42.4335 
41.9158 
41.4106 
40.9174 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.00727 
.00756 
i)0785 
.00814 
.00844 


.00727 
.00756 
.00785 
.00815 
.0C844 


137.507 
132.219 
127.321 
122.774 
118.540 

114.589 
110.892 
107.426 
104.171 
101.107 


.02472 
.02501 
.02530 
.02560 
.02589 


-02473 
.02502 
.02531 
.02560 
.02589 


40.4358 
39.9655 
39-5059 
39^0568 
38^6177 


35 
34 
33 
32 
31 


30 

31 

32 

33 

34 


.00873 
.00902 
.00931 
.00960 
.00989 

.01018 
.01047 
•01076 
.01105 
•01134 

.01164 
•01193 

.01222. 

.01251 

.01280 

.01309 
.01338 
•01367 
•01396 
•01425 

•01454 
■01483 
• 01513^ 
.01542 
.01571 


.00873 
.00902 
.00931 
.00960 
.00989 


.02618 
.02647 
.02676 
.02705 
.02734 


.02619 
.02648 
.02677 
.02706 
•02735 


38.1885 
37.7686 
37.3579 
36.9560 
36-5627 


.30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.99995 
.99995 
.99994 
.99994 
.99994 


.01018 
.01047 
.01076 
.01105 
.01135 


98.2179 
95.4895 
92.9085 
90.4633 
88.1436 


.02763 
.02792 
.02821 
.02850 
•02879 


.99962 
.99961 
.99960 
.99959 
•99959 

•99958 
•99957 
•99956 
.99955 
.99954 

.99953 
.99952 
.99952 
.99951 
.99950 


•02764 
.02793 
.02822 
.02851 
•02881 

•02910 
.02939 
.02968 
.02997 
.03026 

•03055 
•03084 
.03114 
.03143 
.03172 


36-1776 
^5-8006 
35-4313 
35-0695 
34.7151 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.99993 
.99993 
.99993 
.99992 
.99992 


.01164 
.01193 
.01222 
.01251 
.01280 


85-9398 
83 • 8435 
81.8470 
79.9434 
78-1263 


•02908 
•02938 
•02967 
•02996 
•03025 


34-3678 
34.0273 
33-6935 
33-3652 
33-0452 


30 

19 
18 
17 
16 


45 
46 
47 
48 

49 


.99991 
.99991 
.99991 
.99990 
.99990 

.99989 
.99989 
.99989 
.98988 
.99988 


.01309 
.01338 
.01367 
.01396 
.01425 


76-3900 
74.7292 
73.1390 
71-6151 
70.1533 


•03054 
•03083 
•03112 
•03141 
•03170 


32-7303 
32-4213 
32-1181 
31-8205 
31-5234 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.01455 
.01484 
.01513 
.01542 
.01571 


38-7501 
67.4019 
66-1055 
84.8580 
63-6567 

62.4992 
61.3829 
60.3058 
59.2659 
58.2612 


-03199 
.03228 
•03257 
.03286 
.03316 

-03345 
.03374 
.03403 
.03432 
.03461 


.99949 
.99948 
.99947 
•99948 
.99945 


.03201 
.03230 
.03259 
•03288 
.08317 


31-2416 
30 -.9599 
30-6833 
30.4116 
30.1446 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.01600 

01629 

01658 

•01687 

•01716 


.99987 
.99987 
.99986 
.99986 
.99985 

.99985 
Sin. 


.01600 
.01629 
.01658 
-01687 
.01716 

.01746 


.99944 
.99943 
.99942 
.99941 
.99940 
.99939 


.03346 
.03376 
.03405 
.03434 
.03463 


29.8823 
29.6245 
,09.3711 
^(3.1220 
28-8771 


5 
4 
3 
2 
1 


60 


•01745 


57.2900 


.03490 


.03492 


28-6363 





1 


Cos. 


Cot. Tan. j 


Cos. 1 


Sin. 


Cot. 


Tan. 


/ 



89' 



753 



88*^ 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS.. AND COTANGENTSt 







2° 










3^ 








/ 


Sin.. 


Cos. 


Tan. 
•03492 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 





•03490 


•99939 


28.6363 


.05234 


.99863 


.05241 


19.0811 


60 


1 


.03519 


•99938 


•03521 


28 


3994 


.05263 


•99861 


.05270 


18 


9755 


59 


2 


.03548 


.99937 


.03550 


28 


1664 


•05292 


•99860 


.05299 


18 


8711 


58 


3 


.03577 


.99936 


.03579 


27 


9372 


•05321 


•99858 


.05328 


18 


7678 


57 


4 


.03606 
•03635 


.99935 


.03609 


27 


7117 


•05350 
•05379 


.99857 


.05357 


18 


6656 


56 


5 


•99934 


.03638 


27 


4899 


.99855 


.05387 


18 


5645 


55 


6 


.03664 


.99933 


.03667 


27 


2715 


•05408 


.99854 


.05416 


18 


4645 


54 


7 


.03693 


.99932 


.03696 


27 


0566 


.05437 


.99852 


.05445 


18 


.3655 


53 


8 


.03723 


.99931 


.03725 


26 


8450 


.05466 


.99851 


.05474 


18 


.2677 


52 


g 


•03752 
•03781 


.99930 
.99929 


.03754 


26 
26 


6367 
4316 


.05495 
.05524 


.99849 


.05503 


18 


1708 


51 


10 


.03783 


.99847 


.05533 


18 


0750 


50 


11 


•03810 


.99927 


.03812 


26 


.2296 


.05553 


.99846 


.05562 


17 


.9802 


49 


12 


•03839 


.99926 


.03842 


26 


0307 


.05582 


.99844 


.05591 


17 


.8863 


48 


3.3 


.03868 


.99925 


.03871 


25 


8348 


.05611 


.99842 


.05620 


17 


.7934 


47 


14 


.03897 


•99924 


.03900 


25 


6418 


.05640 
.05669 


.99841 


.05649 


17 


.7015 


46 


15 


•03926 


•99923 


.03929 


25 


4517 


.99839 


.05678 


17 


.6106 


45 


Ifi 


•03955 


•99922 


.03958 


25 


• 2644 


.05698 


.99838 


.05708 


17 


• 5205 


44 


17 


.03984 


•99921 


.03987 


25 


.0798 


.05727 


.99836 


.05737 


17 


.4314 


43 


18 


•04013 


•99919 


.04016 


24 


.8978 


.05756 


.99834 


.05766 


17 


.3432 


42 


19 


.04042 
.04071 


.99918 


.04046 


24 


.7185 


.05785 


.99833 


.05795 


17 


.2558 


41 


30 


•99917 


•04075 


24 


• 5418 


.05814 


.99831 


.05824 


17 


.1693 


40 


21 


.04100 


• 99916 


•04104 


24 


.3675 


.05844 


.99829 


.05854 


17 


0837 


39 


22 


.04129 


•99915 


•04133 


24 


.1957 


.05873 


.99827 


.05883 


16 


■ 9990 


38 


23 


.04159 


.99913 


•04162 


24 


.0263 


.05902 


.99826 


.05912 


16 


.9150 


37 


24 


.04188 


•99912 


•04191 


23 


.8593 


•05931 


.99824 


.05941 


16 


.8319 


36 


9.^ 


.04217 


•99911 


•04220 


23 


.6945 


•05960 


.99822 


.05970 


16 


.7496 


35 


26 


•04246 


•99910 


•04250 


23 


5321 


•05989 


.99821 


.05999 


16 


• 6681 


34 


P7 


•04275 


•99909 


•04279 


23 


3718 


•06018 


.99819 


.06029 


16 


.5874 


33 


28 


•04304 


•99907 


.04308 


23 


2137 


.06047 


.99817 


.06058 


16 


• 5075 


32 


29 


•04333 


.99906 
•99905 


.04337 


23 


0577 


.06076 


.99815 


•06087 
•06116 


16 


.4283 


31 


30 


•04362 


.04366 


22 


9038 


.06105 


^99813 


16 


• 3499 


30 


31 


•04391 


•99904 


.04395 


22 


7519 


.06134 


.99812 


.06145 


16 


• 2722 


29 


32 


•04420 


•99902 


.04424 


22 


6020 


.06163 


.99810 


•06175 


16 


.1952 


28 


33 


•04449 


.99901 


.04454 


22 


4541 


.06192 


.99808 


•06204 


16 


■ 1190 


27 


34 


.04478 
.04507 


.99900 


.04483 


22 


3081 


.06221 


.99806 


.06233 


16 


.0435 


26 


35 


.99898 


.04512 


22 


.1640 


•06250 


.99804 


.06262 


15 


.9687 


25 


36 


.04536 


•99897 


.04541 


22 


0217 


.06279 


•99803 


•06291 


15 


• 8945 


24 


37 


.04565 


•99896 


.04570 


21 


8813 


.06308 


•99801 


.06321 


15 


.8211 


23 


38 


-04594 


.99894 


.04599 


21 


7426 


.06337 


•99799 


.06350 


15 


• 7483 


22 


39 


,04623 


•99893 


.04628 
.04658 


21 


6056 


.06366 


.99797 
•99795 


•06379 


15 
15 


6762 


21 


40 


.04653 


•99892 


21 


4704 


.06395 


-06408 


6048 


30 


41 


.04682 


•99890 


.04687 


21 


3369 


.06424 


•99793 


•06437 


15 


5340 


19 


42 


.04711 


•99889 


.04716 


21 


2049 


.06453 


•99792 


•06467 


15 


4638 


18 


43 


.04740 


.99888 


.04745 


21 


0747 


.08482 


•99790 


.06496 


15 


3943 


17 


44 


.04789 


.99886 


.04774 
.04803 


20 


9460 


-06511 
.06540 


•99788 


.06525 


15 
i5 


3254 
2571 


16 


45 


.04798 


•99885 


20 


8180 


•99786 


•06554 


15 


46 


•04827 


.99883 


.04833 


20 


6932 


.06569 


•99784 


.06584 


15 


1893 


14 


47 


.04856 


.99882 


04862 


20 


5691 


.06598 


•99782 


.06613 


15 


1222 


13 


48 


•04885 


.99881 


.04891 


20 


4465 


.06627 


•99780 


•06642 


15 


0557 


12 


49 


.04914 


99879 
•99878 


.04920 
. 04949 


20 


3253 


.06656 
.06685 


.99778 


•06671 
•06 700 


14 
14 


9898 
9244 


11 


50 


.04943 


20 


2058 


•99776 


, 10 


51 


•04972 


•99876 


.04978 


20 


0872 


.06714 


•99774 


•06730 


14 


8596 


9 


32 


.05001 


.99875 


.05007 


19 


9702 


.06743 


•99772 


•06759 


14 


7954 


8 


53 


.05030 


•99873 


.05037 


19 


8546 


.06773 


.99770 


•06788 


14 


7317 


7 


54 


.05059 


•99872 


•05066 


19 


7403 


.06802 


•99768 


•06817 
•06847 


14 


6685 


6 


55 


•05088 


•99870 


•05095 


19 


6273 


.06831 


.99766 


14 


6059 


5 


56 


•05117 


•99869 


•05124 


19 


5156 


•06860 


.99764 


•06876 


14 


5438 


4 


57 


•05146 


•99867 


•05153 


19 


4051 


.06889 


.99762 


•06905 


14 


4823 


3 


i38 


•05175 


•99866 


•05182 


19 


2959 


.06918 


•99760 


•06934 


14 


4212 


2 


P9 


•05205 
.05234 


•99864 
•99863 


•05212 
.05241 


19 
19 


1879 
0811 


•06947 


.99758 
. 99756 


.06963 
.06993 


14. 

14. 


3607 
3007 


1 


60 


•06976 





/ 


Cos. 


Sin. 


Cot. 


Tan. [ 


Cos. 


Sin. 


Cot. 


T 


an. 


/ 



§7 



754 



86° 



lABLE IX.— NATURAL SINES, COSINES. TANGENTS. AND COTANGENTa 







4^ 








5° 






1 


Sin. 


Cos. 


Tan. 


Cot^ 


Sin. 


Cos. Tan. 1 


Cot. 


t 




1 

2 
S 

4 


.06976 
.07005 
.07034 
.07063 
.07092 


.99756 
.99754 
.99752 
.99750 
.99748 


.06993 
•07022 
.07051 
.07080 
.07110 


14.3007 
14.2411 
14.1821 
14.1235 
14. 0655 


.08716 
.08745 
.08774 
■08803 
.08831 


.99619 
.99617 
.99614 
•99612 
.99609 

•99607 
•99604 
.99602 
.99599 
.99596 


.08749 
.08778 
•08807 
.08837 
•08866 


11.4301 
11.3919 
11 •3540 
11.3163 
11.2789 


60 

59 
58 
57 
56 


5 
6 
V 
8 
9 


.07121 
.07150 
.07179 
.07208 
.07237 


.99746 
.99744 
.89742 
.99740 
.99738 


.07139 
.07168 
•07197 
.07227 
.07256 


14-0079 
13-9507 
13-8940 
13.8378 
13.7821 


•08860 
.08889 
.08918 
•08947 
.08976 


•08895 
•08925 
•08954 
•08983 
.09013 


11.2417 
11.2048 
11.1681 
11.1316 
11.0954 


55 
54 
53 
52 
51 


10 

11 

is 
IS 
14 


.07266 
.07295 
.07324 
.07353 
.07382 


.99736 
.89734 
.99731 
.99729 
.99727 


.07285 
.07314 
.07344 
.07373 
.07402 


13 •7267 
13.6719 
13.6174 
13.5634 
13.5098 


.08005 
.09034 
.09063 
.09092 
.08121 


.99594 
.99591 
.99588 
.99586 
.99583 

•99580 
•99578 
•98575 
.9S572 
• JJ9570 


.09042 
.09071 
.09101 
.09130 
.09159 


11^0594 
11^0237 
10^9882 
10^9528 
10.9178 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.07411 
.07440 
.0V469 
.07498 
.07527 

.07556 
.07585 
.07614 

.07R43 
.07672 

.07701 
.07730 
.07759 
.07788 
.07817 


.99725 
.99723 
.89721 
.99719 
.99716 


.07431 
.07461 
.07490 
.07519 
.07548 


13.4566 
13.4039 
13.3515 
13.2996 
13.2480 


.08150 
.09178 
.08208 
-08237 
.08266 


.09189 
.09218 
.09247 
.08277 
•08306 


10.8829 
10.8483 
10.8139 
10^7797 
10.7457 


4S 
44 
43 
42 
41 


20 

21 

22 

23 

24 


.99714 
.99712 
.99710 
.99708 
.99705 

.99703 
.99701 
.99699 
.99696 
.99694 

.99692 
.99689 
.89687 
.99685 
.99683 


.07578 
.07607 
.07636 
.07665 
.07695 


13.1969 
13.1461 
13.0958 
13.0458 
12.9962 


.09295 
.09324 
.09353 
.09382 
.09411 


.99567 
.99564 
.99562 
• 88559 
.88556 


.08335 
.09365 
•08394 
.08423 
.08453 


10.7119 
10.6783 
10.6450 
10.6118 
10.5789 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.07724 
.07753 
.07782 
.07812 
.07841 


12.9469 
12^8981 
12^8496 
12^8014 
12^7536 


.09440 
.09469 
.09498 
.09527 
.09556 


•88553 
•98551 
.99548 
.99545 
•99542 


.09482 
.08511 
.08541 
•09570 
•08600 

•09628 
•08658 
•09688 
•09717 
•09746 


10-5462 
10.5136 
10-4813 
10.4491 
10.4172 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.07846 
.07875 
.07904 
.07933 
.07962 


.07870 
.07899 
.07929 
.07858 
.07887 


'12-7062 
12.6591 
12.6124 
12.5660 
12.5188 


•09585 

•09614 
.09642 
.09671 
•09700 


.99540 
•99537 
•99534 
.99531 
.99528 

.99526 
.98523 
.99520 
.99517 

.99514 

.99511 
.99508 
•99506 
•99503 
.98500 


10.3854 
10.3538 
10^3224 
10.2813 
10.2602 


30 

29 
28 
27 
28 


35 
86 
37 
38 

39 


.07991 
.08020 
.08049 
.08078 
.08107 

.08136 
.08165 
.08194 
.08223 
.08252 

•08281 
.08310 
.08339 
.08368 
.08397 


.99680 
.99678 
.99676 
.89673 
.99671 

.99668 
.99666 
.99664 
.99661 
.99659 


.08017 
.08046 
.08075 
.08104 

.08134 

.08163 
•08192 
•08221 
.08251 
.08280 


12.4742 
12 •4288 
12^3838 
12^33S0 
12.2946 

12.2505 
12.2067 
12-1632 
12-1201 
12.0772 


•09728 
•08758 
•08787 
•08816 
.09845 

.08874 
•08803 
•08932 
•09961 
.09990 

•10019 
•10048 
•10077 
.10106 
.10135 


-09776 
-09805 
•08834 
.09864 
.09893 


10.2284 
10.1988 
10 •1683 
^0^1381 
10.1080 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.09823 
•09952 
.09981 
•10011 
. 10040 

.10069 
.10099 
.10128 
.10158 
•10187 


10.0780 
10 •0483 
10 0187 
9^88931 
9.96007 

9.93101 
9.90211 
9.87338 
9.84482 
9.81641 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.99657 
.99654 
.99652 
.99649 
.98647 


.08309 
•08339 
.08368 
•08397 
.08427 


12-0346 
11^9923 
11^8504 
11^8087 
11.8673 


•99497 
.99494 
.99491 
.89488 
.98485 

.99482 

.99479 

.99476' 

•98473 

-99470 


15 
14 
13 
12 
11 


60 

51 
52 
53 
54 


.08426 
.08455 
.08484 
.08513 
08542 


.99644 
.99642 
.99639 
.99637 
.99635 


.08456 
.08485 
.08514 
.08544 
•08573 


11 •8262 
11.7853 
11.7448 
11.7045 
11 • 6645 


.10164 
.10192 
.10221 
.10250 
.10279 


•10216 
.10246 
.10275 
.10305 
•10334 


9.78817 
9.76009 
9-73217 
9 - 70441 
9.67680 


10 

9 
8 
7 
6 


55 
56 
67 
58 
59 


.08571 
.08600 
.08629 
.08658 
.08687 


.99632 
.99630 
.99627 
.99625 
.99622 


.08602 
•08632 
•08661 
.08690 
.03720 


11.6248 
11.5853 
11.5461 
11.5072 

11.4685 


•10308 
•10S37 
•30S66 
.10395 
• 10424 


: 99467 
•99464 
•99461 
•99458 
•99455 


.10363 
.10393 
.10422 
.10452 
•10481 


8.64935 
9.62205 
9.59490 
9 56781 
8.54106 


5 
4 
3 
2 
1 


60 


.08716 


.99619 


•08748 


11.4301 


.10453 


•99452 


•10510 


8.51436 





/ 


Cos. 


Sin. 


Cot. 


Tau. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



85° 



755 



84'= 



TABLE rx. - 


XATURAL SINES, COSINES. TANGENTS. AND COTANGENTS. 
6° 7" 


/ 


Sin, 


(^.os. 1 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 
8.14435 


"60; 





. 10453 


•89452 


-10510 


9.51436 


•12187 


.99255 


.12278 


1 


•10482 1 


.99449 


. 10540 


9.48781 


-12216 


•99251 


.12308 


8.12481 


59 


2 


.10511 


•99448 


.10569 


9.46141 


.12245 


.99248 


.12338 


8.10536 


58 


3 


.10540 


•99443 


.10599 


9.43515 


.12274 


.99244 


.12367 


8.08600 


57 


4 


.10569 


- 99440 


•10628 


9.40904 


.12302 


.99240 


.12397 


3.06674 
8 04756 


56 
55J 


5 


.10597 


•99437 


.10657 


9.38307 


•12331 


.99237 


.12426 


6 


.10626 


.99434 


-10687 


9.35724 


.12360 


.99233 


.121:56 


8 02848 


54; 


7 


.10655 


.99431 


.10716 


9.33155 


.12389 


.99230 


.12485 


8.00948 


53 


8 


.10684 


.99428 


.10746 


9.30599 


.12418 


.99226 


.12515 


7.99058 


52 


9 


-10713 


.99424 
.99421 


.10775 


9.28058 


•12447 


.99222 


.12544 
.12574 


7.97176 


51/ 
50 


10 


. 10742 


.10805 


9.25530 


•12476 


•99219 


7-95302 


11 


.10771 


.99418 


.10834 


9 23016 


.12504 


.99215 


.12603 


7- 93438 


49 


12 


. 10800 


.99415 


.10863 


9.20516 


.12533 


.99211 


.12633 


7-91582 


48 


13 


.10829 


.99412 


.10893 


9.18028 


.12562 


.99208 


.12662 


7.89734 


47, 


14 


.10858 
.10887 


.99409 
.99406 


.10922 
•10952 


9.15554 


12591 


.99204 
.99200 


.12692 


7.87895 
7-86064 


46 1 
45 1 


15 


9.13093 


.12620 


.12722 


16 


.10916 


.99402 


.10981 


9.10646 


.12649 


.99197 


.12751 


7.84242 


44' 


17 


. 10945 


.99399 


.11011 


9.08211 


.12678 


.99193 


.12781 


7.82428 


43 


18 


.10973 


.91396 


.11040 


9.05789 


.12706 


.99189 


.12810 


7.80622 


42 


19 


.11002 
.11031 


.99393 
.99390 


.11070 


9.03379 


.12735 
•12764 


•99186 
•99182 


.12840 
.12869 


7.78825 


40 


30 


.11099 


9-00983 


7.77C35 


21 


.11060 


•99386 


.11128 


8-98598 


•12793 


•99178 


.12899 


7.75254 


39 


22 


.11089 


•99383 


.11158 


8.96227 


.12822 


99175 


.12929 


7-73480 


38 


23 


11118 


•99380 


.11187 


8-93867 


.12851 


•99171 


.12958 


7.71715 


37 


24 


. 11147 


.99377 


.11217 


8.91520 


.12880 


.99167 


.12988 


7.69957 


36 


25 


•11176 


.99374 


.11246 


8.89185 


.12908 


•99163 


.13017 


7.68208 


35 


26 


.11205 


•99370 


.11276 


8.86862 


.12937 


-99160 


.13047 


7.86466 


34 


27 


-11234 


-99367 


.11305 


8-84551 


.12966 


•99156 


.13076 


7.64732 


33 


28 


-11263 


•99364 


.11335 


8-82252 


.12995 


•99152 


.13106 


7.83005 


32 


29 


.11291 


.99360 
•99357 


.11364 


8.79964 


.13024 
.13053 


.99148 


.13136 


7.61287 
7.59575 


31 


30 


.11320 


.11394 


8.77689 


-99144 


.13165 


.30 


31 


.11349 


•99354 


.11423 


8.75425 


.13081 


-99141 


.13195 


7.57872 


29 


32 


.11378 


•99351 


.11452 


8.73172 


.13110 


-99137 


.13224 


7.56176 


28 


33 


•11407 


•99347 


.11482 


8.70931 


.13139 


.99133 


.13254 


7-54487 


27 


34 


•11436 


•99344 
.99341 


.11511 


8.68701 


.13168 


.99129 


.13284 


7-52806 
7-51132 


26 


35 


.11465 


.11541 


8.66482 


.13197 


.99125 


.13313 


25 


36 


.11494 


•99337 


.11570 


8.64275 


•13226 


.99122 


.13343 


7-49465 


24 


37 


.11523 


•99334 


.11600 


8.62078 


•13254 


.99118 


.13372 


7-47806 


23 


38 


.11552 


•99331 


•11629 


8.59893 


.13283 


.99114 


.13402 


7-46154 


22 


39 


.11580 


.99327 


.11659 
.11688 


8.57718 
8.55555 


.13312 


.99110 


.13432 


7-44509 
7-42871 


21 


40 


•11609 


•99324 


•13341 


.99106 


.13461 


20 


41 


11638 


•99320 


.11718 


8.53402 


•13370 


.90102 


.13491 


7-41240 


19 


42 


11667 


.99317 


.11747 


8.51259 


-13399 


.99098 


.13521 


7-39616 


18 


43 


.11696 


.99314 


.11777 


8.49128 


.13427 


.99094 


.13550 


7-37999 


17 


44 


■11725 
.11754 


.99310 


.11806 
.11836 


8.47007 


.13456 
.13485 


.99091 


.13580 


7-36389 


16 


45 


.99307 


8-44896 


.99087 


.13609 


7-34786 


15 


46 


.11783 


.99303 


.11865 


8-42795 


•13514 


-99083 


.13639 


7-33190 


14 


47 


.11812 


.*'9300 


.11895 


8-40705 


•13543 


.99079 


.13669 


7-31600 


13 


48 


-11840 


.99297 


.11924 


8.38625 


.13572 


.99075 


.13698 


7-30018 


12 


49 


•11869 


.99293 


•11954 


8.36555 
8-34496 


•13600 
.13629 


.99071 


.13728 


7-28442 


11 


50 


•11898 


.99290 


•11983 


.99067 


.13758 


7-26873 


10 


51 


•11927 


•99286 


.12013 


8.32446 


.13658 


.99063 


.13787 


7-25310 


9 


52 


•11956 


.99283 


.12042 


8.30406 


•13687 


.99059 


.13817 


7-23754 


8 


53 


•11985 


.99279 


.12072 


8.28376 


.13716 


.99055 


.13846 


7-22204 


7 


54 


•12014 


.99276 


.12101 


8.26355 


.13744 


.99051 


.13876 


7-20661 


6 


55 


•12043 


.99272 


.12131 


8.24345 


.13773 


-99047 


.13906 


7-19125 


5 


56 


.12071 


.99269 


.12160 


8.22344 


.13802 


•99043 


.13935 


7.17594 


4 


57 


.12100 


.99265 


.12190 


8.20352 


•13831 


99039 


.13965 


7.16071 


3 


58 


.12129 


.99262 


.12219 


8-18370 


•13860 


-99035 


.13995 


7.14553 


2 


59 


•12158 


.99258 


.12249 


8-16398 
8.14435 


•13889 
•13917 

Cos. 


•99031 
- 99027 


.14024 


7.13042 
7.11537 


1 


60 


•12187 


.99255 
Sin. 


.12278 


.14054 





/ 


Cos. 


Cot. 


Tan. 


Sin. 


Cot. 


Tan. 


/ 



SS"" 



-756 



88" 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
8° 9° 



t 


Sin. 


Cos. 


Tan. 


Cot. 
7.11537 


Sin, 


. Cos. I 


Tan. 
•15838 


Cot. 


t 





.13917 


•99027 


.14054 


•15643 


.98769 


6 •31375 


60 


1 


.13946 


•99023 


.14084 


7.10038 


•15672 


.98764 


•15868 


6-30189 


59 


2 


•13975 


.99019 


.14113 


7.08546 


•15701 


.98760 


•15893 


6^29007 


58 


3 


.14004 


.99015 


.14143 


7-07059 


.15730 


.98755 


•15928 


6-27829 


57 


4 


.14033 


•99011 


.14173 


7-05579 


.15758 


.98751 


•15958 
•15988 


6-26655 


56 


5 


•14061 


.99006 


.14202 


7^04105 


.15787 


•98746 


6-25488 


55 


6 


.14090 


.99002 


.14232 


7^02637 


.15816 


.98741 


•16017 


6.24321 


54 


7 


•14119 


.98998 


.14262 


7-01174 


.15845 


.98737 


•16047 


6-23160 


53 


8 


• 14148 


.98994 


.14291 


6^99718 


.15873 


.98732 


•16077 


6.22003 


52 


9 


.14177 


.98990 
.98986 


.14321 


6-98268 


15902 


.98728 


•16107 


6.20851 


51 


10 


•14205 


.14351 


6-96823 


.15931 


.98723 


•16137 


6.19703 


50 


n 


•14234 


•98982 


.14381 


6-95385 


.15959 


.98718 


•16167 


6.18559 


49 


12 


.14263 


.98978 


.14410 


6-93952 


.15988 


.98714 


•16198 


6.17419 


48 


13 


.14292 


.98973 


. 14440 


6-92525 


.16017 


.98709 


•16226 


6.16283 


47 


14 


.14320 

•14349 


.98969 


•14470 


6-91104 
6-89688 


•16046 


.98704 


•16256 


6.15151 


46 


15 


.98965 


•14499 


•16074 


•98700 


•16286 


6.14023 


45 


16 


.14378 


•98961 


•14529 


6-88278 


•16103 


•98695 


•16316 


6-12889 


44 


17 


.14407 


•98957 


•14559 


6-86874 


•16132 


•98690 


•16346 


6.11779 


43 


18 


.14438 


•98953 


•14588 


6^85475 


•16160 


•98686 


•16376 


6.10664 


42 


19 


. 14464 
• 14493 


.98948 
.98944 


.14618 
.14648 


6 •84082 


•16189 
•16218 


•98681 


•16405 


6.09552 


41 


SO 


6^82694 


•98676 


.16435 


6.08444 


40 


21 


.14522 


.98940 


.14678 


6-81312 


•16246 


•98671 


.16465 


6-07340 


39 


22 


.14551 


.98936 


.14707 


6.79936 


•16275 


.98667 


.16495 


6-06240 


38 


23 


.14580 


.98931 


•14737 


6.78564 


.16304 


•98662 


.16525 


6-05143 


37 


24 


,J.4608 
.14637 


.98927 


•14767 
.14798 


6-77199 


.16333 


•98657 


.16555 


6-04051 
6-02962 


38 


25 


•98923 


6.75838 


.16361 


.98652 


.16585 


35 


26 


.14666 


•88919 


•14826 


6.74483 


.16390 


.98648 


.16615 


6-0 J 878 


34 


87 


.14695 


•98914 


•14856 


6.73133 


.16419 


.98643 


.16645 


6-00797 


33 


28 


.14723 


•98910 


.14886 


6.71789 


•16447 


.98638 


.16674 


5-99720 


32 


29 


•14752 


•98906 


.14915 


6.70450 


•16476 


.98633 


•16704 


5-98646 


31 


80 


.14781 


•98902 


.14945 


6-69116 


•16505 


.98629 


•16734 


5-97576 


30 


31 


.14810 


•98897 


.14975 


6.67787 


•16533 


.98624 


•16764 


5-96510 


29 


32 


.14838 


•98893 


.15005 


6-66463 


•16562 


•98619 


.16794 


5-95448 


28 


33 


.14867 


•98889 


.15034 


6-65144 


•16591 


.98614 


.16824 


5-94390 


27 


34 


.14896 


•98884 
•98880 


.15064 


6-63831 
6.62523 


.16620 


.98609 


.16854 


5-93335 


28 


85 


•14925 


.15094 


•16648 


.98604 


.16884 


5.82283 


25 


36 


•14954 


•98876 


•15124 


6.61219 


•16677 


.98600 


.16914 


5-91236 


24 


37 


•14982 


.98871 


•15153 


6-59921 


•16706 


.98595 


.16944 


5.90191 


23 


38 


•15011 


.88867 


.15183 


6.58627 


•16734 


98590 


•16974 


5.89151 


22 


39 


•15040 
.15069 


.98863 
.98858 


.15213 
•15243 


6.57339 
6.66055 


•16763 
•16792 


.98585 


.17004 


5.88114 


21 


40 


.98580 


.17033 


5-87080 


^0 


41 


.15097 


.98854 


•15272 


6.54777 


•16820 


.98575 


.17063 


5-86051 


19 


42 


.15126 


.98849 


•15302 


6.53503 


•16849 


.98570 


•17093 


5-85024 


18 


43 


.15155 


.98845 


•15332 


6.52234 


•16878 


.98585 


•17123 


5.84001 


17 


44 


•15184 


•98841 


•15362 


6.50970 


.16906 


.98561 


•17153 


5-82982 
5^81066 


16 


45 


•15212 


.98836 


•15391 


6.49710 


.16935 


.98556 


•17183 


15 


46 


•15241 


•98832 


•15421 


6.48456 


.16964 


.98551 


•17213 


5^80953 


14 


47 


.15270 


•98827 


•15451 


6.47206 


•16992 


•98543 


.17243 


5.79944 


13 


48 


.15299 


•98823 


•15481 


6.45961 


•17021 


•98541 


.17273 


5.78938 


12 


49 


.15327 


•98818 


•15511 


6-44720 


.17050 


.98536 


.17303 


5.77936 


11 


60 


.15356 


•98814 


•15540 


6.43484 


•17078 


•88531 


.17333 


5 •76937 


10 


51 


•15385 


•98809 


•15570 


6.42253 


•17107 


•98526 


.17363 


5-75941 


9 


52 


• 15414. 


•98805 


.15600 


6.41026 


•17136 


•98521 


.17393 


5 . 74949 


8 


53 


•15442 


•98800 


.15630 


6.39804 


•17164 


•98516 


.17423 


5^7S960 


7 


54 


•15471 
.15500 


•98796 
•98791 


•15660 
•15689 


6.38587 
6. 37374 


.17183 
•17222 


.98511 
.98506 


.17453 


5 • 72974 


6 


55 


.17483 


5^71992 


5 


56 


15529 


•98787 


•15719 


6.36165 


•17250 


.98501 


.17513 


5-71013 


4 


57 


.15557 


.98782 


•15749 


6-34961 


•17279 


.98496 


.17543 


5 •70037 


3 


58 


.15586 


•98778 


•15779 


6 -33 '761 


•17308 


.98491 


.17573 


5 •69064 


2 


59 


■15615 


•98773 


•15809 


6-32566 


•17336 


•98486 


.17603 


5 •68094 


1 


60 


•15643 
Cos. 


.98769 

Sin. 


•15838 
Cot. 


6-31375 


.17365 
Cos. 


•98481 


•17633 


5.67128 





f 


Tan. 


Sin. 


Cot. 


Tan. 


r 



81*= 



757 



8(f 



l!A5Lfi IX.— NAttJRAL glNJES, COSINeS, TANGENTS, AND COTANGENTS; 
10° 11° 



f 


Sin. 

.17365 
.17393 
■17422 
.17451 
•17479 


Cos. 

.98481 
.98476 
.98471 
.98466 
.98461 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 


r 




X 

3 

4 


.17633 
.17663 
.17693 
•17723 
.17753 


5 
5 
5 
5 
5 


67128 
66165 
65205 
64248 
63295 


.19081 
.19109 
.19138 
.19167 
-19195 


.93163 
.98157 
-98152 
-98146 
.98140 


•19438 
•19468 
•19498 
.19529 
•19559 


5 
5 
5 
5 
5 


• 14455 
•13658 
•12862 
•12089 
•11279 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


•17508 
.17537 
.17565 
•17594 
•17623 


.98455 
.98450 
.98445 
.98440 
•98435 


.17783 
.17813 
.17843 
.17873 
.17903 


5 
5 
5 
5 
5 


62344 
61397 
60452 
59511 
58573 


-19224 
.19252 
a9281 
.19309 
.19338 


■98135 
■98129 
■98124 
■98118 
98112 


.19589 
•19619 
•19649 
•19680 
•19710 


5 
5 
5 
5 
5 


•10490 
.09704 
.08921 
•08139 
■07360 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


•17651 
•17680 
•17708 
•17737 
•17766 


•98430 
-98425 
•98420 
.98414 
•98409 


17933 
.17963 
.17993 
.18023 
.18053 


5 
5 
5 
5 
5 


57638 
56706 
55777 
54851 
53927 


•19366 
.19395 
•19423 
.19452 
•19481 


■98107 

■98101 

98096 

98090 

.98084 


.19740 
•19770 
•19801 
.19831 
.19861 


5 
5 
5 
5 
5 


•06584 
•05809 
•05037 
■04267 
•03499 


50 

49 
48 
47 
46 


15 
16 
17 
18 
;9 


•17794 
•17823 
.17852 
■17880 
■17909 


•98404 
•98399 
•98394 
.98389 
•98383 

•98378 
.98373 
.98368 
•98362 
•98357 


.18083 
.18113 
.18143 
.18173 
.18203 


5 
5 

5 
5 
5 


53007 
52090 
51176 
50264 
49356 


•19509 
.19538 
-19536 
-19595 

■19623 


■98079 
.98073 
.98067 
•98061 
•98056 

•98050 
.98044 
.98039 
.98033 
.98027 


.19891 
.19921 
.19952 
.19982 
•20012 


5 
5 
5 
5^ 

4 


■02734 
■01971 
■01210 
■00451 
■99695 


45 
44 
43 
42 
41 


30 

21 
22 
23 

24 


17937 

17966 

•17995 

.18023 

•18052 


.18233 
.18263 
.18293 
•18323 
•18353 


5 
5 
5 
5 
5 


48451 
.47548 

46648 
.45751 
.44857 


.19652 
•19680 
•19709 
.19737 
•19766 


• 20042 
•20073 
-20103 
•20133 
•20164 


4 
4 
4 
4 
a. 


■98940 
•98188 
•97438 
•96690 
•95945 


40 

39 
38 
37 
36 


25 
26 
27 
23 
29 


•18081 
•18109 
•18138 
.18166 
•18195 


•98352 
.98347 
•98341 
.98336 
.98331 


•18384 
•18414 
• 18444 
.18474 
.18504 


5 
5 
5 
5 
5 


.43966 
.43077 
•42192 
•41309 

.40429 


•19794 
•19823 
•19851 
■19880 
■19908 


.98021 
•98016 
•98010 
•98004 
■97998 


.20194 
.20224 
•20254 
•20285 
•20315 


4 
4 
4 

4 
4 


•95201 
•94460 
•93721 
•92984 
S2249 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•18224 
■18252 
■18281 
■18309 
•18338 


.98325 
•98320 
•98315 
•98310 
■98304 


•18534 
.18564 
•18594 
.18624 
■18654 


5 
5 
5 
5 

5 


.39552 
.38677 
.37805 
.36936 
.36070 


■19937 
■19965 
•19994 
■20022 
■20051 


•97992 
•97987 
97981 
■97975 
■97969 


•20345 
.20376 
.20406 
.20436 
. 20466 


4 
4 

4 
4 
4 


91518 
90785 
90056 
89330 
88605 


30 

29 
28 
27 
28 


35 
36 
37 
38 
39 


•18367 
18395 
•18424 
.18452 
•18481 


•98299 
•98294 
•98288 
.98283 
.98277 


.18634 
•18714 
•18745 
.18775 
.18805 

•18835 
•18865 
.18895 
•18925 
.18955 


5 
5 
5 
5 
_5. 

5 
5 
5 
5 
5 


.35206 

.34345 

•33487 

32631 

31778 

30928 
30080 
29235 
28393 
27553 


•20079 
•20108 
•20138 
•20165 
^20193_ 

■20222 
■20250 
■20279 
■20307 
■20336 


•97963 
•97958 
•97952 
•97945 
•97940 

•97934 
•97928 
•97922 
•97916 
■97910 


.20497 
•20527 
•20557 
•20588 
•20618 


4 
4 

4 
4 
4 


87882 

87162 
86444 
85727 
85013 


25 
24 
23 
22 
21 


40 

41 

42 
43 
44 


•18509 
•18538 
-18567 
•18595 
•18624 


.98272 
.98267 
.98261 
.98256 
.98250 


-20648 
-20679 
•20709 
•20739 
•20770 

•20800 
•20830 
•20861 
■20891 
20921 


4 
4 
4 
4 
4 


84300 
83590 
82882 

82175 
81471 


20 

19 
18 
17 
18 


45 
46 
47 
48 
49 


•18652 
■18681 
•18710 
■18738 
. 18767 


.98245 
.98240 
.98234 
.98229 
.98223 


•18986 
-19016 
•19046 
•19076 
•19106 


5 
5 
5 
5 
5 


26715 
25880 
25048 
24218 
23391 


•20364 
■20393 
.20421 
.20450 
. 20478 


■97905 
-97899 
-97893 
-97887 
-97881 


4 
4 
4 
A 

4 


80769 
80068 
79370 
78673 
77978 


15 
14 
IS 
12 
11 


50 

51 
52 
53 
54 


■18795 
■18824 
.18852 
•18881 
.18910 


•98218 
.98212 
•98207 
.98201 
.98196 


•19136 
•19166 
.19197 
•19227 
•19257 


5 

5 

5- 

5 

5. 


22566 
21744 
20925 
20107 
19293 


■20507 
■20535 
■20563 
■20592 
■20820 


•97875 
•97869 
■97863 
■97E57 
-97851 


■20952 
■20982 
■21013 
■21043 
■21073 


4 
4 
4 
4 
4 


77286 
76585 
759G8 
75219 
74534 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•18938 
•18967 
•18995 
•19024 
•19052 


.98180 
.98185 
•98179 
•98174 
•98168 


.19287 
•19317 
•19347 
•19378 
.19408 


5. 
5. 
5. 
5. 
5. 


18480 
17671 
16863 
16058 
15256 


■20649 
.20677 
.20708 
■20734 
■20763 


-97845 
-97839 
■97833 
■97827 
-97821 


•21104 
•21134 
■21164 
•21195 
•21225 


4 
4 
4 
4 
4^ 


73851 
73170 
72490 
71813 
71137 


5 

4 
3 
2 

1 


^? 


.19081 


•98163 

Sin. 


■19438 


5. 


14455 


■20791 


-97815 


■21256 


A. 


70463 





*~T~ 


Cos. 


Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 1 


''" 



^9*' 



758 



7S° 



TABLE IX.— NATURAL SINES. COSINES, TANGENTS, AND COTANGENTS. 
12° 13° 



/ 


Sin. 1 


Cos. 


Tan. 




Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
8 

4 


•20791 
.20820 
.20848 
.20877 
■20905 


.97815 
.97809 
.97803 
-97797 
.97791 


.21256 
.21286 
.21316 
.21347 
•21377 

.21408 
.21438 
.21469 
.21499 
.21529 


4 
4 
4 
4 
4 


70463 
69791 
69121 
68452 
67786 


•22495 
•22523 
•22552 
•22580 
•22608 


.97437 
.97430 
.97424 
.97417 
.97411 


•23087 
•23117 
•23148 
.23179 
.23209 

-23240 
.23271 
-23301 
.23332 
■23363 


4 
4 
4 
4 
4 


33148 
32573 
32001 
31430 
30860 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


•20933 
.20962 
•20990 
.21019 
.21047 


.97784 
.97778 
.97772 
.97766 
.97760 


4 
4 
4 
4 
4 


67121 
66458 
65797 
65138 
64480 


.22837 
.22665 
•22693 
.22722 
.22750 

•22778 
•22807 
.22835 
.22863 
.22892 


•97404 
.97398 
.97391 
.97384 
.97378 

.97371 
•97365 
•97358 
•97351 
•97345 


4 
4 
4 
4 
4 


30291 
29724 
29159 
28595 
28032 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.21076 
.21104 
.21132 
.21161 
.21189 


.97754 
.97748 
.97742 
.97735 
.97729 


•21560 
.21590 
•21621 
.21651 
.21682 

.21712 
.21743 
.21773 
.21804 
.21834 

•21864 
.21895 
.21925 
.21956 
.21986 


4 
4 

4 
4 
4 


63825 
63171 
62518 
61868 

61219 


.23393 
-23424 
-23455 
-23485 
.23516 


4 
4 
4 
4 
4 


27471 
26911 
26352 
25795 
25239 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.21218 
.21246 
.21275 
.21303 
•21331 

.21360 
.21388 

.21417 
.21445 
.21474 


.97723 
.97717 
.97711 
.97705 
.97698 

.97692 
.97686 
.97680 
.97673 
.97667 


4 
4 
4 
4 
4 


60572 
59927 
59283 
58641 
58001 


.22920 
.22948 
•22977 
•23005 
.23033 

•23062 
•23090 
.23118 
.23146 
•23175 


•97338 
.97331 
.97325 
.97318 
.97311 


-23547 
.23578 
.23608 
.23639 
.23670 


4 
4 
4 
4 
4 


24685 
24132 
23580 
23030 
22481 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


4 
4 
4 
4 
4 


57363 
56726 
56091 
55458 
54826 


.97304 
.97298 
•97291 
•97284 
■97278 


.23700 
•23731 
-23762 
-23793 
-23823 


4 
4 
4 
4 
4 


21933 
21387 
20842 
20298 
19756 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.21502 
•21530 
.21559 
.21587 
.21616 


.97661 
.97655 
.97648 
.97642 
•97636 


.22017 
.22047 
.22078 
.22108 
.22139 


4 
4 
4 
4 
4 


54196 
53568 
52941 
52316 
51693 


•23203 
■23231 
.23260 
•23288 
•23316 


•97271 
■97264 
.97257 
•97251 
.97244 


•23854 
•23885 
•23916 
•23946 
•23977 

. 24008 
••24039 
•24069 
.24100 
.24131 


4 
4 
4 
4 
4 


.19215 

18675 

18137 

.17600 

.17064 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•21644 
.21672 
•21701 
•21729 
•21758 


.97630 
.97623 
.97617 
.97611 
.97604 


.22169 
.22200 
.22231 
.22261 
•22292 

.22322 
.22353 
•22383 
•22414 
• 22444 


4 
4 
4 
4 
4 


51071 
50451 
49832 
49215 
48600 


.23345 
.23373 
■23401 
■23429 
■23458 


.97237 
.97230 
•97223 
.97217 
.97210 

.97203 
.97196 
.97189 
.97182 
.97176 


4 

4 
4 
4 
4 


16530 
15997 
15465 
14934 
14405 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•21786 
•21814 
•21843 
•21871 
•21899 


.97598 
.97592 
.97585 
.97579 
•97573 


4 
4 
4 
4 
4 


47986 
47374 
46764 
46155 
45548 


23486 
.23514 
•23542 
.23571 
•23599 

•23627 
•23656 
■23684 
■23712 
■23740 


-24162 
-24193 
-24223 
.24254 
.24285 


4 
4 
4 
4 
4 


13877 
13350 
12825 
12301 
11778 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•21928 
•21956 
•21985 
•22013 
.22041 

.22070 
.22098 
•22126 
•22155 
•22183 

.22212 
•22240 
.22288 
.22297 
.22325 


.97566 
.97560 
.97553 
.97547 
•97541 


.22475 
•22505 
.22536 
.22567 
•22597 


4 
4 
4 
4 
4 

4 
4 
4 
4 
4 


44942 
44338 
43735 
43134 
42534 

41936 
41340 
40745 
40152 
39560 


.97169 
.97162 
-97155 
.97148 
■97141 


.24316 
.24347 
.24377 
. 24408 
.24439 


4 
4 
4 
4 
4 


11256 
10736 
10216 
09699 
09182 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.97534 
.97528 
.97521 
.97515 
.97508 

.97502 
.97496 
.97489 
.97483 
.97476 


•22628 
•22658 
.22689 
.22719 
•22750 

.22781 
.22811 
. 22842 
.22872 
.22903 

.22934 
•22964 
•22995 
•23026 
•23056 


•23769 
•23797 
•23825 
.23853 
•23882 


■97134 
■97127 
.97120 
.97113 
.97106 


.24470 
.24501 
-24532 
-24562 
.24593 


4 
4 
4 
4 
4 


08666 
08152 
07639 
07127 
06616 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


4 
4 
4 
4 
4 


38969 
38381 
37793 
37207 
36623 


•23910 
•23938 
.23966 
•23995 
■24023 


.97100 
.97093 
.97086 
.97079 
.97072 


.24624 
.24655 
.24686 
.24717 
-24747 


4 
4 
4 
4 
4 


06107 
05599 
05092 
04586 
04081 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•22353 

•22382 

•22410 

-22438 

l22467_ 

•22495 


.97470 
.97463 
•97457 
•97450 
•97444 

•97437 1 


4 
4. 
4 
4 
4 


36040 
35459 
34879 
34300 
33723 


•24051 
•24079 
•24108 
•24136 
.24164 


.97065 
.97058 
■97051 
.97044 
.97037 


.24778 
.24809 
. 24840 
.24871 
.24902 


4 
4 
4 
4 
4 


03578 
03076 
02574 
02074 
01576 


5 
4 
3 
2 
1 


60 


.23087 


4 


33148 


■24192 


.97030 


.24933 


4 


01078 





/ 


Cos. 


Sin. 


Cot. 




Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



■jr 



759 



7a" 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
14° * 15° 



t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos- 


Tan. 


Cot. 


r 




1 

2 
3 
4 


.24192 
.24220 
.24249 
.24277 
.24305 


•97030 
-97023 
•97015 
•97008 
•97001 


•24933 
•24964 
•24995 
•25026 
•25056 


4.01078 
4.00582 
4. 00086 
3.99592 
3.99099 


-25882 
-25910 
.25938 
-25966 
-25994 


•96593 
.96585 
.96578 
-96570 
-96562 

-96555 
-96547 
-98540 
-96532 
-96524 

-96517 
-96509 
-96502 
-96494 
-96486 


-26795 
-26826 
-26857 
-26888 
•26920 


3 
3 
3 
3 
3 


■73205 
■72771 
■72338 
■71907 
■71476 


60 

59 
58 
57 
58 


5 
6 
7 
8 
9 


.24333 
.24362 
.24390 
.24418 
. 24446 


•96994 
•96987 
•96980 
•96973 
.96966 


.25087 
•25118 
•25149 
.25180 

.25211 


3.98607 
3.98117 
3.97627 
3^97139 
3.96651 


-26022 
-26050 
-26079 
-26107 
.26135 


•26951 
•26982 
•27013 
•27044 
•27076 


3 
3 
3 
3 
3 


■71046 
•70616 
•70188 
■69761 
■69335 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.24474 
.24503 
.24531 
.24559 
.24587 

.24615 
. 24644 
.24672 
.24700 
.24728 


.96959 
.96952 
.96945 
.96937 
.96930 


•25242 
25273 
•25304 
•25335 
•25366 


396185 
3-95680 
3-95196 
3.94713 
3.94232 


.26163 
■26191 
■26219 
■26247 
■26275 


-27107 
-27138 
-27169 
-27201 
-27232 


3 
3 
3 
3 
3 


■68909 
■ 88485 
■68061 
•67638 
•67217 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.96923 
.96916 
96909 
.98902 
.96894 


•25397 
•25428 
.25459 
•25490 
•25521 


3-93751 
3-93271 
3-92793 
392316 
3- 91839 


.26303 
.28331 
.26359 
.26387 

.26415 


-96479 
-96471 
-96463 
-96456 
-96448 


-27263 
-27294 
-27326 
-27357 
-27388 


3 
3 
3 
3 
3 


•66796 
• 66376 
•65957 
•65538 

•65121 


45 
44 
43 
42 
41 


30 

21 

22 

23 

24 


.24756 
.24784 
.24813 
. 24841 
.24869 


96887 
.98880 
.96873 
.96866 
.96358 


.25552 
•25583 
•25614 
•25645 
•25676 


3.91364 
3 90890 
3-90417 
3- 89945 
3. 89474 


.28443 
.26471 
.26500 
.26528 
.26556 


-96440 
-96433 
-96425 
-96417 
-96410 


-27419 
-27451 
-27482 
-27513 
-27545 


3 
3 
3 
3 
3 


•64705 
•64289 
•63874 
•63461 
■ 63048 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.24897 
.24925 
.24954 
.24982 
.25010 


.96851 
.96844 
.96837 
.96829 
•96822 


.25707 
•25738 
•25769 
•25800 
.25831 


3-89004 
3-88536 
3.88068 
3.87601 
3^87136 


.26584 
.26612 
26640 
.26668 
.26698 


-96402 
.96394 
.98386 
.98379 
-96371 


-27576 
-27607 
-27638 
.27670 
.27701 


3 
3 
3 
3 
3 


•62636 
• 62224 
•61814 
•61405 
•60996 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.25038 
•25066 
25094 
•25122 
.25151 


•96815 

•96807 

96800 

96793 

98786 

.96778 
.96771 
96764 
.96756 
.96749 

.96742 
.96734 
•96727 
96719 
.96712 


•25862 
•25893 
.25924 
.25955 
.2598?! 

•26017 
•26048 
•26079 
•26110 
■28141 


3.88671 
3 •86208 ■ 
3 •85745 
3-85284 
3. 84824 


•26724 
■26752 
-26780 
-26808 
-26836 


-96363 

-96355 

96347 

96340 

■96332 


.27732 
.27764 
.27795 
-27826 
-27858 

-27889 
■27921 
■27952 
■27983 
■28015 

-28046 
-28077 
-28109 
-28140 
-28172 


3 
3 
3 
3 
3 


•80588 

•60181 

59775 

59370 

58966 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•25179 
•25207 
•25235 
.25263 
.25291 

.25320 
.25348 
25376 
. 25404 
.25432 


3 •84364 
3-83906 
3- 83449 
3.82992 
3. 82537 


■26864 
■28892 
■28920 
■28948 
.28976 


-96324 
■96316 
■96308 
■96301 
■96293 

-96285 
■96277 
■96269 
■96261 
■96253 


3 
3 
3 
3 
3 


58562 
58160 
57758 
57357 
56957 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•26172 
•26203 
•26235 
•26286 
•28297 


3. 82083 
3. 81630 
3. 81177 
3-80726 
3. 80276 


27004 
■27032 
.27060 
•27088 
.27116 


3 
3 
3 
3 
3 

3 

3 

3 

3- 

3- 

3- 
3- 
3- 
3- 
3- 

3. 
3. 
3. 
3. 
3- 


56557 
56159 
55761 
55364 
549 68_ 

54573 
54179 
53785 
53393 
5300 1_ 

52609 
52219 
51829 
51441 
51053 

50666 
50279 
49894 
49509 
49125 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.25460 
25488 
.25516 
•25545 
•25573 


•96705 
•96697 
.96690 
96682 
•96675 


•26328 
•28359 
.26390 
•26421 
•26452 


3-79827 
3-79378 
3-78931 
3-78485 
3- 78040 


.27144 
.27172 
.27200 
.27228 
.27256 

-27284 
-27312 
•27340 
•27368 
•27396 

•27424 
•27452 
•27480 
•27508 
27536 
•27564 


-96246 
-96238 
-96230 
-96222 
-96214 


-28203 
- 28234 
.28266 
-28297 
-28329_ 

-28360 
■28391 
■28423 
■28454 
-28488 

-28517 
-28549 
-28580 
-28612 
-28643 


15 
14 
13 
12 
11. 


50 

51 
52 
53 
54 


.25601 
•25829 
•25657 
•25685 
.25713 


96667 
• tJ6660 
•96653 
•96645 
•96638 


.28483 
•28515 
.26546 
.28577 
.26608 

.26639 
.26670 
.26701 
.26733 
•26764 


3-77595 
3-77152 
3-76709 
3-76268 
3-75828 


-96206 
-96198 
•96190 
•96182 
•98174 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.25741 
.25769 
.25798 
•25826 
■25854 


•96830 
•96623 
.96615 
•96608 
•96800 


3-75388 
3.74950 
3.74512 
3.74075 
3-73840 


-96166 
-98158 
-96150 
.96142 
.96134 


5 
4 
3 
2 
1 


60 


•25882 


•96593 


•26795 


3-73205 


-96126 


■28675 


-1. 


48741 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



75' 



760 



74"= 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
16° 17° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


1 





.27564 


•96126 


.28675 


3.48741 


-29237 


-95630 


.30573 


3-27085 


no 


1 


.27592 


•96118 


.28706 


3^48359 


•29265 


-95622 


.30605 


3 •26745 


59 


2 


.27620 


•96110 


.28738 


3^47977 


•29293 


-95613 


.30637 


3 •26406 


58 


3 


.27648 


•96102 


.28769 


3.47596 


•29321 


-95605 


.30669 


3^26067 


57 


4 


.27676 


•96094 


.28800 


3^47216 


•29348 


-95596 
-95588 


.30700 


3^25729_ 
3 •25392 


56 


5 


.27704 


•96086 


•28832 


3-46837 


•29376 


.30732 


55 


6 


.27731 


•96078 


.28864 


3 • 46458 


•29404 


.95579 


.30764 


3 • 25055 


54 


7 


.27759 


.96070 


.28895 


3.46080 


•29432 


.95571 


.30796 


3^24719 


53 


8 


.27787 


•96062 


.28927 


3.45703 


•29460 


.95562 


-30828 


3 •24383 


52 


9 


.27815 


•96054 


.28958 


3.45327 


.29487 


^95554_ 
.95545 


-30860 


3 • 24049 
3 •23714 


51 


10 


.27843 


•96046 


.28990 


3.44951 


.29515 


-80891 


50 


11 


.27871 


•96037 


.29021 


3.44576 


-29543 


-95536 


-30923 


3-23381 


49 


12 


.27899 


•96029 


.29053 


3.44202 


-29571 


-95528 


-30955 


3-23048 


48 


13 


.27927 


.96021 


.29084 


3.43829 


-29599 


-95519 


-30987 


3-22715 


47 


14 


.27955 
-27983 


.96013 


.29116 


3.43456 
3.43084 


-29626 


-95511 


-31019 


3 - 22384 


46 


15 


.96005 


.29147 


.29654 


-95502 


-31051 


3-22053 


45 


16 


.28011 


.95997 


.29179 


3^42713 


.29682 


-95493 


-31083 


3 21722 


44 


17 


.28039 


•95989 


•29210 


3-42343 


.29710 


-95485 


-31115 


3-21392 


43 


18 


.28067 


•95981 


.29242 


3-41973 


•29737 


-95476 


-31147 


3-21063 


42 


19 


.28095 


.95972 


.29274 


3.41604 
3-41236 


•29765 


-95467 
-95459 


-31178 
-31210 


3. 20734 


41 


20 


.28123 


•95964 


.29305 


•29793 


3-20406 


40 


21 


.28150 


•95956 


.29337 


3.40869 


-29821 


-95450 


-31242 


3-20079 


3G 


22 


•28178 


•95948 


.29368 


3-40502 


.29849 


-95441 


-31274 


3-19752 


38 


23 


.28206 


.95940 


.29400 


3.40136 


-29878 


-95433 


-31306 


3-19426 


37 


24 


.28234 
.28262 


•95931 


.29432 


3.39771 


-29904 


•95424 


•31338 


3-19100 


36 


25 


•95923 


.29463 


3.39406 


-29932 


•95415 


-31370 


3-18775 


35 


26 


.28290 


•95915 


.29495 


3.39042 


-29960 


-95407 


•31402 


3-18451 


34 


27 


.28318 


.95907 


•29526 


3^38879 


-29987 


-95398 


•31434 


3-18127 


33 


28 


.28346 


.95898 


•29558 


3.38317 


-30015 


•95389 


•31466 


3-17804 


32 


29 


.28374 


.95890 


.29590 


3.37955 


-30043 


•95380 


-314G8 


3-1748] 


31 


30 


.28402 


•95882 


.29621 


3.37594 


-30071 


.95372 


•31530 


3-17159 


30 


31 


.28429 


•95874 


.29653 


3.37234 


-30098 


.95363 


•31562 


3-16838 


29 


32 


.28457 


.95865 


.29685 


3.36875 


-30126 


.95354 


•31594 


3-16517 


28 


33 


.28485 


.95857 


.29716 


3.365.T6 


-30154 


.95345 


•31626 


3-16197 


27 


34 


.28513 


.95849 


.29748_ 
.29780 


3. 36158 


-30182 


.95337 


-31658 


3-15877 


26 


35 


.28541 


.95841 


3-35800 


- 30209 


.95328 


-31690 


3-15558 


25 


36 


.28569 


.95832 


.29811 


3.35443 


-30237 


.95319 


-31722 


3-15240 


24 


37 


.28597 


.95824 


.29843 


3-35087 


-30265 


.95310 


-31754 


3-14922 


23 


38 


.28625 


•95816 


.29875 


3.34732 


-30292 


.95301 


-31786 


3-14605 


22 


39 


.28652 


.95807 


.29906 


3.34377 


-30320 


•95293 
•95284 


•31818 
-31850 


3 14288 
3-13972 


21 


40 


.28680 


•95799 


.29938 


3 •34023 


-30348 


30 


41 


.28708 


•95791 


.29970 


3.33670 


-30376 


.95275 


-31882 


3-13656 


19 


42 


.28736 


.95782 


.30001 


3.33317 


.30403 


.95266 


-31914 


3-13341 


18 


43 


.28764 


.95774 


.30033 


3.32965 


.30431 


-95257 


-31946 


3-13027 


17 


44 


.28792 


.95766 


.30065 


3.32614 


.30459 
.30486 


-95248 


-31978 


3-12713 


16 


45 


•28820 


.95757 


.30097 


3.32264 


-95240 


-32010 


3^12400 


15 


46 


.28847 


.95749 


.30128 


3-31914 


.30514 


-95231 


-32042 


3-12087 


14 


47 


•28875 


.95740 


.30160 


3-31565 


-30542 


-95222 


-32074 


3-11775 


13 


48 


.28903 


•95732 


•30192 


3-31216 


-30570 


-95213 


-32106 


3-11464 


12 


49 


.28931_ 
■28959 


.95724 
•95715 


.30224 


3-30868 
3-30521 


-30597 


.95204 


-32139 
-32171 


3-11153 


11 


60 


.30255 


-30625 


.95195 


3-10842 


10 


51 


.28987 


.95707 


.30287 


3-30174 


-30653 


-95186 


.32203 


3-10532 


9 


52 


.290J5 


.95698 


.30319 


3.29829 


-30680 


-95177 


-32235 


3-10223 


8 


53 


•29042 


.95890 


.30351 


3.29483 


-30708 


-95168 


-32267 


3-09914 


7 


54 


•29070 


.95681 


.30382 


3.29139 
3.28795 


^30736_ , 
-30763 


-95159 


-32299 


3-09606 
3-09298 


6 


55 


.29098 


•95673 


.30414 


-95150 


32331 


5 


56 


•29126 


•95664 


.30446 


3.28452 


-30791 


95142 


-32363 


3-08991 


4 


57 


•29154 


•95656 


.30478 


3-28109 


-30819 


95133 


32396 


3-08685 


3 


58 


.29182 


.95647 


•30509 


3-27767 


.30846 


95124 


32428 


3-08379 


2 


59 


•29209 


•95639 


•30541 


3-27426 


.30874 


95115 


32460 


3-08073 


1 


go. 


.29237 
Cos. 


•95630 


•30573 


3-27085 


-30902 


95106 


32492 


3-07f68 





i ' 


Sin. 


Cot. 


Tan. 


Cos„ 


Sin. 


Cot. 


Tan. 


r - 



73° 



761 



78° 



TABLE IX —NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
18° 19° 



/ 


Sin. 
.30902 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


, 





.95106 


•32492 


307768 


•32557 


.94552 


. 34433 


2-90421 


60 


1 


.30929 


•95097 


•32524 


3 


07464 


•32584 


.94542 


.34465 


2.90147 


59 


2 


.30957 


.95088 


•32556 


3 


07160 


•32612 


.94533 


•34498 


2-89873 


58 


3 


.30985 


.95079 


.32588 


3 


06857 


•32639 


.94523 


.34530 


2. 89600 


57 


4 


.31012 


.95070 


.32621 


3 
3 


p65_5_4__ 
06252 


•32667 
•32694 


.94514 


.34563 . 
•34596 


2. 89327 


56 


5 


.31040 


.95061 


•32653 


.94504 


2.89055 


55 


6 


.31068 


.95052 


.32685 


3 


05950 


•32722 


•94495 


.34628 


2.88783 


54 


7 


.31095 


.95043 


.32717 


3 


05649 


•32749 


•94485 


.34661 


2. 88511 


53 


8 


.31123 


.95033 


.32749 


3 


05349 


•32777 


.94476 


.34693 


2.88240 


52 


9 


.31151 
31178 


.95024 
.95015 


.32782 


3 


05049 


•32804 


. 94466_ 
.94457 


.34726 


2. 87970 


51 


10 


32814 


3 


04749 


•32832 


.34758 


2.87700 


50 


11 


• 31206 


.95006 


.32846 


3 


04450 


•32859 


. 94447 


.34791 


2-87430 


49 


12 


.31233 


.94997 


.32878 


3 


04152 


•32887 


. 94438 


.34824 


2^87161 


48 


13 


.31261 


.94988 


•32911 


3 


03354 


•32914 


.94428 


.34856 


2 •86892 


47 


14 


.31289 
•31316 


.94979 
.94970 


•32943 


3 


03556 


.32942 
.32969 


.94418 


•34889 
•34922 


2^86624_^ 
2^86356 


46 


If) 


•32975 


3 


03260 


.94409 


45 


Ifi 


•31344 


.94961 


.33007 


3 


02983 


.32997 


.94399 


.34954 


2^86089 


44 


17 


•31372 


.94952 


.33040 


3 


02667 


.33024 


.94390 


•34987 


2.85822 


43 


18 


•31399 


.94943 


.33072 


3 


02372 


.33051 


.94380 


•35020 


2.85555 


42 


19 


•31427 
•31454 


.94933 
.94924 


•33104 


_3. 
3 


.02077 


.33079 
•33106 


.94370 
.94P61 


•35052 


2.85289 


41 


30 


•33136 


01783 


•35085 


2.85023 


40 


21 


•31482 


.94915 


•33169 


3 


01489 


•33134 


.94351 


•35118 


2 •84758 


39 


22 


.31510 


.94906 


•33201 


3 


01196 


.33161 


•94342 


•35150 


2-84494 


38 


23 


•31537 


.94897 


•33233 


3 


00903 


•33189 


•94332 


•35183 


2.84229 


37 


24 


•31565 


.94888 


.33266 


3 


00611 


.33216 


.94322 
.94313 


•35216 


2^83965 
2^83702 


36 


25 


•31593 


.94878 


•33298 


3 


00319 


•33244 


•35248 


35 


26 


.31620 


.94869 


•33330 


3 


00028 


•33271 


.94303 


•35281 


2. 83439 


34 


27 


•31648 


.94860 


•33363 


2 


99738 


33298 


.94293 


•35314 


2.83176 


33 


28 


•31675 


.94851 


•33395 


2 


99447 


•33326 


.94234 


.35346 


2.82914 


32 


29 


•31703 


.94842 


•33427 


2 


99158 


.33353 
33381 


.94274 


.35379 


2.82653 
2.82391 


31 


30 


•31730 


.94832 


•33460 


2 


98868 


.94264 


.35412 


30 


31 


•31758 


.94823 


•33492 


2 


98580 


•33408 


.94254 


.35445 


2.82130 


28 


32 


.317&3 


.94814 


.33524 


2 


98292 


•33436 


■94245 


.35477 


2.81870 


28 


33 


.31813 


.94805 


.33557 


2 


98004 


.33463 


.94235 


•35510 


2-81610 


27 


34 


•31841 


.94795 


.33589 


2 


97717 


.33490 


.94225 


.35543 


2-81350 


2% 


35 


31863 


.94786 


.33621 


2 


97430 


.33518 


.94215 


•35576 


2-81091 


25 


36 


•31896 


.94777 


•33654 


2 


97144 


.33545 


.94206 


.35608 


2-80833 


24 


37 


•31923 


•94768 


•33686 


2 


96858 


.33573 


.94196 


.35641 


2-80574 


23 


38 


•31951 


.94758 


•33718 


2 


96573 


.33600 


.94186 


.35674 


2.80316 


22 


39 


•31979 


. 94749 


•33751 


9 


96288 


.33627 


.94176 
•94167 


.35707 


2. 80059 
2.79802 


21 


40 


•32006 


.94740 


33783 


2 


96004 


.33655 


.35740 


30 


41 


•32034 


•94730 


•33816 


2 


95721 


.33682 


•94157 


.35772 


2.79545 


19 


42 


•32061 


.94721 


•33848 


2 


95437 


.33710 


•94147 


.35805 


2.79289 


18 


43 


.32089 


.94712 


•33881 


2 


95155 


.33737 


•94137 


.35838 


2.79033 


17 


44 


.321x6 


.94702 


•33913 


2 


94872 


.33764 


.94127 


.35871 


2.78778 


16 


45 


.32144 


.94693 


.33945 


2 


94591 


.33792 


.94118 


.35904 


2-78523 


15 


46 


•32171 


.94684 


•33978 


2 


94309 


.33819 


.94108 


.35937 


2.78269 


14 


47 


.32199 


•94674 


•34010 


2 


94028 


.33846 


.94098 


.35969 


2.78014 


13 


48 


•32227 


•94665 


•34043 


2 


93748 


.33874 


.94088 


.36002 


2.77761 


12 


49 


•32254 


• 94656- 


•34075 


2 


93468 


.33901 


.94078 
.94068 


.36035 


2.77507 


11 


50 


•32282 


•94646 


•34108 


2 


93189 


.33929 


•36068 


2 . 77254 


10 


51 


•32309 


•94637 


•34140 


2 


92910 


.33956 


.94058 


.36101 


2.77002 


9 


52 


.32337 


•94627 


.34173 


2 


92632 


.33983 


.94049 


•36134 


2.78750 


8 


53 


•32364 


•94618 


•34205 


2 


92354 


.34011 


.94039 


.36167 


2.76493 


7 


54 


.32392 


.94609 


•34238 
•34270 


2 
2. 


92076 


.34038 


.94029 


.36199 


2.76247 


6 


55 


.32419 


.94599 


91799 


•34065 


.94019 


.36232 


2.75996 


5 


56 


.32447 


•94590 


•34303 


2 


91523 


.34093 


.94009 


•36265 


2.75746 


4 


57 


.32474 


.94580 


•34335 


2. 


91246 


•34120 


•93999 


•36298 


2-75496 


3 


58 


32502 


.94571 


•34368 


2. 


90971 


.34147 


.93989 


•36331 


2^75246 


2 


59 


•32529 


•94561 


• 34400 


2 


90696 


.34175 


.93979 


.36364 


2.74997 


1 


60 


• 32557 


•94552 


.34433 


2. 


90421 


.34202 


.93969 


•36397 


2.74748 





/ 


Cos. 


Sin. 


Cot. 


Tan. J 


Cos. 


Sin. 


Cot. 


Tan, 


/ 



71"= 



762 



70" 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 



30° 



31' 



/ 


Sin. 


Cos. 


Tan. 


1 Cot. ( Sin. 


Cos. 


Tan. 


Cot. 

2-60509 
2-60283 
2-60057 
2-59831 
2-59606 


/ 




1 

2 
3 

4 


•34202 
.34229 
.34257 
•34284 
•34311 

.34339 
.34366 
.34393 

.34421 
. 34448 


.93969 
•93959 
•93949 
.93939 
.93929 


•36397 
•36430 
•36463 
•36496 
•36529 


2 •74748 
2-74499 
2-74251 
2-74004 
2-73756 


•35837 
.35864 
•35891 
•35918 
.35945 


•93358 
•93348 
•93337 
•93327 
-93316 


-38386 
•38420 
-38453 
-38487 
-38520 


60 

59 
58 
57 
5fi 


5 
6 
7 
8 
9 


.93919 
.93909 
.93899 
.93889 
.93879 


•38562 
•36595 
•36628 
•36661 
•36694 


2-73509 
2-73263 
2-73017 
2-72771 
2-72526 


.35973 
•36000 
•36027 
•36054 
•36081 


.93306 
-93295 
-93285 
•93274 
•93264 


•38553 
•38587 
.38620 
•38654 
-38687 


2-59381 
2-59156 
2-58932 
2-58708 
2-58484 


55 
54 
53 
52 
51 


10 

7.1 
12 
13 
14 


.34475 
.34503 
.34530 
.34557 
.34584 


.93869 
.93859 
.93849 
.93839 
.93829 


.36727 
.36760 
•36793 
•36826 
.36859 


2^72281 
2-72036 
2-71792 
2-71548 
2-71305 


•36108 
•36135 
•36162 
•36190 
•36217 


•93253 
•93243 
•93232 
•93222 
•93211 


-38721 
•38754 
-38787 
•38821 
•38854 


2-58261 
2-58038 
2-57815 
2-57593 
2-57371 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.34612 
34639 
.34666 
.34694 
•34721 


.93819 
.93809 
.93799 
.93789 
.93779 


•36892 
•36925 
•36958 
.36991 
•37024 


2-71062 
2-70819 
2-70577 
2-70335 
2-70094 


.36244 
•36271 
•36298 
.36325 
•36352 

•36379 
•36406 
•36434 
•36461 
.36488 


-93201 
-93190 
-93180 
-93169 
•93159 


•38888 
•38921 
•38955 
•38988 
•39022 


2-57150 
2-56928 
2-56707 
2-56487 
2.56266 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.34748 
.34775 
.34803 
.34830 
.34857 


.93769 
.93759 
.93748 
.93738 
.93728 


.37057 
.37090 
•37123 
.37157 
.37190 


2-69853 
2-69612 
2-69371 
2^69131 
2^68892 


-93148 
-93137 
-93127 
.93116 
-93106 


•39055 
-39089 
-39122 
-39156 
-39190 


2-56046 
2-55827 
2-55608 
2-55389 
2-55170 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.34884 
•34912 
.34939 
.34966 
.34993 


.93718 
.93708 
.93698 
.93688 
.93677 


•37223 
•37256 
.37289 
.37322 
.37355 


2 •68653 
2-68414 
2.68175 
2-67937 
2.67700 


.36515 
•36542 
.36569 
.36596 
.36623 


-93095 
-93084 
.93074 
.93063 
.93052 


-39223 
39257 
.39290 
.39324 
.39357 


2-54952 
2.54734 
2.54516 
2.54299 
2-54082 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.35021 
.35048 
•35075 
.35102 
•35130 


.93667 
.93657 
•93647 
•93637 
•93626 


•37383 

•37422 
•37455 
•37488 
•37521 


2.67462 
2.67225 
2-66989 
2-66752 
2.66518 


.36650 
.36677 
•36704 
•36731 
•36758 


.93042 
•93031 
•93020 
•93010 
•92999 


■39391 
•39425 
•39458 
•39492 
•39526 


2^53865 
2^ 53648 
2-53432 
2-53217 
2-53001 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•35157 
•35184 
•35211 
•35239 
•35266 


.93616 
.93606 
.93596 
•93585 
•93575 

•93585 
.93555 
.93544 
.93534 
.93524 


■37554 
■37588 
■37621 
-37654 
•37687 

•37720 
•37754 
•37787 
•37820 
•37853 


2-66281 
2 •66046 
2-65811 
2-65576 
2-65342 

2-65109 
2-64875 
2-64642 
2-64410 
2-64177 


•36785 
•36812 
•36839 
•36867 
•36894 

•36G21 
•36948 
-36975 
•37002 
-37029 


-92988 
-92978 
-92967 
•92956 
•92945 


•39559 
■39593 
•39626 
•39660 
■39694 


2-52786 
2-52571 
2-52357 
2-52142 
2-51929 

2-51715 
2-51502 
2-51289 
2-51076 
2-50864 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•35293 
•35320 
•35347 
•35375 
•35402 


-92935 
-92924 
•92913 
•92902 
•92892 


•39727 
•39761 
•39795 
•39829 
•39862 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•35429 
•35456 
■35484 
•35511 
•35538 


.93514 
•93503 
•93493 
.93483 

.93472 


•37887 
.37920 
•37953 
•37986 
•38020 


2-63945 
2-63714 
2-63483 
2-63252 
2-63021 

2-62791 
2-62561 
2-62332 
2 •62103 
2^61874 

2.61646 
2-61418 
2-61190 
2-60963 
2-60736 


-37056 
■37083 
■37110 
•37137 
•37164 


•92881 
•92870 
•92859 
•92849 
•92838 


•39896 
•39930 
•39963 
•39997 
•40031 


2-50652 
2 - 50440 
2-50229 
2-50018 
2-49807 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•35565 
.35592 
• 3561-9 
•35647 
■35674 


.93462 
.93452 
.93441 
.93431 
.93420 


.38053 
.38086 
.38120 
•38153 
•38186 


•37191 
.37218 
.37245 
•37272 
■37299 


-92827 
-92816 
-92805 
-92794 
-92784 


•40065 
•40098 
•40132 
-40166 
-40200 


2-49597 
2-49386 
2-49177 
2-48967 
2-48758 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•35701 
■35728 
•35755 
•35782 
.35810 


•93410 
• 93400 
•93389 
.93379 
•93368 

.93358 


•38220 
•38253 
•38286 
•38320 
•38353 

38386 

Cot. 


■37326 
•37353 
37380 
■37407 
■37434 


-92773 
•92762 
■92751 
•92740 
•92729 


-40234 
-40267 
-40301 
•40335 
-40369 


2-48549 
2-48340 
2-48132 
2-47924 
2-47716 


5 
4 
3 
2 
1 


60 


.35837 


2-60509 
Tan. 


■37461 


•92718 


40403 


2-47509 





/ 


Cos. 


Sin. 


Cos. 


Sin. 


Cot. 


Tan. 


t * 



69^ 



763 



68" 



TABLE IX.— NATURAL 'SINES, COSINES, TANGENTS, AND COTANGENTS. 



33° 



23' 



1 


Sin. 


Cos. 


Tan. 


Cot. ( 


Sin. I 


Cos. . 


Tan. 

.42447 
.42482 
.42516 
.42551 
.42585 




Cot. 


t 




1 

2 
3 
4 


-37461 
•37488 
-37515 
-37542 
.37569 


•92718 
.92707 
.92697 
.92686 
.92675 

.92664 
.92653 
.92642 
.92631 
92620 


.40403 
.40436 
.40470 
.40504 
.40538 


2 
2 
2 
2 
2 


47509 
47302 
47095 
46888 
46682 


•39073 
.39100 
.39127 
.39153 
•39180 


•92050 
•92039 
•92028 
•92016 
-92005 


2 
2 
2 
2 
2 


35585 
35395 
35205 
35015 
34825 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


-37595 
-37622 
.37649 
.37676 
-37703 


.40572 
•40606 
.40640 
.40674 
•40707 


2 
2 
2 
2 
2 


46476 
46270 
46065 
45860 
45655 


.39207 
.39234 
.39260 
.39287 
.39314 


•91994 
•91982 
•91971 
.91959 
.91948 


.42619 
.42654 
.42688 
.42722 
.42757 


2 
2 
2 
2 
2 


.34636 
.34447 
.34258 
.34069 
.33881 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


-37730 
-37757 
-37784 
-37811 
-37838 


.92609 
.92598 
.92587 
.92576 
.92565 


.40741 
.40775 
•40809 
•40843 
-40877 


2 
2 
2 
2 
2 


45451 
■45246 

45043 
.44839 
.44636 


.39341 
.39367 
.39394 
.39421 
. 39448 


.91936 
•91925 
•91914 
•91902 
.91891 


.42791 
.42826 
.42860 
.42894 
.42929 


2 
2 
2 
2 
2 


33693 

33505 

.33317 

.33130 

32943 


50 

49 
48 
47 
46 


15 
16 
17 
18 

19 


-37865 
-37892 
-37919 
-37946 
-37973 

-37999 
.38026 
-38053 
.38080 
.38107 


.92554 
.92543 
.92532 
.92521 
•92510 

.92499 
.92483 
.92477 
.92466 
.92455 


•40911 
•40945 
.40979 
.41013 
.41047 

.41081 
.41115 
.41149 
.41183 
.41217 


2 
2 
2 

2 

2 


.44433 

.44230 

.44027 

43825 

43623 


. 39474 
•38501 
•39528 
•39555 
.39581 

39608 
•39635 
•39661 
.39688 
.39715 


.91879 
.91868 
•91856 
•91845 
•91833 

.91822 
.91810 
.91799 
.91787 
.91775 


.42963 
.42998 
.43032 
.43067 
.43101 


2 
2 
2 
2 
2 


32756 
32570 
32383 
32197 
32012 


45 
44 
43 
42 
._41 


30 

21 
22 
23 

24 


2 
2 
2 
2 
2 


.43422 
.43220 
.43019 
.42819 
42618 


.43136 
.43170 
.43205 
.43239 
.43274 


2 
2 
2 
2 
2 


31826 
31641 
31456 
31271 
31086 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


-38134 
-38161 
-38188 
-38215 
.38241 


.92444 
.92432 
.92421 
.92410 
.92399 


.41251 
.41285 
.41319 
.41353 
.41387 

.41421 
.41455 
.41490 
.41524 
.41558 


2 
2 
2 
2 
2 


.42418 
.42218 
.42019 
.41819 
•41620 


.39741 
.39768 
•39795 
•39822 
•39848 


.91784 
.91752 
.91741 
.91729 
•91718 


.43308 
.43343 
.43378 
.43412 
.43447 


2 
2 
2 
2 
2 


.30902 
30718 ' 
30534 
30351 
30167 


35 
34 
33 
32 
31 


30 

31 
32 
SP 

34 


-38268 
-38295 
-3832? 
.33349 
.38376 


.92388 
.92377 
.92366 
.92355 
.92343 


2 
2 
2 
2 
2 


41421 

41223 

•41025 

.40827 

40629 


•39875 
-39902 
-39928 
-39955 
.39982 


•91706 
.91694 
•91683 
.91671 
:91660 


.43481 
.43516 
.43550 
.43585 
.43620 


2 
2 
2 
2 
2 


29984 
29801 
29619 
29437 
29254 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


■38403 
-38430 
-38456 
-38483 
.38510 


.92332 
.92321 
.92310 
.92299 
-92287 


.41592 
.41626 
.41660 
.41694 
.41728 


2 
2 
2 
2 
2 


40432 
40235 
40038 
39841 
39645 


.40008 
•40035 
40062 
•40088 
•40115 

•40141 
•40168 
•40195 
.40221 
.40248 


.91648 
.91636 
.91625 
.91613 
•91601 

•91590 
•91578 
•91566 
.91555 
.91543 


.43654 
.43689 
•43724 
.43758 
.43793 

.43828 
•43862 
•43897 
.43932 
.43966 


2 
2 
2 
2 
2 


29073 
28891 
28710 
28528 
28348 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


-38537 
.38564 
.38591 
.38617 
-38644 


-92276 
.92265 
.92254 
.92243 
.92231 


.41763 
.41797 
.41831 
.41865 
.41899 


2 
2 
2 
2 
2 

2 
2 
2 
2 
2 


39449 
39253 
39058 
38863 
38668 

38473 
38279 
38084 
37891 
37697 


2 
2 
2 
2 
2 


28167 
27987 
27806 
27626 
27447 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


-38671 
-38698 
. 3^7.25 
.33752 
.38778 


.92220 
.92209 
.92198 
.92186 
.92175 • 


.41933 
.41963 
.42002 
.42036 
.42070 

.42105 
.42139 
•42173 
.42207 
•42242 


.40275 
40301 
.40328 
.40355 
•40381 


.91531 
.91519 
.91508 
.91496 

.91484 


.44001 
.44036 
. 44071 
.44105 
.44140 


2 
2 
2 
2 
2 


27267 
27088 
26909 
26730 
26552 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.38805 
.38832 
.38859 
.38886 
•38912 

.38939 
.38966 
.38993 
.39020 
.39046 


.92164 
.92152 
.92141 
.92130 
.92119 


2 
2 
2 
2 
2 


37504 
37311 
37118 
36925 
36733 


•40408 
•40434 
•40461 
.40488 
.40514 


•91472 
•91461 
•91449 
•91437 
•91425 

.91414 
.91402 
.91390 
.91378 
L?1366_ 

.91355 


.44175 
.44210 
.44244 
.44279 
.44314 

.44349 
.44384 
.44418 
.44453 
. 44488 


2 
2 
2 
2 
2 


26374 
26196 
26018 
25840 
25663 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.92107 
.92096 
.92085 
.92073 
.92062 

.92050 


•42276 
.42310 
.42345 
.42379 
.42413 


2 
2 
2 
2 
2 


36541 
36349 
36158 
35967 
35776 


.40541 
.40567 
.40594 
.40621 
•40847 


2 
2 
2 
2 
2 


25486 
25309 
25132 
24956 
24780 


5 
4 
3 
2 
1 


60 


-39073 


•42447 
Cot. 


2 


35585 


•40674 


•44523 


2 


24604 





f 


Cos. 


Sin. 


Tan. j Cos. 


Sin. 


Cot. 


Tan. 1 


/ 



67° 



764 



66" 



TABLE IX.— NATURAL SINES. COSINES, TANGENTS, AND COTANGENTS. 







24' 






35° 






/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 
-42262 


Cos. 


Tan. 


Cot. 
2.14451 


9 





.40674 


.91355 


.44523 


2.24604 


•90631 


-46631 


60 


1 


.40700 


.91343 


.44558 


2.24428 


-42288 


•90618 


.46666 


2-14288 


59 


n 


.40727 


.91331 


-44593 


2.24252 


.42315 


•90606 


.46702 


2.14125 


58 


3 


.40753 


.91319 


.44627 


2.24077 


•42341 


.90594 


.46737 


2.13963 


57 


4 


.40780 


.9i307 


.44662 


2.23902 


•42367 


.90o82 
.90569 


.46772 


2.13801 


56 


5 


.40806 


.91295 


.44697 


2.23727 


•42394 


.46808 


2.13639 


55 


6 


.40833 


.91283 


•44732 


2.23553 


•42420 


.90557 


-46843 


2.13477 


54 


7 


.40860 


.91272 


•44767 


2-23378 


•42446 


.90545 


-46879 


2.13316 


53 


8 


.40886 


.91260 


.44802 


2.23204 


•42473 


.90532 


-46914 


2-13154 


52 


9 


.40913 


.91248 
.91236 


.44837 


2.23030 


■42499 


.90520 


-46950 


2-12993 


51 


10 


.40939 


.44872 


2.22857 


■42525 


.90507 


-46985 


2-12832 


.50 


11 


.40966 


.91224 


.44907 


2.22683 


•42552 


-90495 


.47021 


2-12671 


49 


12 


.40992 


.91212 


.44942 


2.22510 


•42573 


-90483 


-47056 


2-12511 


48 


13 


.41019 


.91200 


.44977 


2-22337 


•42604 


-90470 


-47092 


2-12350 


47 


14 


.41045 
.41072 


•91188 


.45012 


2-22164 


.42631 
.42657 


■90458 
- 90446 


-47128 


212190 


46 


15 


•91176 


.45047 


2.21992 


-47163 


2-12030 


45 


16 


.41098 


•91164 


.45082 


2.21819 


.42683 


•90433 


-47199 


2-11871 


44 


17 


•41125 


•91152 


.45117 


2-21647 


.42709 


-90421 


-47234 


2-11711 


43 


18 


.41151 


•91140 


.45152 


2.21475 


.42736 


-90408 


-47270 


2-11552 


42 


19 


.41178 


•91128 


•45187 


2.21304 


.42762 
.42788 


•90396 


-47305 


2-11392 
2-11233 


41 


30 


.41204 


•91116 


.45222 


2.21132 


•90383 


-47341 


40 


21 


.41231 


•91104 


.45257 


2-20961 


-42815 


.90371 


-47377 


2-11075 


39 


22 


.41257 


•91092 


.45292 


2-20790 


-42841 


.90358 


-47412 


2-10916 


38 


23 


-41284 


•91080 


.45327 


2-20619 


-42867 


.90346 


-47448 


2.10758 


37 


24 


.41310 


.91068 


.45362 


2-20449 


.42894 
.42920 


•90334 
-90321 


-47483 


2.10600 


36 


25 


.41337 


•91056 


.45397 


2-20278 


-47519 


2 . 10442 


35 


26 


.41363 


•91044 


•45432 


2-20108 


.42946 


-90309 


.47555 


2.10284 


34 


27 


•41390 


•91032 


.45467 


2.19938 


.42972 


•90296 


.47590 


2.10126 


33 


28 


.41416 


•91020 


.45502 


2.19769 


.42999 


.90284 


-47626 


2.09969 


32 


29 


.41443 


.91008 


■45538 


2.19599 


•43025 


-90271 


-47662 


2-09811 


31 


30 


.41469 


.90996 


.45573 


2-19430 


.43051 


.90259 


.47698 


2-09654 


30 


31 


.41496 


.90984 


■45608 


2-19261 


.43077 


.90246 


. 7733 


2-09498 


29 


32 


.41522 


.90972 


.45643 


2.19092 


.43104 


.90233 


.47769 


2-09341 


28 


33 


.41549 


.90960 


.45678 


2.18923 


.43130 


.90221 


.47805 


2-09184 


27 


34 


.41575 
.41602 


•90948 


.45713 


2.18755 


•43156 


•90208 


-47840 


2.09028 


26 


35 


•90936 


.45748 


2.18587 


•43182 


•90196 


-47876 


2-08872 


25 


36 


.41628 


•90924 


.45784 


2.18419 


•43209 


•90183 


-47912 


2-08716 


24 


37 


•41655 


•90911 


•45819 


2.18251 


•43235 


•90171 


-47948 


2-08560 


23 


38 


.41681 


•90899 


.45854 


2-18084 


•43261 


•90158 


-47984 


2-08405 


22 


39 


•41707 


■90887 


.45889 


2.17916 


.43287 


•90146 


-48019 
-48055 


2-08250 
2.08094 


21 


40 


■41734 


.90875 


.45924 


2-17749 


.43313 


•90133 


30 


41 


•41760 


•90863 


.45960 


2.17582 


.43340 


•90120 


-48091 


2.07939 


19 


42 


.41787 


90851 


.45995 


2.17416 


43366 


•90108 


-48127 


2.07785 


18 


43 


•41813 


•90839 


-46030 


2-17249 


.43392 


-90095 


-48163 


2-07630 


17 


44 


■41840 


■90826 


.46065 


2.17083 


•43418 


-90082 


-48198 


2.07476 


IB 


45 


•41866 


•90814 


.46101 


2.16917 


•43445 


-90070 


-48234 


2.07321 


15 


48 


•41892 


.90802 


.46136 


2.16751 


.43471 


•90057 


-48270 


2.07167 


14 


47 


.41919 


.90790 


.46171 


2.16585 


.43497 


.90045 


-48306 


2.07014 


13 


48 


.41945 


.90778 


.46206 


2-16420 


.43523 


.90032 


-48342 


2. 06860 


12 


49 


•41972 


•90766 


•46242 


2.16255 


.43549 


•90019 


-48378 


2-06706 


11 


60 


•41998 


.90753 


•46277 


2.16090 


.43575 


•90007 


-48414 


2-06553 


10 


51 


•42024 


90741 


•46312 


2.15925 


.43602 


•89994 


-48450 


2.06400 


9 


52 


■42051 


•90729 


•46348 


2.15760 


.43628 


•89931 


-48486 


2.06247 


8 


53 


.42077 


.90717 


•46383 


2.15596 


.43654 


•89968 


-48521 


2.06094 


y 


54 


.42104 


90704 


.46418 


2.15432 


.43680 


-89956 


-48557 


2.05942 


6 


55 


.42130 


.90692 


.46454 


2-15268 


.43706 


-89943 


-48593 


2.05790 


5 


56 


.42156 


.90680 


.46489 


2-15104 


.43733 


-89930 


.48629 


2.05637 


4 


57 


.42183 


.90668 


.46525 


2.14940 


.43759 


-89918 


.48665 


2.05485 


8 


58 


.42209 


.90655 


.46560 


2 14777 


.43785 


-89905 


.48701 


2.05333 


^ 


59 


.42235 


.90643 


.46595 


2^14614 


•43811 


89892 


•48737 


2.05182 


i 


60 


.42262 


.90631 


.46631 


2 •14451 


•43837 


-89879 
Sin. 


-48773 


2.05030 


..J2 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Cot. 


Tan. 


t 



65° 



76.5 



64*= 



TABLE IX.~NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS. 



36° 



27' 



1 


Sin. 


Cos. 


Tan. 
.48773 


Cot. 


Sin. 
-45399 


Cos. 


Tan. 
.50953 


Cot. 


/ 


V 


.43837 


•89879 


2.05030 


.89101 


1-96261 


60 


1 


.43863 


.89867 


.48809 


2.04879 


.45425 


.89087 


.50989 


1-96120 


59 


2 


-43889 


.89854 


.48845 


2.04728 


-45451 


.89074 


.51026 


1.95979 


58 


3 


.43916 


.89841 


.48881 


2.04577 


-45477 


.89061 


.51063 


1.95838 


57 


4 


.43942 


•89828 


.48917 
.48953 


2^04426 
2^04276 


-45503 


.89048 


.51099 


1.95698 
1.95557 


56 


5 


.43968 


.89816 


•45529 


.89035 


.51136 


55 


6 


.43994 


.89803 


.48989 


2.04125 


•45554 


.89021 


.51173 


1.95417 


5^ 


7 


.44020 


.89790 


.49026 


2.03975 


-45580 


.89008 


.51209 


1.95277 


53 


8 


. 44046 


.89777 


.49062 


2.03825 


.45606 


.88995 


.51246 


1.95137 


52 


9 


.44072 


.89764 


.49G98_ 
.49134 


2.03675 


.45632 


.88981 


.51283^ 
.51319 


1-94997 


51 


10 


.44098 


.89752 


2.03526 


.45658 


.88968 


1-94858 


50 


11 


•44124 


.89739 


.49170 


2.03376 


.45684 


.88955 


.51356 


1-94718 


49 


12 


.44151 


.89726 


.49206 


2.03227 


.45710 


.88942 


.51393 


1-94579 


48 


13 


.44177 


.89713 


.49242 


2.03078 


.45736 


.88928 


.51430 


1 . 94440 


47 


14 


.44203 


.89700 
.89687 


.49278 
.49315 


2.02929 


.45762 
.45787 


.88915 
.88902 


.51467 
.51503 


1-94301 


46 


15 


.44229 


2.02780 


1.94162 


45 


16 


.44255 


.89674 


.49351 


2.02631 


.45813 


.88888 


.51540 


1.94023 


44 


17 


.44281 


.89662 


.49387 


2.02483 


.45839 


.88875 


.51577 


1.93885 


43 


18 


.44307 


.89649 


.49423 


2-02335 


-45865 


.88862 


.51614 


1.93746 


42 


19 


.44333 


•89636 


.49459 


2. 02187 


.45891 


.88848 


.51651 


1.93608 


41 


30 


.44359 


.89623 


•49495 


2-02039 


-45917 


-88835 


.51688 


1-93470 


40 


21 


.44385 


.89610 


.49532 


2.01891 


-45942 


-88822 


.51724 


1.93332 


39 


22 


.44411 


.89597 


•49568 


2-01743 


-45968 


.88808 


.51761 


1.93195 


38 


23 


.44437 


.89584 


.49604 


2-01596 


45994 


-88795 


.51798 


1.93057 


37 


24 


. 44464 


.89571 


.49640 
•49677 


2 - 01449_ 
2-01302 


-46020 
.46046 


-88782 
-88768 


.51835 
.51872 


l-9?920 


36 


25 


•44490 


.89558 


1-92782 


35 


26 


•44516 


89545 


.49713 


2.01155 


•46072 


.88755 


.51909 


1.92645 


34 


21 


.44542 


-89532 


.49749 


2.01008 


.46097 


.88741 


.51946 


1.92508 


33 


28 


.44568 


.89519 


.49786 


2.00862 


-46123 


.88728 


.51983 


1.92371 


32 


29 


J:4594_ 
.44620 


.89506 
•89493 


•49822 


2^00715 
2-00569 


46149 
-46175 


.88715 


.52020 


1.92235 


31 


30 


.49858 


.88701 


.52057 


1.92098 


30 


81 


.44646 


•89480 


•49894 


2 00423 


-46201 


.88688 


.52094 


1.91962 


29 


32 


.44672 


.89467 


•49931 


2-00277 


-46226 


-88674 


.52131 


1.91826 


28 


33 


.44698 


.89454 


.49987 


2.00131 


-46252 


-88661 


.52168 


1.91690 


27 


84 


.44724 


.89441 


. 50004 _ 
. 50040 


1.99986 


-46278 


-88647 


. 52205_ 
.52242 


1-91554 


26 


35 


.44750 


.89428 


1.99841 


-46304 


-88634 


1-91418 


25 


36 


.44776 


.89415 


.50076 


1.99695 


-46330 


-88620 


.52279 


1.91282 


24 


S7 


•44802 


.89402 


.50113 


1.99550 


-46355 


-88607 


.52316 


1.91147 


23 


38 


.44828 


.89389 


.50149 


1.99406 


-46381 


-88593 


.52353 


1.91012 


22 


39 


-44854 


•89376 


.50185 

• 50222 


1.99261 


.46407 


-88580 


-52390 
-52427 


1-9087R 


21 


40 


.44880 


•89363 


1.99116 


-46433 


-88566 


1-90741 


30 


41 


.44906 


.89350 


•50258 


1.98972 


-46458 


-88553 


-52464 


1-90607 


19 


42 


•44932 


.89337 


.50295 


1.98828 


.46484 


-88539 


-52501 


1-90472 


18 


43 


•44958 


.89324 


•50331 


1.98684 


-46510 


-88526 


-52538 


1-90337 


17 


44 


.44984 


.89311 


• 5036l8_ 
- 50404 


1.98540 


-46536 


-88512 


-52575 
- 52613 


1-90203 


16 


45, 1 


•45010 


•89298 


1.98396 


.46561 


-88499 


1-90069 


15 


46 


•45036 


.89285 


• 50441 


1.98253 


-46587 


-88485 


-52650 


1-89935 


14 


47 


•45062 


•89272 


•50477 


1.98110 


-46613 


- 88^72 


-52687 


1-89801 


13 


48 


•45088 


•89259 


•50514 


1-97966 


-46639 


-88458 


-52724 


1-89667 


Jf 


49 


•45114 


•89245 
•89232 


.50550 
•50587 


1-97823 
1-97681 


- 46664 _ 
-46690 


-88445 


-52761 
.52798 


1-89533 


50 


.45140 


-88431 


1-89400 


10 


51 


•45168 


.89219 


• 50623 


1.97538 


-46716 


88417 


-52836 


1-89266 


9 


52 


^45192 


.89206 


50660 


1.97395 


.46742 


-88404 


-52873 


1-89133 


8 


53 


.45218 


•89193 


.50696 


1-97253 


-46767 


-88390 


-52910 


1-89000 


7 


54 


.45243 
•45269 


.89180 
89167 


.50733 
.50769 


1.9711L 
1.96969 


.46793 
.46819 


-88377 


-52947 


1-88867 


6 


55 


-88363 


-52985 


1-88734 


5 


56 


.45295 


•89153 


.50806 


1.96827 


-46844 


.88349 


-53022 


1. 88602 


4 


57 


45321 


.89140 


.50843 


1.96685 


.46870 


-88336 


-53059 


1.88469 


3 


58 


.45347 


.89127 


50879 


1-96544 


46896 


-88322 


.53096 


1-88337 


2 


59 


;_45373 
•45399 

Cos. 


89114 

89101 

Sin. 


50918 


1-96402 

1-96261 

Tan. 


.46921 
46947 


-88308 


-53134 


1-88205 


1 


60 


50953 

Cot. 


-88295 


-53171 


1-88073 





$ 


Cos. 


Sino 


Cot. 


Tan. 


/ 



63° 



766 



6!?^ 



TABLE IX —NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
38° 29° 



-1 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 

55431 
•55469 
•55507 
•55545 
•55583 


Cot. 


t 




1 

2 
3 
4 


.46947 
.46973 
•46939 
.47024 
.47050 


.88295 
•88281 
•88267 
•88254 
•88240 


-53171 
-53208 
53246 
.53283 
^3b20_ 

. 53358 
.53395 
.53432 
53470 
-53507 


1. 88073 
1.87941 
1^ 87809 
1^ 87677 
1^87546 


•48481 
.48506 
•48532 
•48557 
•48583 


87462 
-87448 
-87434 

87420 
.874.06 


1 80405 
1.80281 
1.80158 
1 80034 
1.79911 


60 

59 
58 
57 
5fi 


5 
6 
7 
8 
9 


.47076 
.47101 
•47127 
.47153 
.47178 


88226 

88213 

•88199 

•88185 

88172 


1.87415 
1.87283 
1.87152 
1.87021 
1.86891 


•48608 
.48634 
•48659 
48684 
•48710 


.87391 
-87377 
-87363 
.87349 
.87335 

•87321 
•87306 
.87292 
.87278 
.87264 


•55621 
•55659 
•55697 
■55736 
.55774 


1 79788 
1 79665 
1.79542 
1.79419 
1.79296 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


.47204 
.47229 
.47255 
.47281 
•47306 


.88158 
•88144 
•88130 
.88117 
•88103 


53545 
-53582 
.53620 
.53657 
.53694 


1.86760 
1.86630 
1.86499 
1.86369 
1.86239 

1.86109 
1.85979 
1-85850 
1-85720 
1-85591 


•48735 
•48761 
.48786 
.48811 
•48837 

.48862 
•48888 
.48913 
.48938 
•48964 


•55812 
•55850 
•55888 
•55926 
•55964 


1.79174 
17905] 
i 1.78929 
1-78807 
1-78685 


60 

49 
48 
47 
46 


15 
16 
17 
18 
J9 


-47332 
•47358 
.47383 
•47409 
•47434 


.88089 
.88075 
.88062 
•88048 
.88034 


.53732 
.53769 
-53807 
.53844 
.53882 

.53920 
.53957 
.53995 
.54032 
.54070 


•87250 
•87235 
•87221 
•87207 
-87193 


-56003 
-56041 
-56079 
.56117 
•56156 


1.78563 
1-78441 
1-78319 
1-78198 
1-78077 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


•47460 
•47486 
-47511 
•47537 
-47562 


.88020 
88006 
-87993 
.87979 
•87965 


1-85462 
i^85333 
1^85204 
1^ 85075 
1-84946 


48989 
•49014 
•49040 
•49065 
•49090 


.87178 
.87134 
•87150 
•87136 
.87121 


•56194 
•56232 
•56270 
•56309 
•56347 


1^77955 
1^ 77834 
1-77713 
1-77592 
1-77471 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•47588 
•47614 
47639 
•47665 
.47690 


•87951 
•87937 
•87923 
•87909 
•87896 


.54107 

•54145 
•54183 

• 54220 
54258 

•54296 
•54333 
•54371 

• 54409 

• 54446 


1.84818 
1.84689 
1.84561 
1.84433 
1-84305 


.49116 
•49141 
•49166 
•49192 
.49217 


•87107 
87093 
.87079 
.87064 
•87050 


-56385 
-56424 
•56462 
•56501 
.56539 


1-77351 
1. 77230 
1 77110 
1-76990 
1-76869 


35 
34 
33 
32 
31 


30 

81 

32 

33 

34 


•47716 
• 47741 
•47767 
•47793 
•47818 


•87882 
•87868 
•87854 
.87840 
•87826 


1-84177 
1 - 84049 
1-83922 
1-83794 
1.83667 


•49242 
•49268 
•49293 
.49318 
.49344 


•87036 
•87021 
•87007 
.86993 
•86978 


•56577 
•56616 
-56654 
-56693 
-58731 


1-76749 
1.76629 
1.76510 
1.76390 
1.76271 


30 

29 
28 
27 
26 


35 
S6 
37 
38 
39 


•47844 
•47869 
.47895 
•47920 
•47946 

•47371 
.47997 
•48022 
•48048 
•48073 


.87812 
•87798 
.87784 
•87770 
•87756 


• 54484 
•54522 
•54560 
•54597 
•54635 


1.83540 
1-83413 
1-83286 
1-83159 
1-83033 


.49369 
-49394 
-49419 
-49445 
•49470 


•86964 
•86949 
.86935 
.86921 
•86906 


.56769 
.56808 
-56846 
-56885 
■56923 


1-76151 
1-76032 
1-75913 
1-75794 
1-75675 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


-87743 
-87729 
.87715 
.87701 
-87687 


.54673 
•54711 
•54748 
•54786 
•54824 


1-82906 
1-82780 
1-82654 
1-82528 
1-82402 


•49495 
•49521 
•49546 
•49571 
49596 


•86892 
•86878 
•86863 
.86849 
•86834 


-56962 
-57000 
.57039 
.57078 
-57116 

-57155 
-57193 
- 57^32 
•57271 
■57309 


1-75556 
1 75437 
1-75319 
1-75200 
1-75082 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•48099 
•48124 
•48150 
.48175 
•48201 


.87673 
-87659 
•87645 
•87631 
.87617 


•54862 
•54900 
.54938 
.54975 
.55013 


1-82276 
1-82150 
1-82025 
1-81899 
1-81774 


•49622 
-49647 
-49672 
-49697 
-49723 


.86820 
.86805 
.86791 
•86777 
•86762 


1-74964 
1-74846 
1-74723 
1-74610 
1-74492 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


• 48226 
-48252 
.48277 

• 483,03 
•48328 


•87603 
•87589 
•87575 
•87561 
•87546 


.55051 
.55089 
.55127 
.55165 
•55203 


1-81649 
1-81524 
1-81399 
1-81274 
1-81150 


.49748 
.49773 
.49798 
.49824 
-49849 


•86748 
-86733 
-86719 
-8fc!704 
•86690 


■57348 
-57386 
-57425 
■57464 
■57503 


1-74375 
1..4257 
1.74140 
1 . 74022 
1-73905 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.48354 
•48379 
•48405 
•48430 
• 48456 


•87532 
•87518 
-87504 
-^^7490 
.87476 

87462 


-55241 
-55279 
-55317 
.55355 
•55393 


1-81025 
1-80901 
1-80777 
1-80653 
1.80529 


.49874 
.49899 
•49924 
•49950 
•49975 


-86675 
-86661 
-86648 
-86632 
•86617 

•86603 

Sin. 


-57541 
-57580 
-57619 
-57657 
■57696 


1-73788 
1-73671 
1-73555 
1-73438 
1. 73321 

1-73205 


5 
4 
3 
2 
1 


60 


• 48481 . 


•55431 


1^80405 


-50000 


57735 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Co.s, 


Cot. 


Tan. 


1 



61° 



767 



60° 



TABLE IX.- 


-NATURAL SINES, COSINiES. TANGENTS, AND COTANGENTS 
SO"* 31° 


/ 


Sin. 


Cos. 


Tan. 


Cot. 


Pin. 


Cos. 


Tan. 


Cot. 


60 

59 ; 

58 
57 1 
56 



1 

2 
3 

4 


.50000 
.50025 
.50050 
.50076 
.50101 

.50126 
.50151 
.50176 
.50201 
.50227 


•86603 
•86588 
•86573 
.86559 
•86544 

.86530 
.86515 
•86501 
•86488 
•86471 


•57735 
.57774 
•57813 
.57851 
•57890 


1.73205 
1.73089 
1.72973 
1.72857 
1.72741 


-51504 
•51529 
•51554 
•51579 
•51604 


•85717 
•85702 
.85687 
•85672 

• 85657 

•85642 
•85627 
•85612 
•85597 
•85582 

•85567 
•85551 
•85536 
•85521 
. 85506 

•85491 
•85476 
•85461 

• 85446 
■85431 


•60086 
•60126 
•60165 
•60205 
• 60245 


1-66428 
1-66318 
1-66209 
1-66099 
1-65990 


5 
6 
7 
8 
9 


•57929 

• 57968 
•58007 
•58046 
.58085 

•58124 
•58162 
•58201 

• 58240 
•58279 


1.72625 
1-72509 
1-72393 
1.72278 
1-72163 


•51628 
•51653 
• 51678 
•51703 
•51728 


•60284 
-60324 
■60364 
. 60403 
• 60443 


1^65881 
1-65772 
1 - 65663 
l-655.'^4 
1-65445 


55 1 
54 J 
53 ! 
52 ; 
51 


10 

11 
12 
13 

14 


.50252 
.50277 
50302 
.50327 
.50352 


•86457 

•86442 
•86427 
•86413 
•86398 


1-72047 
1^71932 
!• 71817 
1.71702 
1. 71588 


•51753 
•51778 
.51803 
•51828 
•51852 


.60483 
.60522 
.60562 

• 60602 
•60642 

•60681 
-60721 
.60761 
.60801 

• 60841 


1-65337 
1^65228 
1^65120 
1-65011 
1 - 64903 

1-64795 
1-64687 
1-64579 
1-64471 
1-64363 


50 

49 
48 
47 
48 


15 
16 
17 
18 
19 


.50377 
. 50403 
.50428 
.50453 
. 50478 


•86384 
•86369 
•86354 
•86340 
•86325 


•58318 
•58357 
•58396 
•58435 
•58474 

•58513 
•58552 
•58591 
58631 
•58670 


1 . 71473 
1^71358 
1.71244 
1-71129 
1^71015 


.51877 
.51902 
•5.927 
•51952 
• 519:77 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.50503 
.50528 
.50553 
•50578 
.50603 


•86310 
.86295 
.86281 
•86266 
•86251 


1^70901 
1.70787 
1.70673 
1.70560 
1 . 70446 


•52002 
52026 
- 52051 
-52076 
-52101 


.85416 
. 85401 
.85385 
•85370 
•85355 


•60881 
.60921 
.60960 
.61000 
61040 


1.64256 
1-64148 
1.64041 
1.63934 
1.63826 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


■50628 
.50654 
. 50679 
50704 
.50729 

■50754 

.50779 

50804 

50829 

50854 


.86237 
.86222 
.86207 
•86192 
•86178 


58709 

•58748 

58787 

58826 

•58865 

.58905 
. 58944 
.58983 
.59022 
•59061 

.59101 
-59140 
•59179 
59218 
.59258 


1.70332 
1.70219 
1.70106 
1.69992 
1.69879 


-52126 
-52151 
-52175 
■52200 
•52225 

•52250 
•52275 
52299 
■ 52324 
.52349 


.85340 
•85325 
•85310 
•85294 
■85279 

•85264 
■85249 
■85234 
.85218 
.85203 


.61080 
.61120 
.61160 
. 61200 
•61240 


1.63719 
1.63612 
i. 63505 
1.63398 
1.63292 


35 
34 
33 
32 
31 


30 

31 
32 
S3 
34 


•86163 
•86148 
• 86133 
•86119 
•86104 


!• 69766 
1.69653 
1.69541 
1.69428 
1.69316 


.61280 
•61320 
.61360 
.61400 
.61440 


1.63185 
1.63079 
1.62972 
1.62866 
1.62760 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


■50879 
.50904 
■50929 
■50954 
•50979 


•86089 
.86074 
•86059 
•86045 
. 86030 


1 . 69203 
1.69091 
1.68979 
1.68866 
1.68754 


•52374 
52399 
■52423 
■ 52448 
•52473 


•85188 
.85173 
.85157 
-85142 
-85127 


.61480 
.61520 
.61551 
-61601 
•61641 


1.62654 
1.62548 
1-62442 
1-62336 
1-62230 


25 
24 
23 

22 
21 


4CI 

41 

42 

43 

44 


■51004 
51029 

■51054 
51079 

■51104 


.86015 
.86000 
85985 
•85970 
•85956 


.59297 
.59336 
.59376 
-59415 
•59454 


1^ 68643 
1^68531 
1.68419 
1.68308 
1^68196 


52498 
•52522 
■ 52547 
■52572 
■52597 


.85-112 
•85096 
•85081 

• 85066 

• 85051 


-61681 
.61721 
.61761 
.61801 
- 61842 

.61882 

• 61922 
61962 

• 62003 
•62043 


1.62125 
1.62019 
1.61914 
1.61808 
.1.61703 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


51129 

51154 

■51179 

■51204 

■51229 


.85941 
.85928 
.85911 
•85896 
•85881 


•59494 
59533 
•59573 
•59612 
.59651 

•59691 
•59730 
•59770 
• 59809 
•59849 

•59888 
•59928 
59937 
.60007 
•60046 

•60086 


1^ 68085 
1.67974 
1.67863 
1.67752 
1-67641 


•52621 
•52646 
•52671 
•52696 
•52720 


•85035 

• 85020 
•85005 
•84989 
- 84974 

•84959 

• 84943 
•84928 
■84913 

• 84897 


1.61598 
1 .61493 
1. 61388 
1.61283 
1.61179 


15 
14 
13 
12 
11 


50 

51 

52 

53 

54 


■51254 
•51279 
■51304 
■51329 
■51354 


•85866 
•85851 
-85836 
•85821 
•85806 


1-67530 
1 - 67419 
1-67309 
1-67198 
1.67088 

1.66978 
1.66867 
1-66757 
1. 66647 
1-66538 

!• 66428 


•52745 
•52770 
•52794 
•52319 
•52844 


•62083 
62124 
.62164 
.62204 
- 62245 


1.61074 
1.60970 
1 60865 
1.60761 
1.60657 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.51379 
•51404 
. 51429 
-51454 
•51479 


•85782 
•85777 
•85762 
•85747 
85732 


.52869 1 

.52893 

•52918 

• 52943 

• 52967 


• 84882 
84866 
•84851 
.84836 
•84820 


.62285 
.62325 
.62366 
•62406 
- 62446 

-62487 
Cot. 


1.60553 
1.60449 
1-60345 
1-60241 
1.60137 

1 . 60033 


S 
4 
3 
2 
1 


6Gi 


•51504 


•85717 


•52992 


.84805 





/" 


Cos. 1 Sin. 1 Cot. 


Tan. 


Cos. 


Sin. 


Tan. 


"'"' 



59? 



768 



58° 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS. AND COTANGENm 

33* sa"* 


1 

*0~ 
1 
2 

3 

! 4 

5 

1 6 
! 7 

; 8 

9 
10 

11 
12 
13 
14 


Sin. 


Cos 


Tan. 


Cot 


Sin. 


Cos. 


Tan. 


Cot. 


* 


.52992 
.53017 
■53041 
.53066 
.53091 


.54805 
.84789 
.84774 
.84759 
. 84743 


.62487 
.62527 
•62568 
.62608 
•62649 


1-60033 
!• 59930 
1-59826 
1-59723 
1^59620 

1.59517 
1.59414 
1.59311 
1.59208 
1.59105 

1.59002 
1.58900 
1.58797 
1.58695 
1-58593 


-54464 
-54488 
-54513 
-54537 
-54561 


-83867 
-83851 
-83835 
-83819 
-83804 


•64941 
•64982 
.65024 
•65065 
.65106 


1.53986 
1.53888 
1.53791 
1-53693 
1^53595 


6(1 

59 
58 
57 
56 


.53115 
.53140 
.53164 
.53189 

.53214 


.84728 
.84712 
.84697 
.84681 

.84666 


.62689 
•62730 
.62770 
.62811 
•62852 


^4585 

046 10 

-54635 

-54659 

-54683 


-83788 
•83772 
•83756 
.83740 
.83724 

.83708 
•83692 
.83676 
•83660 
•83645 


.65148 
•65189 
•65231 
•65272 
•65314 


1-53497 
1^53400 
1.53302 
1.53205 
1.53107 
1.53010 
1.52913 
1.52816 
1.52719 
1.52622 


55 
54 
53 
5i; 
51 


.53238 
.53263 
.53288 
.53312 
.53337 


.84650 
.84635 
.84619 
.84604 
.84588 

.84573 
.84557 
.84542 
.84526 
.84511 


.62892 
.62933 
•62973 
•63014 
.63055 


•54708 
-54732 
-54756 
.54781 
.54805 


•65355 
.65397 
•65438 
•65480 
i65521_ 

• 65^563 
.65604 
.65646 
.65688 
•65729 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.53361 
.53386 
.53411 
.53435 

.53460 


.63095 
.63136 
.63177 
.63217 
.63258 

.63299 
.63340 
•63380 
.63421 
. 63462- 


1.58490 
1.58388 
1.58286 
1.58184 
1.58083 


.54829 
.54854 
.54878 
.54902 
.54927 


•83629 
•83613 
-83597 
-83581 
•83565 


1.52525 
1.52429 
1.52332 
1.52235 
1.52139 


45 
44 
43 
42 

41 


30 

21 
22 

23 
24 


.53484 
.53509 
.53534 
.52558 
.53583 

.53607 
.53632 
.53656 
.53681 
.53705 


.84495 
.84480 
.84464 
. 84448 
.84433 

.84417 
.84402 
.84386 
.84370 
-84355 


1.57981 
1.57879 
1.57778 
1.57676 
1.57575 


54951 
.54975 
.54999 
.55024 
.55048 


•83549 
•83533 
•83517 
.83501 
.83485 


.65771 
.65813 
.65854 
.65896 
.65938 


1.52043 
1.51948 
1.51850 
1.51754 
1-51658 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•63503 
.63544 
-63584 
-63625 
•83666 


1.57474 
1.57372 
1.57271 
1.57170 
1.57069 


•55072 
•55097 
•55121 
•55145 
•55163 


.83469 
•83453 
•83437 
•83421 
•83405 


.65980 
.66021 
.66063 
.66105 
.66147 


1.51562 
1.51466 
1.51370 

1-51275 
1-51179 


35 
34 
33 
32 
31 


30 

31 
32 
S3 
34 


.53730 
.53754 
.53779 
.53804 
.53828 


.84339 
.84324 
.84308 
.84292 
.84277 


.63707 
-63748 
.63789 
.63830 
•63871 


1.56969 
1.56868 
1.56767 
1.56667 
1.56566 


.55194 
.55218 
•55242 
•55266 
•55291 


•83389 
•83373 
•83356 
•83340 
•83324 


.66189 
.66230 
.66272 
.66314 
.66356 

.66398 
. 66440 
.66482 
.66524 
.66566 


1.51084 
1.50988 
1.50893 
1.50797 
1.50702 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.53853 
.53877 
.53902 
.53926 
.53951 

.53975 
.54000 
.54024 
.54049 
.54073 


.84261 
.84245 
.84230 
.84214 
.84198 


•63912 
.63953 
.63994 
.64035 
-64076 


1.56466 
1.56366 
1.56265 
1.56165 

1.56065 


.55315 
•55339 
.55363 
.55388 

-55412 

-55436 
-55460 
-55484 
-55509 
-55533 


•83308 
.83292 
.83276 
•83260 

•83244 

.83228 
•83212 
•83195 
.83179 
•83163 


1-50607 
1.50512 
1.50417 
1.50322 
1-50228 


• 25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.84182 
.84167 
.84151 
.84135 
.84120 


.64117 
•64158 
•64199 
. 64240 
-64281 

.64322 
.64363 
. 64404 
. 64446 
-64487 


1.55966 
1.55866 
1.55766 
1.55666 
1.55567 


.66608 
.66650 
.66692 
.66734 
.66776 


1.50133 
1.50038 
1.49944 
1.49849 
1-49755 

1.49661 
1.49566 
1.49472 
1.49378 
1-49284 

.1-49190 
1.49097 
1.49003 
1.48909 
1-48816 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.54097 
.54122 
.54146 
.54171 
.54195 


.84104 
.84088 
. 84072 
.84057 
•84041 


1.55467 
1.55368 
1.55269 
1.55170 
1.55071 


.55557 
.55581 
•55605 
.55630 
.55654 


•83147 
•83131 
•83115 
-83098 
-83082 

-83066 

-83050 

.83034 

83017 

83001 


.66818 
.66860 
.66902 
.66944 
.66986 


15 
14 
13 
12 
11 


50 

'51 

m 

53 
54 


.54220 
. 54244 
.54269 
.5429a 
.54317 

.54342 
■54366 
.54391 
.54415 
. 54440 

.54464 

Cos. 


.84025 
.84009 
.83994 
.83978 
.83962 


-64528 
-64569 
- 64610 
-64652 
.64693 

• 64734 
. 64775 
. 64817 
.64858 
64899 


1.54972 
1.54873 
1.54774 
1.54675 
1.54576 


.55678 
-55702 
.55726 
.55750 
.55775 

.55799 
.55823 
.55847 
.55871 
.55895 


.67028 
.67071 
.67113 
.67155 
•67197 


.10 

9 
8 
7 
6 


55 
56 
57 
58 

59 


.83946 
-83930 
•83915 
.83899 
.83883 

83867 


1.54478 
1.54379 
1.54281 
1.54183 
1.54085 


82985 

82969 

-82953 

.82938 

•82920 


.67239 
.67282 
.67324 
.67366 
•67409 


1.48722 
1.48629 
1.48536 
1.48442 
1 - 48349 


5 
4 
3 
2 
1 


60 


-64941 


1-53986 


55919 


82904 


•67451 


1-48256 


O 


' / 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



51' 



769 



6S° 



TABLE IX.- 


-Nt^tural s nes, cosines, tangents, and cotangents. 

34° 35° 




Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 




Cot. ' 1 




1 
2 
3 

.4 


.55919 
•55943 
.55968 
.55992 
.56016 


•82904 
.82887 
.82871 
.82855 
82839 


•67451 
.67493 
.67536 
.67578 
•67620 




48256 
48163 
48070 
47977 
47885 


•57358 
.57381 
.57405 
.57429 
•57453 


•81915 
•81899 
.81882 
-81865 
•81848 


•70021 
•70064 
•70107 
•70151 
.70194 




42815 
42726 
42638 
42550 
42462 


60 

59 
58 
57 
58 

5S 
54 
53 
52 
51 1 

50 

49 
48 
47 
46 

45 
44 
43 
42 
_41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 


5 
6 
7 
6 
9 


.56040 
.56064 
.56088 
.56112 
.56136 


.82822 
.82806 
.82790 
.82773 
.82757 


.67663 
.67705 
.67748 
.67790 
.67832 


■1 


47792 
47699 
47607 
47514 
47422 


.57477 
.57501 
.57524 
.57548 
.57572 


•81832 
.81815 
.81798 
.81782 
.81765 


•70238 
.70281 
.70325 
•70368 
.70412 

.70455 
.70499 
•70542 
•70586 
•70629 




42374 
42286 
42198 
42110 
42022 


10 

11 
12 
13 

14 


.56160 
.56184 
.56208 
.56232 
.56256_ 

. 56280 
.56305 
.56329 
.56353 
.56377 


.82741 
.82724 
.82708 
.82692 
.82675 


•67875 
.67917 
.67960 
.68002 
•68045 


- 

X 


47330 
47238 
47146 
47053 
46962 


•57596 
•57619 
•57643 
.57667 
.57691 


.81748 
.81731 
.81714 
■81698 
■81681 




41934 
41847 
41759 
41672 
41584 


15 
16 
17 
18 
19 


.82659 
.82643 
.82626 
.82610 
•82593 


.68088 
.68130 
.68173 
.68215 
.68258 




46870 
46778 
46686 
46595 
46503 


.57715 
.57738 
.57762 
.57786 
•57810 


.81664 
.81647 
.81631 
.81614 
■81597 


•70673 
•70717 
•70760 
•70804 
. 70848 




41497 
41409 
41322 
41235 
41148 


20 

21 
22 
23 
24 


.56401 
.56425 
. 56449 
•56473 
■56497 


.82577 
.82561 
.82544 
.82528 
•82511 


.68301 
.68343 
.68386 
.68429 
•68471 




46411 
46320 
46229 
46137 
46046 


•57833 
•57857 
•57881 
•57904 
•57928 


.81580 
.81563 
.81546 
.81530 
.81513 


.70891 
.70935 
.70979 
.71023 
.71066 




41061 
40974 
40887 
40800 
40714 


25 
26 
27 
28 
29 


•56521 
•56545 

• 56569 

• 56593 
56617 


•82495 
.82478 
. 82462 
.82446 
•82429 


•68514 
•68557 
.68600 
•68642 
•68685 




45955 
45864 
45773 
45682 
45592 


•57952 
•57976 
•57999 
•58023 
•58047 


.81496 
•81479 
.81462 
.81445 
.81428 


.71110 
•71154 
•71198 
•71242 
•71285 




.40827 
40540 
40454 
40367 
40281 


30 

31 
32 
33 
34 


• 56641 
.56665 

• 56689 

• 56713 
56736 


•82413 
•82396 
.82380 
.82363 
.82347 


.68728 
.68771 
.68814 
.68857 
.68900 




45501 
45410 
45320 
45229 
45139 


•58070 
•58094 
•58118 
•58141 
•58165 


.81412 
.81395 
.81378 
.81361 
•81344 


•71329 
•71373 
•71417 
.71461 
.71505 




40195 
40109 
40022 
89936 
39850 


30 

29 
28 
27 
26 


85 
36 
37 
88 


• 56760 
.5678<i 
.56803 
.56832 

• 568.'i6 


.82330 
.82314 
.82297 
.82281 
•82264 


.68942 
68985 
•69028 
•69071 
•69114 




45049 
.44958 
•44868 
•44778 

44688 


•58189 
•58212 
•58236 
•58260 
•58283 

•58307 
•58330 
•58354 
-58378 
■58401 


•81327 
•81310 
•81293 
.81276 
.81259 


•71549 
•71593 
•71637 
.71681 
.71725 




39764 
39679 
395'3 
39507 
39421 


25 
24 
23 
22 
21 


40 

41 

42 
43 
44 


.56880 
. 56904 
.56928 
.56952 
.56976 


•82248 
.82231 
•82214 
•82198 
•82181 


•69157 
•69200 
•69243 
•69286 
■69329 




•44598 
.44508 
.44418 
.44329 
.44239 


.81242 
•81225 
•81208 
•81191 
•81174 


.71769 
.71813 
.71857 
.71901 
•71946 




39336 
39250 
39165 
39079 
38994 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.57000 
.57024 
.57047 
.57071 
•57095 


•82165 
•82148 
.82132 
.82115 
•82098 


•69372 
•69416 
.69459 
.69502 
•69545 




•44149 
44060 
43970 
43881 
43792 


■58425 
- 58449 
■58472 
•58496 
■58519 

■58543 
.58567 
■58590 
■58614 
■58637 


•81157 
•81140 
•81123 
81106 
•81089 


■71990 
•72034 
•72078 
•72122 
•72167 




38909 
38824 
38738 
38653 
38588 


15 
14 
13 
12 
11 


60 

51 
52 
53 
54 


•57119 
.57143 
.57167 
.57191 
•57215 


•82082 
.82065 
.82048 
.82032 
•82015 


•69588 
•69631 
•69675 
•69718 
•69761 




43703 
43614 
43525 
43436 
43347 


•81072 
•81055 
•81038 
•81021 
•81004 


•72211 
•72255 
•72299 
•72344 
•72388 




38484 
38399 
38314 
38229 
38145 


10 

9 
8 
7 
6 


55 
56 
57 
58 

59 


•57238 
•57262 
•57286 
•57310 
•57334 


.81999 

•81982 

■81965 

81949 

81932 


•69804 
.69847 
•69891 
•69934 
•69977 




43258 
43169 
43080 
42992 
42903 


■58661 
■58684 
■58708 
■58731 
■58755 


•80987 
•80970 
•80953 
•80936 
■80919 


•72432 
•72477 
■72521 
•72565 
•72610 




38060 
37976 
37891 
37807 
37722 


5 

4 
3 
2 
1 


60 


•57358 


•81915 


.70021 


1 


42815 


■58779 


■80902 


•72654 


1 


37638 





-V- 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 1 


/ 



55" 



770 



54*= 



TABLE IX - 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
36" 37° 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 1 
-60182 ! 


Cos. 


Tan. 


Cot- 


r 


58779 


.80902 


.72654 


1.37638 


79864 . 


75355 


1-32704 


60 


1 


.58802 


.80885 


.72699 


1.37554 60205 1 


79846 


75401 


1-32624 


59 


2 


58826 


.80867 


.72743 


1.37470 


■60228 


79829 


75447 


1-32544 


58 


3 


•58849 


.80850 


.72788 


1.37386 


-60251 


79811 


75492 


1.32464 


57 


4 
5 


.58873 
58896 


.80833 
.80816 


.72832 


1.37302 


- 60274 


79793 


75538 
.75584 


1. 32384 


56 


.72877 


1.37218 


- 60298 


.79776 


1.32304 


55 


6 


•58920 


.80799 


.72921 


1.37134 


.60321 


.79758 


.75629 


1.32224 


54 


7 


•58943 


.80782 


•72966 


1.37050 


.60344 


-79741 


.75675 


1.32144 


53 


1 8 


•58967 


.80765 


•73010 


1-36967 


-60367 


-79723 


-75721 


1.32064 


52 


1 9 
10 


58990 


.80748 


•73055 


1.36883 


-60390 


-79706 


-75767 


1.31984 


51 


•59014 


.80730 


•73100 


1-36800 


.60414 


-79688 


.75812 


1.31904 


50 


In 


•59037 


.80713 


.73144 


1.36716 


.60437 


.79671 


-75858 


1.31825 


49 


112 


• 59061 


.80696 


•73189 


1.36633 


.60460 


.79653 


-75904 


1.31745 


48 


13 


. 59084 


.80679 


.73234 


1.36549 


.60483 


-79635 


-75950 


1-31666 


47 


15 


.59108 


.80662 
. 80644 


.73278 


1.36466 


60506 
.60529 


.79618 
•79600 


-75996 


1.31586 


46 


•59131 


.73323 


1.38383 


.76042 


1.31507 


45 


16 


.59154 


.80627 


.73368 


1.36300 


.60553 


.79583 


.76088 


1.31427 


44 


17 


•59178 


.80610 


■73413 


1.362ir 


.60576 


.79565 


•76134 


1.31348 


43 


18 


•59201 


.80593 


.73457 


1.36134 


.60599 


-79547 


.76180 


1.31269 


42 


19 


•59225 


.80576 
.80558 


.73502 
.73547 


1.36051 


.60622_ 
.60645 


-79530 


76226 
.76272 


1.31190 


41 


20 


-59248 


1.35968 


79512 


1.31110 


40 


21 


•59272 


.80541 


.73592 


1.35885 


.60668 


.79494 


.76318 


1-31031 


39 


22 


•59295 


.80524 


.73637 


1.35802 


.60691 


-79477 


.76364 


1-30952 


38 


23 


•59318 


.80507 


.73681 


1.35719 


.60714 


-79459 


.76410 


1-30873 


37 


24 
25 


•59342 


.80489 


.73726 


1-35637 


.60738 


. 79441 


-76456 


1. 30795 


36 


•59365 


.80472 


.73771 


1.35554 


.60761 


. 79424 


-76502 


1-30716 


35 


26 


•59389 


80455 


.73816 


1.35472 


.60784 


.79406 


-76548 


1.30637 


34 


27 


.59412 


.80438 


.73861 


1.35389 


.60807 


.79388 


-76594 


1.30558 


33 


28 


•59436 


•80420 


.73906 


1.35307 


.60830 


.79371 


-76640 


1.30480 


32 


29 
30 


•59459 
.59482 


.80403 


.73951 


1.35224 


.60853 


•79353 


.76686 
.76733 


1 . 30401 


31 


-80336 


.73996 


1.35142 


.60876 


-79335 


1.30323 


30 


31 


•59506 


.80368 


.74041 


1-35060 


.60899 


.79318 


-76779 


1.30244 


29 


32 


.59529 


.80351 


•74088 


1.34978 


•60922 


.79300 


-76825 


1.30166 


28 


33 


•59552 


.80334 


•74131 


1.34896 


.60945 


.79282 


-76871 


1.30087 


27 


34 


.59576 


.80316 


•74176 


1.34814 


.80968 


.79264 


-76918 


.l^ 30009 


26 


S5 


.59599 


.80299 


•74221 


1.34732 


-60991 


79247 


.76964 


1.29931 


25 


36 


.59622 


.80282 


•74267 


1.34650 


.61015 


.79229 


.77010 


1.29853 


24 


37 


.59646 


.80264 


•74312 


1.34568 • 61033 


-79211 


.77057 


1-29775 


23 


38 


.59669 


.80247 


•74357 


1-34487 61061 


-79193 


-77103 


1.29696 


22 


39 


59693 
.59716 


.80230 


. 74402 
• 74447 


1.34405 -61084 


.79176 


.77149 


1.29618 


21 


40 


.80212 


1-34323 


•61107 


-79158 


.77196 


1.29541 


30 


41 


.59739 


.80195 


•74492 


1-34242 


•61130 


- 79140 


-77242 


1.29463 


19 


42 


.59763 


•80178 


•74538 


1-34160 


.61153 


-79122 


-77289 


1.29385 


18 


43 


.59786 


.80160 


.74583 


1-34079 


•81176 


-79105 


-77335 


1.29307 


17 


44 


.59809 


.80143 


.74628 
.74674 


1.33998 -61199 
1-33916 -61222 


.79087 


.77382 


1.29229 


16 


45 


59832 


.80125 


-79069 


.77428 


1.29152 


15 


46 


•59856 


•80108 


.74719 


1-33835 .61245 


-79051 


.77475 


1.29074 


14 


47 


•59879 


.80091 


.74764 


1.33754 -61268 


.79033 


-77521 


1.28997 


13 


48 


•59902 


.80073 


.74810 


1.33673 .61291 


-79016 


-77568 


1-28919 


12 


49 


•59926 
•59949 


.80056 
.80038 


.74855 


1.33592 


.61314 
.61337 


-78998 
-78980 


-77615 


1-28842 


11 


/>0 


.74900 


1.33511 


-77661 


1.28764 


10 


51 


•59972 


.80021 


.74946 


1-33430 


.61360 


-78962 


-77708 


1.28687 


9 


52 


•59995- 


- .80003 


.74991 


1.33349 


1-61383 


-78944 


-77754 


1.28610 


8 


53 


•60019 


.79986 


.75037 


1.33288 '-61406 


-78926 


-77801 


1-28533 


'/ 


54 


.60042 


.79968 


.75082 


1-33187 f. 61429 


-78908 


-77848 


1.28456 


6 


55 


.60065 


.79951 


.75128 


1-33107 


-61451 


-78891 


-77895 


1.28379 


5 


56 


.60089 


.79934 


.75173 


1-33026 


.61474 


-78873 


-77941 


1.28302 


4 


57 


.60112 


•79916 


.75219 


1-32946 


.61497 


-78855 


-77988 


1-28225 


3 


58 


.60135 


.79899 


.75264 


1-32865 


.61520 


-78837 


-78035 


1-28148 


2 


59 


.60158 
.60182 


.79881 


. .75310 
.75355 
Cot. 


1.32785 

1-32704 

Tan. 


.61543 

.61566 

Cos. 


78819 

-78801 

Sin. 


-78082 


1.28071 


1 


60 


.79864 


-78129 


1.27994 
Tan. 





/ 


Cos. 


Sin. 


Cot. 





sa" 



7n 



sa" 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS. 
38'' 39" 


/ 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 1 


1 
/ 

60 

59 
58 
57 
56 




1 
2 
3 
4 


-61566 
•61589 
•61612 
.61635 
.61658 


.78801 
.78783 
.78765 
.78747 
.78729 


.78129 
.78175 
•78222 
.78269 
•78316 


1 
1 
1 

1 
1 


27994 
27917 
27841 
27764 
27688 


•62932 
•62955 
•62977 
•63000 
•63022 


.77715 
•77696 
. 77678 
.77660 
•77641 


•80978 
•81027 
.81075 
.81123 
.81171 




23490 
23416 
23343 
23270 
23196 


5 
6 
7 
8 
9 


■61681 
•61704 
•61726 
•61749 
.61772 

•61795 
•61818 
•61841 
•61864 
•61887 


•78711 
.78694 
.78676 
•78658 
• 78640 


•78363 
.78410 
.78457 
.78504 
.78551 


1 

1 

1^ 

1. 

1 


27611 
27535 
27458 
27382 
27306 


•63045 
•63068 
•63090 
•63113 
•63135 


.77623 
•77605 
77586 
.77568 
.77550 


.81220 
.81268 
•81316 
•81364 

•81413 




23123 
23050 
22977 
22904 
22831 


55' 

53 
52' 
51: 

50 

49 
48 
47; 
48 

45 
44 
43 
42 
41 


10 

11 
12 
13 
14 


•78622 
•78604 
.78586 
.78568 
.78550 


•78598 
•78645 
•78692 
.78739 
.78786 


1. 
1. 
1. 
1. 
1 


27230 
27153 
27077 
27001 
26925 


•63158 
•63180 
•63203 
•63225 
•63248 


.77531 
.77513 
.77494 
.77476 
.77458 


•81461 
•81510 
•81558 
•81606 
•81655 


1 


22758 
22685 
22612 
22539 
22467 


15 
16 
17 
18 
19 


•61909 
•61932 
•61955 
•61978 
•62001 


•78532 
.78514 
•78496 

• 78478 

• 78460 


■78834 
.78881 
•78928 
•78975 
•79022 


1 
1 
1 

1 
1 


26849 
26774 
26698 
26622 
26546 


•63271 
•63293 
•63316 
63338 
•63361 


.77439 
.77421 
•77402 
•77384 
•77366 


•81703 
•81752 
•81800 
•81849 
•81898 




22394 
22321 
22249 
22176 
22104 


20 

21 
22 
23 
24 


• 62024 
•62046 
•62069 
•62092 
.62115 


• 78442 

• 78424 
•78405 
•78387 
•78369 


•79070 
•79117 
•79164 
•79212 
•79259 


1 
1 
1 

1 
1 


26471 
26395 
26319 
26244 
26169 


•63383 
. 63406 
•63428 
.63451 
•63473 


•77347 
•77329 
•77310 
•77292 
•77273 


■81948 
•81995 
•82044 
•82092 
•82141 




22031 
21959 
21886 
21814 
21742 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•62138 
•62160 
•62183 
•62206 
•62229 


•78351 
•78333 
•78315 
•78297 
•78279 


•79306 
•79354 

• 79401 

• 79449 
•79496 


1 

1 

1 

1. 

1 


26093 
26018 
25943 
25867 
25792 


■63496 
•63518 
■63540 
•63563 
•63585 


•77255 
•77236 
•77218 
•77199 
•77181 


•82190 
•82238 
•82287 
•82336 
•82385 




21670 
21598 
21526 
21454 
21382 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•62251 
•62274 
.62297 
.62320 
•62342 


•78261 
•78243 
•78225 
•78206 
-78188 


•79544 
•79591 
•79639 
.79686 
•79734 


1 
1 
1 
1 
1 


25717. 

25642 

25567 

25492 

25417 


•63608 
•63630 
•63653 
•63675 
•63698 


•77162 
•77144 
•77125 
•77107 
•77088 


•82434 
•82483 
•82531 
•82580 
•82629 




21310 
21238 
21166 
•21094 
21023 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•62365 
•62388 
•62411 
• 62433 
.62456 


•78170 
•78152 
•78134 
•78116 
•78098 


•79781 
•79829 
•79877 
•79924 
•79972 


1 
1 
1 
1 
J. 


25343 
25268 
25193 
25118 
25044 


63720 
•63742 
•63765 
■63787 
■63810 


•77070 
•77051 
•77033 
•77014 
•76996 


•82678 
•82727 
•82776 
■82825 
•82874 




•20951 
•20879 
•20808 
•20736 
•20665 


25 
24 
23 
22 
, 21 


40 

41 
42 
43 
44 


•62479 
•62502 
•62524 
•62547 
- 62570 


•78079 
•78061 
.78043 
.78025 
.78007 


•80020 
•80067 
•80115 
•80163 
•80211 


1 
1 
1 
1 
1 


•24969 
•24895 
•24820 
.24746 
•24672 


•63832 
•63854 
■63877 
-63899 
■63922 


•76977 
•76959 
•76940 
•76921 
•76903 


•82923 
•82972 
•83022 
•83071 
•83120 




•20593 
•20522 
•20451 
•20379 
•20308 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•62592 
•62615 
• 63638 
•62660 
•62683 


.77988 
.77970 
•77952 
•77934 
•77916 


•80253 
•8030e 
•80354 
•80402 
•80450 


1 
1 
1 
1 

1 


•24597 

•24523 

• 24449 

24375 

24301 


63944 
63966 

■63989 
64011 

•64033 


•76884 
■76866 
■76847 
■76828 
■76810 


•83169 
•83218 
•83268 
•83317 
■83366 




•20237 

•20166 

•20095 

20024 

19953 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•62706 
•62728 
•62751 
•62774 
•62796 


•77897 
•77879 
•77861 
•77843 
•77824 


•80498 
•80546 
•80594 
•80642 
•80690 


1 
1 

1 
1 

1 


24227 
24153 
24079 
24005 
23931 


•64056 
64078 
•64100 
•64123 
•64145 


•76791 
•76772 
•76754 
•76735 
•76717 


•83415 
•83465 
•83514 
•83564 
•83613 




19882 
19811 
19740 
19669 
19599 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•62819 
•62842 
•62864 
•62887 
•62909 


•77806 
•77788 
•77769 
•77751 
•77733 


•80738 
•80786 
.80834 
.80882 
•80930 


1 
1 
1 
1 
1 


23858 
23784 
23710 
23637 
23563 


•64167 
■64190 
•64212 
.64234 
•64256 


•76698 
•76679 
•76661 
. 76642 
•76623 


83662 
•83712 
•83761 
•83811 

83860 




19528 
19457 
19387 
19316 
19246 


5 

4: 

3 
2 
1 


60 


•62932 


•77715 


80978 


1 


23490 


■64279 


•76604 


•83910 


1 


19175 





t 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



51*= 



772 



50° 



FABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 



40° 



41= 



* 

1 

2 
3 

4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 


t 


.64279 
•64301 
•64323 
•64346 
•64368 

•64390 
. 64412 
• 64435 
•64457 
. 64479 


■76604 
■76586 
■76567 
■ 76548 
■76530 


•83910 
■83960 
■ 84009 
■84059 
■84108 




19175 
19105 
19035 
18964 
18894 


•65606 
■65628 
■65650 
■65672 
■65694 


■75471 
■75452 
■75433 
■75414 
■75395 


■86929 
■88980 
■87031 
■87082 
•87133 




15037 
14969 
14902 
14834 
14767 


60 

59 
58 
57 
5fi 


■76511 
■76492 
■76473 
■76455 
■76436 


•84158 
•84208 
■84258 
■84307 
■84357 




18824 
18754 
18684 
18614 
18544 


■65716 
■65738 
■65759 
■65781 
■65803 


•75375 
■75356 
•75337 
•75318 
•75299 


•87184 
•87238 
•87287 
•87338 
■87389 




14699 
14632 
14565 
14498 
14430 


55 
54 
53 
52 
51 


•64501 
■64524 
■64546 
■64568 
■64590 


•76417 
■76398 
■76380 
■76361 
•76342 

.76323 
•76304 
•76286 
•76267 
•76248 

•76229 
•76210 
.76192 
■78173 
■76154 

■76135 
■76116 
■76097 
■76078 
■78059 


■ 84407 

■84457 
•84507 
■84556 
■84806 




18474 
18404 
18334 
18264 
18194 


•65825 
•65847 
•65869 
■65891 
■65913 


•75280 
•75261 
•75241 
• 75222 
•75203 


■87441 
■87492 
•87543 
•87595 
•87846 




14363 
14298 
14229 
14162 
14095 


50 

49 
48 
47 
4fi 


15 
16 
17 
18 
19 


■64612 
■64635 
•64657 
■64679 
•64701 

•64723 
• 64746 
•64768 
•64790 
.64812 


■84656 
-84706 
■84756 
■84806 
■84856 


™L 


18125 
18055 
17986 
17916 
17846 

17777 
17708 
■17638 
17569 
17500 


■65935 
■65956 
■65978 
■66C00 
•66022 

■ 66044 
■66066 
■66088 
■66109 
■66131 


•75184 
•75165 
•75146 
•75126 
■75107 

■75088 
75069 
•75050 
•75030 
•75011 

•74992 
.74973 
•74953 
•74934 
•74915 


•87698 
•87749 
•87801 
•87852 
•87904 




14028 
13981 
13894 
13828 
13761 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


•84906 
• i;4958 
•85006 
•85057 
•85107 


•87955 
•88007 
•88059 
•88110 
•88162 

•88204 
•88265 
•88317 
•88369 
•88421 




13694 

13627 

■13561 

■13494 

13428 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


■ 64834 
■64856 
■64878 
■64901 
■64923 


•85157 
•85207 
•85257 
■85308 
■85358 




■17430 
-17361 
■17292 
■17223 
■17154 


■66153 
■66175 
■66197 
•66218 
■66240 




■13381 
■13295 
■13228 
■13162 
13096 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


■64945 
■64967 
■64989 
■65011 
•65033 

■65055 
■65077 
■65100 
•65122 
■65144 


• 76041 
■76022 
■76003 
■75984 
■75965 


■85408 
■85453 
■85509 
■85559 
■85609 




■17085 
■17016 
■16947 
•16878 
•16309 


■66262 
•66284 
•68306 
•66327 
•66349 

•68371 
•66393 
•66414 
■66436 
■66458 

■ 66480 
■66501 
■66523 
■66545 
•66566 


•74896 
•74876 
•74857 
•74838 
■74818 


•88473 
•88524 
•88576 
•88628 
■88680 




■13029 
•12963 
•12897 
•12831 
■12765 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


■75946 
■75927 
■75908 
■75889 
•75870 

■75851 
■75832 
■75813 
■75794 
■75775 


■85660 
■85710 
■85761 
■85811 
•85862 

■85912 
■85963 
■86014 
■86064 
■86115 




•16741 
■16672 
■16603 
■16535 
■16466 


■74799 
■74780 

■ 74760 

■ 74741 

■ 74722 


■88732 
■88784 
■88836 
■88888 
•88940 




■12699 
■12633 
■12587 
■12501 
■12435 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


■65166 
65188 
■65210 
■65232 
•65254 




■16398 
■16329 
■16261 
■16192 
16124 


■ 74703 

■ 74883 

■ 74664 

■ 74644 

■ 74625 


•88992 
•89045 
•89097 
•89149 
•89201 




•12369 
-12303 
•12238 
•12172 
•12106 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


■65276 
■65298 
■65320 
■65342 
■65364 


•75756 
•75738 
•75719 
•75700 
■75680 


■86166 
86216 
■86267 
■86318 
■86368 




■16056 
■15987 
■15919 
•15851 
■15783 


■66588 
■66610 
•66632 
■68653 
■66675 


■74806 
■74586 
■74567 
■ 74548 
■74528 


■89253 
■89306 
■89358 
■89410 
■89463 




•12041 
•11975 
•11909 
•11844 
•11778 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


■65386 

■ 65408. 
■65430 

■ 65452 
•65474 


•75661 
•75642 
•75623 
•75604 
•75585 


■86419 
■86470 
■86521 
■86572 
■86623 




■15715 
■15647 
■15579 
■15511 
■15443 


■66697 
66718 
■66740 
■66762 
■66783 


■ 74509 

■ 74489 
• 74470 
•74451 
•74431 


■89515 
■89567 
■89820 
■89672 
•89725 




•11713 

■11648 

11582 

11517 

11452 


10 
9 
8 
7 
6 


55 
56 
57 
58 
59 


•65496 
•65518 
•65540 
•65562 
•65584 


•75566 
•75547 
.75528 
.75509 
•75490 


■86674 
■86725 
■86776 
■86827 
■86878 




■15375 
■15308 
■15240 
■15172 
•15104 


■86805 
•66827 
•66848 
•66870 
•66891 


•74412 
■74392 
■74373 
■74353 
■74334 


•89777 
•89830 
•89883 
•89935 
■89988 




11387 
11321 
11256 
11191 
11126 


5 
4 
3 
2 
1 


60 


.65606 


.75471 


■86929 


1 


•15037 


■66913 

Cos. 


■74314 


■90040 


1 


11081 





*""/ 


Cos. 


Sin. 


Cot. 


Tan. 1 


Sin. 


Cot. 


Tan. 


f 



49° 



773 



48" 



TABLE IX- 


-NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS 
43° 43° 


/ 


Sin. 


Cos. 


Tan, 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


"ec 

59 
58 
57 

5e 

55 
54] 
53, 
52 
51 

50 

49 
48 
47 
461 




1 
2 
3 

4 


.66913 
66935 
.66956 
.66978 
.66999 


•74314 
.74295 
•74276 
.74256 
•74237 


. 90040 
•90093 
.90146 
.90199 
.90251 


1.11061 
1 10996 
1.10931 
1.10867 
1.10802 


■68200 
.68221 
.68242 
-68264 
■68285 


-73135 
-73116 
-73096 
-73076 
.73056 


•93252 
•93306 
•93360 
•93415 
.93469 


1-07237 
1.07174 
107112 
1^07049 
1^06987 


5 
6 
7 
8 
9 


.67021 
67043 
.67064 
.67086 
.67107 


.74217 
•74198 
74178 
.74159 
.74139 


.90304 
.90357 
.90410 
.90463 
.90516 


1.10737 
1.10672 
1.10607 
1.10543 
1.10478 


68306 
-68327 
.68349 
.68370 
.68391 


-73036 
.73016 
.72996 
.72976 
.72957 

.72937 
•72917 
.72897 
.72877 
.72857 


.93524 
.93578 
.93633 
.93688 
.93742 


1.06925 
1^ 06862 
1-06800 
1.06738 
1-06676 


10 

11 
12 
13 
34 


.67129 
.67151 
.67172 
.67194 
.67215 


.74120 
.74100 
. 74080 
. 74061 
. 74041 


•90569 
.90621 
.90674 
90727 
.90781 

.90834 
.90887 
.90940 
.90993 
.91046 


1.10414 
1.10349 
1.10285 
1.10220 
1.10156 


.68412 
. 68434 
. 68455 
.68476 
. 68497 


.93797 
.93852 
.93906 
.93961 
.94016 


1-06613 
1-06551 
1^ 06489 
1^ 06427 
1- 06365 


15 
16 
17 
18 
19 


.67237 
.67258 
.67280 
.67301 
.67^23 


. 74022 
. 74002 
.73983 
.73963 
. 73944 


.1.10091 
1.10027 
1.09963 
1.09899 
1-09834 


.68518 
.68539 
.68561 
.68582 
.68603 


.72837 
.72817 
.72797 
.72777 
.72757 


.94071 
.94125 
.94180 
.94235 
.94290 


1.06303 
1.06241 
1.06179 
1.06117 
1-06056 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


•67344 
.67366 
.67387 
.67409 
•67430 


.73924 
. 73904 
. 73885 
.73865 
.73846 


.91099 
.91153 
.91206 
.91259 
.91313 


1.09770 
1.09706 
1-09642 
1.09578 
1.09514 


.68624 
.68645 
.68666 
.68688 
. 68709 


.72737 
.72717 
-72697 
■ 72677 
.72657 


.94345 
. 94400 
-94455 
.94510 
.94565 


1.05994 
1.05932 
1-05870 
1-05809 
1-05747 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


67452 
.67473 
•67495 
•67516 
.67538 


.73826 
.73806 
•73787 
•73767 
•73747 


.91366 
.91419 
.91473 
•91526 
•91580 


1.09450 
1.09386 
1.09322 
1.09258 
1-09195 


.68730 
.68751 
. 68772 
-68793 
.63814 


.72637 
.72617 
-72597 
.72577 
.72557 


.94620 
.94676 
.94731 
.94786 
-94841 


1-05685 
1-05624 
1.05562 
1-05501 
1-05439 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.67559 
•67580 
•67602 
•67623 
•67645 


.73728 
.73708 
•73688 
•73669 
.7^849 


.91633 
.91687 
.91740 
•91794 
•91847 


1^09131 
1.09067 
1.09003 
1-08940 
1-08876 


-68835 
68857 
-68378 
-68899 
-68920 


.72537 
.72517 
.72497 
. 72477 
.72457 

.72437 
.72417 
.72397 
.72377 
.72357 


-94896 
■94952 
-95007 
■95062 
95118 


1-05378 
1-05317 
1-05255 
1-05194 
1. 05133 

1-05072 
1-05010 
1 - 04949 
1-04888 
1-04827 


30 

29 
28 
21 
26 


35 
36 
37 
SB 


•67666 
67688 

•67709 
67730 
67752 


•73629 
.73610 
•73590 
73570 
•73551 


•91901 
•91955 
•92008 
92062 
•92116 


1-08813 
1-08749 
1-08688 
1-08622 
1-08559 


- 68941 
-68962 
-68983 

- 69004 
-69025 


■95173 
-95229 
-95284 
-95340 
.95395 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.67773 
•67795 
•67816 
•67837 
•67859 


.73531 
.73511 
.73491 
. 73472 
.73452 


•92170 
.92224 
.92277 
•92331 
•92385 


1-08496 
1-08432 
1-08369 
1-08306 
1-08243 


-69046 
-69067 
- 69088 
-69109 
-69130 


-72337 
.72317 
.72297 
.72277 
.72257 


.95451 
.95506 
.95562 
.95618 
.95673 


1-04766 
1-04705 
1 - 04644 
1-04583 
1-04522 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•67880 
■67901 
•67923 
•67944 
•67965 


73432 
. 73413 
•73393 
.73373 
■73353 


•92439 
.92493 
.92547 
.92601 
•92655 


1-08179 
1-08116 
1-08053 
1-07990 
1-07927 


-69151 
-69172 
•69193 
-69214 
-69235 


.72236 
•72216 
.72196 
.72176 
.72156 


.95729 
.95785 
.95841 
.95897 
.95952 


1-04461 
1-04401 
1-04340 
1-04279 
1-04218 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


. 67987 
•68008 
• 68029 
■68951 
•68072 


•73333 
73314 
•73294 
•73274 
•73254 

•73234 
•73215 
•73195 
.73175 
.73155 


•92709 
.92763 
.92817 
.92872 
.92926 


1-07864 
1-07801 
1-07738 
1.07676 
1.07613 


-69256 
.69277 
.69298 
.69319 
.69340 


.72136 
.72116 
.72095 
.72075 
.72055 


.96008 
•96064 
•96120 
•96176 
•96232 


1-04158 
1.04097 
1.04036 
1.03976 
1.03915 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•68093 
.68115 
•68136 
■68157 
.68179 


.92980 
.93034 
.93088 
•93143 
•93197 
•93252 
Cot. 


1.07550 
1.07487 
1.07425 
1.07362 
1-07299 


.69361 
69382 
- 69403 
. 69424 
■69445 


.72035 
.72015 
•71995 
.71974 
•71954 


•96288 
•96344 
•96400 
96457 
-96513 


1.03855 
1.03794 
1.03734 
1.03674 
1.03613 


5 
4 
3 
2 
1 


60 


■68200 


.73135 
Sin. 


1. 07237 


69466 


71934 


-96589 


1-03553 


<l 


■ / 


Cos. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan.. 





47* 



774 



46"* 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS 

44° 44° 



r 

"o" 

1 

2 
3 

4 

1 5 
6 
7 

8 
9 

10 

11 
12 
13 
14 


Sin. 


Cos. 

•71934 
•71914 
.71894 
•71873 
71853 


Tan. 


Cot. 


' J_l 


Sin. 


Cor. 


Tan. 


I Cot. 


r 


69466 
•69487 
•69508 
•69529 
•69549 


•96569 
•98625 
.96681 
.96738 
.96794 


1^03553 
1^03493 
!• 03433 
1^ 03372 
1^03312 


60 

59 
58 
57 
56 


30 

SI 
32 
33 

34 

35 
36 
37 
38 
39 


.70091 
.70112 
.70132 
.70153 
•70174 


•71325 
•71305 
•71284 
•71264 
•71243 




98270 
98327 
98384 
98441 
98499 




01761 
01702 
01642 
01583 
01524 


30 

29 
28 
27 
26 


•69570 
•69591 
•69612 
• 69633 
•69654 

•69675 
•69696 
•69717 
•69737 
•69758 

•69779 
.69800 
•69821 
.69842 
.69862 


•71833 
.71813 
.71792 
•71772 
.71752 


.96850 
.96907 
.96963 
.97020 
.97076 


1^03252 
1.03192 
1.03132 
1^03072 
1^03012 


55 
54 
53 
52 
51 


•70195 
.70215 
.70236 
•70257 
•70277 


•71223 
•71203 
•71182 
•71162 
,•71141 

•71121 
-71100 
•71080 
•71059 
•71039 




98556 
.98613 
•98671 
.98728 
•98786 




01465 
01406 
01347 
01288 
01229 

01170 
01112 
01053 
00994 
00935 


25 
24 
23 
22 
21 


.71732 
.71711 
•71691 
71671 
; 71650 

•71630 
•71610 
.71590 
.71569 
•71549 

•71529 
.71508 
•71488 
•71468 
•71447 


.97133 
.97189 
.97246 
•97302 
.97359 


1 02952 
1.02892 
1-02832 
1.02772 
1.02713 


50 
49 
48 
47 
46 


40 

41 
42 
43 
44 


•70298 
•70319 
•70339 
•70360 
• 70381 




98843 
•98901 
•98958 
.99016 
.99073 


20 

19 
18 
17 
1» 


15 
16 
17 
18 
19 


.97416 
.97472 
.97529 
.97586 
.97643 

■97700 
•97756 
.97813 
.97870 
•97927 


1.02653 
1.02593 
1.02533 
1.02474 
!• 02414 


45 
44 
43 
42 
41 

40 

39 
38 
37 
36 


45 
46 
47 
48 
49 


• 70401 

• 70422 
70443 

•70463 

• 70484 


•71019 
•70998 
•70078 
•70957 
70937 




.99131 
•99189 
•99247 
.99304 
•99362 




00876 
00818 
00759 
00701 
00642 


15 
14 
13 
12 
11 


80 

21 
22 
23 
24 


•69883 
•69904 
•69925 
•69946 
69966 


1^02355 
1.02295 
1.02236 
1.02176 
1.02117 


50 

51 
52 
53 

54 


. 70505 
.70525 
.70546 
•70567 
• 70587 


•70916 
•70896 

• 70875 

• 70855 
. 70834 




•99420 
•99478 
•99536 
.99594 
.99652 




•00583 
00525 
00467 
00408 
00350 


lO 

9 
8 
7 

6 


25 
26 
27 
28 
29 


•69987 

• 70008 

• 70029 

• 70049 
•70070 


.71427 
. 71407 
.71386 
.71366 
.71345 

• 71325 


•97984 
98041 
•98098 
•98155 
•98213 
.98270 


1.02057 
1.01998 
1.01939 
1.01879 
1^ 01820 


35 

34 
83 
32 
31 

30_ 

/ 


55 
56 
57 
58 
59 


•70608 
•70628 
. 70649 
•70670 
• 70690 


•70813 
•70793 

• 70772 

• 70752 
•70731 




.99710 
99768 
99826 
99884 

99942 


1 


00291 
00233 
00175 
00116 
00058 


5 
4 
3 
2 
1 


20- 


•70091 


1.01761 


60 


-70711 

Cos. 


•70711 


!• 


00000 


1. 


00000 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


' 1 


Sin. 


Cot. 


Tan. 1 


""'^ 



4^" 



775 



^° 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTg ] 
0° 1° 3° 3"* 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


^ Vers. 


Ex. sec. 

.00061 
.00062 
.00063 
.00064 
-00065 

'•00066 
•00067 
•00068 
•00069 
-00070 


Vers. 


Ex. sec. 


1 

i 
s 

'4 

•i 

£ 
IC 

11 

13 
14 

15 

le 

17 
18 
19 

20 

21 
22 
23 
24 

25 
28 
27 
28 
29 




1 
2 
3 
4 


•00000 
•00000 
•00000 
•00000 
•00000 


.00000 
.00000 
.00000 
.00000 
•00000 


.00015 
.00016 
.00016 
.00017 
•00017 




00015 
00016 
00016 
00017 
00017 


-00061 
-00062 
-00063 
-00064 
-00065 


•00137 
•00139 
•00140 
•00142 
■00143 


•00137 
•00139 
.00140 
•00142 
•00143 


5 
6 
7 
8 
9 


•00000 
.00000 
•00000 
•00000 
■00000 


.00000 
.00000 
.00000 
.00000 
.00000 

.00000 
.00001 
.00001 
.00001 
•00001 


•00018 
.00018 
•00019 
.00020 
•00020 




00018 
00018 
00019 
00020 
00020 


-00066 
.00067 
-00068 
-00069 
•00070 


•00145 
•00146 
•00148 
•00150 
•00151 


•00145 
.00147 
•00148 
.00150 
•00151 

•00153 
•00155 
•00156 
•00158 
•00159 


10 

11 
12 
13 
14 


•00000 
•00001 
■00001 
•00001 
•00001 


-00021 
-00021 
.00022 
-00023 
-00023 




•00021 
.00021 
.00022 
.00023 
-00023 


-00071 
-00073 
•00074 
•00075 
■00076 


-00072 
•00073 
•00074 
•00075 
•00076 


•00153 
•00154 
•00156 
•00158 
•00159 


15 
16 
17 
18 
19 


•00001 
•00001 
•00001 
•00001 
•00002 


.00001 
.00001 
.00001 
.00001 
.00002 


.00024 
.00024 
-00025 
-00026 
•00026 




.00024 
•00024 
•00025 
.00026 
-00026 


■00077 
•00078 
•00079 
•00081 
•00082 


•00077 
•00078 
•00079 
•00081 
-00082 


•00161 
•00162 
•00164 
•00166 
•00168 


•00161 
.00163 
•00164 
.00166 
.00168 


^0 

21 
22 
23 
24 


•00002 
•00002 
00002 
•00002 
■00002 


.00002 
.00002 
.00002 
.00002 
-00002 

.00003 
.00003 
.00003 
.00003 
.00004 


-00027 
•00028 
-00028 
-00029 
•00030 




.00027 
00028 
00028 
00029 
00030 


-00083 

00084 

00085 

■00087 

-00083 


-00083 
-00084 
-00085 
■00087 
■00088 


•00169 
• 00171 
•00173 
•00174 
■00176 


.00169 
•00171 
•00173 
•00175 
-00176 


25 
26 
27 
28 
29 


•00003 
00003 
. 00003 
•00003 
■00004 


-00031 
•00031 
-00032 
•00033 
00034 




.00C31 
00031 
00032 
00033 
00034 


-00089 
00090 
-00091 
■00093 
■00094 


-00089 
•00090 
.00091 
-00093 
-00094 


•0017o 
•00179 
00181 
•00183 
•00185 


•00178 
•00180 
•00182 
■00183 
-00185 


30 

31 
32 
33 
34 


•00004 
•00004 
• 00004 
•00005 
•00005 


.00004 
.00004 
.00004 
.00005 
.00005 


■00034 
•00035 
■00036 
•00037 
•00037 




00034 
00035 
00036 
00037 
00037 


00095 
-00096 
-00098 
-00099 
-00100 


-00095 
■00097 
•00098 
.00099 
•00100 


00187 
00188 

•00190 
00192 

■00194 


•00187 
•00189 
•00190 
■00192 
■00194 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


00005 
-00005 
•00006 
•00006 
•00006 


.00005 
.00005 
.00006 
00006 
•00006 


•00038 
-00039 
-00040 
-00041 
-00041 




00038 
00039 
00040 
00041 
00041 


-00102 
■00103 
-O0104 
-00106 
■00107 


•00102 
•00103 
•00104 
.00106 
•00107 


•00196 
•00197 
•00199 
•00201 
•00203 


•00196 
•00198 
•00200 
•00201 
00203 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•00007 
00007 

•00007 
00008 

■00008 


•00007 
•00007 
•00007 
.00008 
00008 


- 00042 
•00043 
00044 
-00045 
-00046 




00042 
00043 
00044 
00045 
00046 


■00108 
-00110 
-00111 
-00112 
■00114 


•00108 
•00110 
-00111 
-00113 
00114 


•00205 
•00207 
•00208 
•00210 
00212 


.00205 
-00207 
-00209 
-00211 
_j.00213 

00215 
.'0216 
•00218 
•00220 
•00222 

•00224 
•00226 
.00228 
•00230 
•00232 

•00234 
•00236 
•00238 
•00240 
■ 00242 


40 

41 
42 
43 
44 


45 
46 
47 
47 
49 


00009 

00009 

•00009 

•00010 

00010 


-00009 
.00009 
.00009 
•00010 
•00010 


-00047 
00048 
•00048 
• 0G049 
•00050 




00047 
00048 
00048 
00049 
00050 


-00115 
-00U7 
■00118 
-00119 
■00121 


•00115 
00117 
•00118 
•00120 
•00121 

•00122 
•00124 
•00125 
•00127 
•00128 


■00214 
■00216 
■00218 
•00220 
•00222 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•00011 
00011 

•00011 
00012 

■00012 


.00011 
.00011 
.00011 
.00012 
.00012 


•00051 
•00052 
•00053 
-00054 
-00055 


C0051 
00052 
00053 
00054 
00055 


■00122 
-00124 
-00125 
-00127 
-00128 


•00224 
•00226 
-00228 
00230 
•00232 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


■00013 
•00013 
00014 
•00014 
• 00015 


.00013 
.00013 
•00014 
.00014 
.00015 


00056 
•00057 
•00058 
•00059 
•00060 




00056 
00057 
00058 
00059 
00060 


■00130 
■00131 
■00133 
-00134 
•00136 


•00130 
.00131 
00133 
•00134 
•00136 


-00234 
-00236 
00238 
-00240 
-00242 


55 
56 
57 
58 
59 


SO 


•00015 


•00015 


•00061 


.00061 1 


00137 


•00137 


•00244 


.00244 


60 



776 



tABLE X— NATURAL VERSED SINES AND EXTERNAL SECANTS. 





4° 




•5° 






6" 






r 






1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


1 


1 

2 
3 
4 


.00244 
.00246 
.00248 
.00250 
00252 

•00254 
.00256 
.00258 
.00260 
.00262 




00244 
00246 
00248 
00250 
002 5 2_ 

00254 
00257 
00259 
00261 
00263 

00265 
00267 
00269 
00271 
00274 

00276 
00278 
00280 
00282 
00284 


•00381 
00383 
•00386 
•00388 
•00391 




00382 
00385 
00387 
00390 
00392 


•00548 
•00551 
•00554 
•00557 
•00560 




00551 
00554 
00557 
00560 
00563 


■00745 
•00749 
00752 
•00756 
•00760 




00751 
00755 
00758 
00762 
00765 




1 

2 
3 
4 


5 
6 
7 
8 
9 


•00393 
•00396 
•00398 
•00401 

• 00404 

• 00406 
•00409 
•00412 
•00114 
•00417 

•00420 
•00422 
•00425 
•00428 
•00430 




00395 
00397 
00400 
00403 
00405 


-00563 
•00566 
•00569 
•00572 
•00576 




00566 
00569 
00573 
00576 
00579 


•00763 
00767 
00770 

.00774 
00778 




00769 
00773 
C0776 
00780 
00734 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


00264 
.00266 
.00269 
.00271 

00273 




00408 
00411 
00413 
00416 
00419 


00579 
•00582 
00585 
00588 
00591 




00582 
00585 
00588 
00592 
00595 


•00781 
•00785 
•00789 
00792 
•00796 




00787 
00791 
00795 
00799 
00802 


10 

11 
12 
13 
14 


15 
16 
]7 
18 
19 


.00275 
.00277 
.00279 
.00281 
.00284 




00421 
•00424 
00427 
00429 
00432 


00594 
•00598 
•00601 
•00604 
•00607 




00598 
00601 
00604 
00608 
00611 


•00800 
• 00803 
•00807 
•00811 
•00814 




00806 
00810 
00813 
00817 
C0821 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


00286 

00288 

.00290 

.00293 

.00295 




00287 
00289 
00291 
00293 
00296 


•00433 
•00436 
•00438 

• 00441 

• 00444 




00435 
•00438 
•00440 
•00443 

00446 


•00610 
-00614 
•00617 
•00620 
00623 




00614 
00617 
00621 
00624 
00627 


•00818 
•00822 
•00825 
■00829 
00833 




C0825 
00828 
00832 
00836 
00840 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.00297 
.00299 
•00301 
.00304 
00306 




.00298 
.00300 
.00302 
.00305 
•00307 


•00447 
■ 00449 
•00452 
•00455 
•00458 




•00449 
00451 

•00454 
00457 
00460 


•00626 
•00630 
•00633 
•00636 
. 00640 




00630 
•00634 
•00637 
•00640 
•00644 


00837 

00840 

.00844 

•00848 

•00852 




00844 
00848 
00851 
00855 
00859 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


00308 

.00311 

•00313 

00315 

00317 




•00309 
•00312 
.00314 
•00316 
•00318 


•00460 
•00463 
•00466 
•00469 
00472 




•004b3 
•00465 
.00468 
•00471 
•00474 


.00643 

.00646 

.00649 

00653 

00656 




•00647 
•00650 

00654 
•00657 

00660 


•00856 
00859 
•00863 
•00867 
•00871 




00863 
00867 
00871 
00875 
00878 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


00320 

00322 

•00324 

.00327 

.00329 




•00321 
•00323 
•00328 
•00328 
•00330 


•00474 
•00477 
•00480 
•00483 
.00486 




00477 
•00480 
•00482 
•00485 
•00488 


00659 
.00663 
•00666 
•00669 
.00673 




•00664 
•00667 
•00671 
•00674 
•00677 


00875 
■00878 
00882 
00886 
00890 




00882 
00886 
008SO 
00894 
00&98 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.00332 

00334 

00336 

.00339 

■00341 




•00333 
•00335 
•00337 
•00340 
•00342 


•00489 
•00492 
•00494 
•00497 
• 00^500 




•00491 
•00494 
•00497 
•00500 
•00503 


.00676 
00680 

■00683 
00686 

■00690 




•00681 
•00884 
•00688 
•00691 
•00695 


00894 
00898 
00902 
.00906 
00909 


- 


00902 
00906 
00910 
00914 
00918 

00922 
00926 
00930 
00934 
00938 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•00343 

•00346 

•00348 

C0351 

00353 

•00356 
•00358. 
00361 
•0036d 
•00365 




•00345 
•00347 
00350 
•00352 
•00354 


•00503 
•00506 
•00509 
.00512 
.00515 




•00506 
•00509 
•00512 
00515 
.00518 


■00693 
00607 
.00700 
.00703 
.00707 




.00698 

•00701 

•00705 

00708 

00712 


.00913 
00917 
00921 
00925 

■00929 


45 
46 
47 
48 
49 


50 

5i 
52 
53 
54 




•00357 
•00359 
•00362 
•00364 
•00367 


00518 
•00521 
•00524 
.00527 
•00530 




•00521 
•00524 
.00527 
•00530 
•00533 


.00710 
•00714 
.00717 
■00721 
.00724 




.00715 
•00719 
•00722 
•C0726 
•00730 


■00933 
00937 
-00941 
.00945 
■00949 




00942 
00946 
00950 
00954 
■00958 


50 

51 
52 
53 
54 


55 
5b 
57 
58 
59* 


•00368 
•00370 
•00373 
•00375 
.00378 




•00369 

•00372 

00374 

00377 

•00379 


•00533 
•00536 
•00539 
•00542 
00545 




.00536 
•00539 
•00542 
•00545 
00548 


.00728 
•00731 
•00735 
•00738 
•00742 




•00733 
•00737 
•00740 
• 0074 4 
.00747 


.00953 
.00957 
.00961 
00965 
.00969 




00962 
■00966 

00970 
.00975 

00979 


55 
58 
57 
58 
59 


60 


•00381 


•00382 


00548 




•00551 


•00745 




•00751 


.00973 




■00983 


60 



777 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL 
8° 0° 10° ll^ 


SECANTa, 

! 


/ 


Vers. 


Ex. sec. 

.00983 
.00987 
.00991 
.00995 
.00999 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


? 

2 
3 

4 



1 

2 
3 

4 


.00973 
.00977 
-00981 
.00985 
.00989 


.01231 
.01236 
.01240 
.01245 
.01249 




01247 
01251 
01256 
01261 
01265 


-01519 
-01524 
•01529 
.01534 
.01540 


.01543 
.01548 
.01553 
.01558 
•01564 


•01837 
•01843 
•01848 
.01854 
•01860 




01872 
01877 
01883 
01889 
01895 


5 
6 
7 
8 
9 


00994 

00998 

•01002 

.01008 

.01010 


•01004 
.01008 
.01012 
•01016 
.01020 


.01254 
.01259 
•01263 
•01268 
01272 




.01270 
01275 
01279 
01284 
01289 


•01545 
•01550 
•01555 
•01560 
-01565 


.01569 
•01574 
.01579 
.01585 
.01590 


.01865 

01871 

.01876 

.01382 

01888 




.01901 
.01906 
.01912 
.01918 
.01924 


Si 

6 

7 

8 

9 


10 

11 
12 
13 
14 


•01014 
•01018 
•01022 
•01027 
•01031 


.01024 
.01029 
•01033 
.01037 
.01041 


.01277 
.01282 
•01288 
-01291 
-01296 




01294 
01298 
01303 
01308 
01313 


-01570 
.01575 
.01580 
.01586 
.01591 


.01595 
.01601 
•01606 
•01611 
•01616 


.01893 
.01899 
•01904 
.01910 
•01916 




.01930 
.01936 
.01941 
.01947 
-01953 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


• 01035 
01039 
01043 
01047 

•01052 


.01046 
.01050 
.01054 
.01059 
.01063 


-01300 
-01305 
-01310 
-01314 
01319 

-01324 
-01329 
-01333 
01338 
-01343 




01318 
01322 
01327 
01332 
01337 


•01596 
01601 
•01606 
.01612 
•01617 


•01622 
•01627 
•01633 
.01638 
.01643 

.01649 
.01654 
.01659 
.01665 
•01670 


•01921 
-01927 
.01933 
-01939 
-01944 

•01950 
.01956 
.01961 
-01967 
•01973 




.01959 
.01965 
.01971 
.01977 
.01983 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•01056 

01060 

01064 

•01069 

•01073 


.01067 
.01071 
.01076 
.01080 
.01084 




01342 
01346 
01351 
01356 
01361 


.01622 
.01627 
.01632 
.01638 
.01643 


- 


.01989 
.01995 
.02001 
.02007 
•02013 

• 02019 
.02025 
.02031 
.02037 
• 02043_ 

.02049 
.02055 
.02061 
.02067 
.02073 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.01077 
.01081 
•01086 
•01090 
•01094 


.01089 
.01093 
.01097 
.01102 
.01106 


.01348 
.01352 
.01357 
.01362 
.01367 




01366 
01371 
01376 
01381 
01386 


-01648 
-01653 
-01659 
•01664 
•01669 


.01676 
.01681 
01687 
.01692 
•01698 


01079 
•01984 
•01990 
•01996 
.02002 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•01098 
.01103 
01107 
• 01111 
•01116 


.01111 
.01115 
.01119 
.01124 
.01128 


.01371 
.01376 
.01381 
.01386 
-01391 


- 


01391 
01395 
01400 
01405 
01410 

01415 
01420 
01425 
01430 
01435 


.01675 
.01680 
•01685 
.01690 
•01696 


.01703 
.01709 
.01714 
.01720 
•01725 


•02008 
-02013 
-02019 
-02025 
•02031 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•01120 
•01124 
•01129 
.01133 
•01137 


.01133 
•01137 
.01142 
.01146 
-01151 


-01396 
-01400 
.01405 
.01410 
-01415 


.01701 
.01706 
.01712 
■01717 
•01723 


.01731 
.01736 
.01742 
.01747 
.01753 


•02037 

• 02042 

• 02048 
•02054 
■02060 




•02079 
•02085 
•02091 
.02097 
02103 

02110 
02116 
02122 
02128 
02134 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


01142 
•01146 
•01151 
•01155 
•01159 


.01155 
.01160 
•01164 
.01169 
01173 


■01420 
-01425 
-01430 
■01435 
-01439 




01440 
01445 
01450 
01455 
01461 


•01728 
01733 
•01739 
.01744 
-01750 


.01758 
.01764 
•01769 
•01775 
•01781 

01786 
.01792 
•01793 
.01803 
.01809 

.01815 
.01820 
.01826 
.01832 
01837 


•02066 
-02072 
-02078 
-02084 
-02090 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•01164 
•01168 
01173 
.01177 
•01182 


.01178 
.01182 
.01187 
.01191 
.01196 


-01444 
.01449 
-01454 
-01459 
-01464 




.01466 
01471 
01476 
01481 
01486 


-01755 
-01760 
01766 
-01771 . 
-01777 


.02095 
-02101 
-02107 
-02113 
02119 


- 


02140 
02146 
02153 
02159 
02165 

02171' 

02178 

02184 

02190 

02196 


45 
46 
47 
48 
49 


60 

51 
R2 
53 
54 


01183 
■01191 
•01195 
•01200 
•01204 


.01200 
-.01205 
.01209 
•01214 
.01219 


-01469 
■01474 
.01479 
.01484 
.01489 




01491 
01496 
01501 
01506 
01512 


■01782 
.01788 
.01793 
.0179-. 
01804 


-02125 
-02131 
-02137 
-02143 
-02149 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.01209 
•01213 
•01218 
-01222 
• 01227 


.01223 
-01228 
.01233 
.01237 

•01242 


.01494 
.01499 
.01504 
.01509 
01514 




01517 
01522 
01527 
01532 
01537 


.01810 
.01815 
•01821 
.01826 
.01832 

.01837 


.0184d 
.01849 
.U1854 
.01860 
.01866 


-02155 
-02161 
.02167 
.02173 
.02179 

02185 




02203 
02209 
02215 
G2221 
02228 


55 
56 
57 
58 
59 


60 


01231 


.01247 


.01519 




01543 


•01872 




02234 


60 



778 



TABLE X— NATURAL VERSED SINES AND EXTERNAL 
13° 13° 14° 15° 


SECANTg. 


t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 

.03061 
•03069 
.03076 
•03084 
•03091 


Vers. . 


Ex. sec. j 


/ 




\ 

3 

4 

5 
6 
7 
8 
9 


.02185 
• 02191 
02197 
•02203 
•02210 




02234 
02240 
02247 
02253 
02259 


.02563 
.02570 
.02576 
.02583 
.02589 


.02630 
.02637 
.02644 
.02651 
.02658 


.02970 
.02977 
.02985 
•02992 
.02999 


-03407 
.03415 
•03422 
•03430 
•03438 




03528 
03536 
03544 
03552 
03560 

0356b 
03576 
03584 
03592 
03601 




1 

2 
3 

4 


.02216 
.02222 
.02228 
.02234 
•02240 




02266 
02272 
02279 
02285 
02291 


•02596 
.02602 
•02609 
.02616 
.02622 


.02665 
.02672 
.02679 
.02686 
.02693 


•03006 
.03013 
.03020 
.03027 
.03034 


•03099 
.03106 
•03114 
•03121 
_^03129 

•03137 
•03144 
•03152 
•03159 
.03167 


•03445 
•03453 
03460 
•03468 
•C3476 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


•02246 
•02252 
•02258 
.02265 
•02271 




02298 
02304 
02311 
02317 
02323 


.02629 
■02635 
.02642 
.02649 
.02655 


.02700 
•02707 
■02714 
.02721 
•02728 


.03041 
.03048 
.03055 
.03063 
.03070 


• 03483 
. 03491 
•03498 
•03506 
03514 




03609 
03617 
03625 
03633 
03642 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


02277 
.02283 
-02289 
•02295 
.02302 




02330 
02336 
02343 
02349 
02356 


.02662 
.02669 
.02675 
.02682 
.02689 


.02735 
.02742 
.02749 
.02756 
•02763 


•03077 
.03084 
.03091 
.03098 
.031C6 

.03113 
.03120 
•03127 
.03134 
.03142 


.03175 
•03182 
•03190 
•03198 
•03205 


■03521 
■03529 
•03537 
•03544 
•03552 

•03560 
•03567 
•03575 
•03583 
•03590 




03650 
03658 
03666 
03674 
03683 


15 
16 
17 
18 

19 


30 

21 
22 
23 
24 


.02308 
.02314 
.02320 
.02327 
.02333 




02362 
02369 
02375 
02382 
02388 


.02696 
.02702 
.02709 
.02716 
.02722 


.02770 
.02777 
.02784 
.02791 
.02799 


•CS213 
•03221 
•03228 
•03236 
•03244 




03691 
03699 
03708 
03716 
■03724 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.02339 
.02345 
.02352 
02358 
.02364 




02395 
02402 
02408 
02415 
02421 


.02729 
.02736 
.02743 
.02749 
.02756 


.02806 
.02813 
.02820 
.02827 
.02834 


•03149 
•03156 
•03163 
03171 
•03178 


•03251 
.93259 
•03267 
•03275 
•03282 


.03598 
.03606 
.03614 
.03621 
.03629 




.03732 
03741 
.03749 
•03758 
.03766 


25 
26 
27 
28 
29 


30 

31 

32 

33 

^34 


•02370 
02377 
•02383 
.02389 
■02396 




02428 
02435 
02441 
02448 
02454 


.02763 
.02770 
.02777 
.02783 
.02790 


.02842 
.02849 
•02856 
.02863 
•02870 


-03185 
•03193 
.03200 
.03207 
•03214 


•03290 
•03298 
•03306 
•03313 
•03321 


.03637 
.03645 
.03653 
.03660 
.03668 




•03774 
.03783 
•03791 
•03799 
•03808 


30 

31 
32 
33 
34 


3u 
36 
37 
38 

39 


.02402 
02408 
.02415 
-02421 
•02427 

•02434 
•02440 
•02447 
•02453 
.02459 




02461 
02468 
02474 
02481 
02488 


•02797 
•02804 
•02811 
•02818 
•02824 


.02878 
.02885 
.02892 
.02899 
.02907 


•03222 
•03229 
•03236 
•03244 
•03251 


•03329 
•03337 
•03345 
•03353 
03360 


.03676 
■03684 
.03692 
.03699 
•03707 




•03816 
03825 
.03833 
•03842 
•03850 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 




02494 
02501 
02508 
02515 
02521 


•02831 
•02838 
•02845 
02852 
•02859 


.02914 
.02921 
.02928 
.02936 
.02943 


•03258 
•03266 
•03273 
•03281 
•03288 

•03295 
•03303 
•03310 
•03318 
•03325 

03333 
•03340 
•03347 
•03355 
.03362 


•03368 
.03376 
.03384 
.03392 
•03400 


03715 
•03723 
•03731 
•03739 
•03747 




•03858 
•03867 
.03875 
.03884 
•03892 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•02466 
•02172 
•02479 
•02485 
■02492 




02528 
02535 
02542 
02548 
02555 


-02866 
.02873 
.02880 
.02887 
.02894 


.02950 
.02958 
.02965 
.02972 
.02980 


•03408 
•03416 
•03424 
•03432 
•03439 


•03754 
•03762 
03770 
•03778 
■03786 




.03901 
•03909 
.03918 
.03927 
•03935 


45 
46 
47 
48 
49 


60 

51 
52 
53 
54 


•02498 
.02504 
•02511 
•02517 
•02524 

•02530 
•02537 
.02543 
•02550 
•02556 




02562 

02569 

.02576 

.02582 

.02589 


.02900 
.02907 
.02914 
•02921 
•02928 


•02987 
.02994 
•03002 
.03009 
.03017 


•03447 
•03455 
. 03463 
.03471 
.03479 


•03794 
•03802 
.03810 
.03818 
•03826 




•03944 
•03952 
•03961 
•03969 
■03978 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 




.02596 
.02603 
.02610 
.02617 
•02624 


•02935 
•02942 
•02949 
•02956 
•02963 


.03024 
.03032 
.03039 
03046 
.03054 


.03370 
.03377 
03385 
.03392 
.03400 


.03487 
-03495 
.03503 
.03512 
.03320 


.03834 
.03342 
•03850 
.03858 
■03866 

.03874 




•03987 
.03995 
.04004 
04013 
.04021 


55 
56 
57 
58 
59 


60 


•02563 




.02630 


.02970 


.03061 


03407 


.03528 




.04030 


60 



779 



TABLE X.- 
1€ 


-NATURAL VERSED SINES AND EXTERNAL 
;° 17" 18° 19° 


i 
SECANTS, ! 


t 


Vers. 


Px. sec. 


Vers. 


Ex. sec. 


Vers. 

04894 
04903 
04912 
04921 
04930 


Ex. sec. 


Vers. 


Ex. sec 




1 
2 
3 
4 

5 

6 : 

7' 
8 

9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 ! 
23 
24 ! 

25 
26 
27 
28 
29 

30 

31 

32 
33 
34 




1 
2 
3 
4 


■03874 
•03882 
■03890 
03898 
■03906 




04030 
04039 
04047 
04056 
04065 


•04370 
•04378 
•04387 
•04395 
• 04404 




04569 
04578 
04588 
04597 
04606 




05146 
05156 
05166 
05176 
05186 


•05448 
•05458 
.05467 
.05477 
•05486 




05762 
05773 
05783 
05794 
05805 


5 
6 
7 
8 
9 


■03914 
■03922 
.03930 
.03938 
■03946 




04073 
04082 
04091 
04100 
04108 


•04412 
•04421 
•04429 
•04438 
• 04446 




04616 
04625 
04635 
04644 
04653 • 


.04939 
04948 
.04957 
.04967 
•04976 




05196 
05206 
05216 
05226 
05236 


•05496 
•05505 
•05515 
•05524 
•05534 




05815 
05826 
05836 
05847 
05858 


10 

11 
12 
13 
14 


■03954 
•03963 
.03971 
•03979 
•03987 




04117 
04126 
04135 
04144 
04152 


.04455 
.04464 
•04472 
•04481 
.04489 




04663 
04672 
04682 
04691 
04700 


.04985 
-04994 
.05003 
.05012 
•05021 




05246 
05256 
05266 
05276 
05286 


.05543 
•05553 
•05562 
•05572 
•05582 




05869 
05879 
05890 
05901 
05911 


15 
16 
17 
18 
19 


•03995 
•04003 
•04011 
■04019 
.04028 




04161 
04170 
04179 
04188 
04197 


•04498 
•04507 
•04515 
•04524 
•04533 




04710 
04719 
04729 
04738 

04748 


•05030 
•05039 
•05048 
•05057 
•05067 




05297 
05307 
05317 
05327 
05337 


•05591 
•05601 
056]0 
•05620 
■05630 




05922 
05933 
05944 
05955 
05965 


20 

21 
22 
23 
24 


■04036 
■ 04044 
■04052 
04060 
■04069 




04206 
04214 
04223 
04232 
04241 


•04541 
•04550 
•04559 
•04567 
.04576 




04757 
04767 
04776 
04786 
04795 


•05076 
•05085 
•05094 
•05103 
•05112 




05347 
05357 
05367 
05378 
.05383 


•05639 
•05649 
•05658 
.05668 
•05678 




05976 
05987 

.05998 
06009 

•06020 


25 
26 
27 
28 
29 


•04077 
•04085 
•04093 
04102 
•04110 




04250 
04259 
04268 
04277 
04286 


•04585 
•04593 
•04602 
•04611 
•04620 




04805 
04815 
•04824 
04834 
04843 


•05122 
•05131 
•05140 
•05149 
•05158 




.05398 
.05408 
.05418 
.05429 
.05439 


•05687 
.05697 
05707 
.05716 
•05726 




•06030 
.06041 
.06052 
.06063 
.06074 


30 

31 
32 
33 
34 


•04118 
•04126 
■04135 
04143 
•04151 




04295 
04304 
04313 
04322 
04331 


•04628 
•04637 
•04646 
04655 
•04663 




04853 
04863 
04872 
04882 
.04891 


•05168 
•05177 
•05186 
.05195 
.05205 




.05449 
.05460 
.05470 
.05480 
.05490 


•05736 
•05746 
•05755 
.05765 
•05775 




.06085 
•06096 
-06107 
-06118 
•06129 


35 
36 
37 
38 
39 


•04159 
■04168 
■04176 
■04184 
■04193 




04340 
04349 
04358 
04367 
04376 


.04672 
.04681 
•04690 
•04699 
•04707 




.04901 
.04911 
04920 
•04930 
•04940 


.05214 
-05223 
•05232 
•05242 
•05251 




.05501 
•05511 
•05521 
-05532 
-05542 


•05785 
•05794 
•05804 
•05814 
■05824 




-06140 
-06151 
-06162 
-06173 
•06184 


35 
86 
37 
38 
89 


40 

41 
42 
43 
44 


•04201 
■04209 
• 04218 
•04226 
•04234 




.04385 
.04394 
.04403 
.04413 
.04422 


•04716 
•04725 
. 04734 
. 04743 
•04752 




•04950 
•04959 
•04969 
•04979 
.04989 


•05260 
•05270 
•05279 
•05288 
•05298 




.055'32 
•05563 
.05573 
•05584 
•05594 


■05833 
•05843 
•05853 
•05863 
•05873 




•06195 
-06206 
•06217 
-06228 
•06239 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•04243 
•04251 
04260 
•04268 
■04276 




•04431 
. 04440 
. 04449 
.04458 
.04468 


•04760 
•04769 
•04778 
•04787 
•04796 




.04998 
.05008 
.05018 
.05028 
.05038 


-05307 
.05316 
.05326 
.05335 
.05344 




•05604 
.05615 
.05625 
.05636 
•05646 


•05882 
•05892 
•05902 
.05912 
.05922 




-06250 
•06261 
•06272 
-06283 
.06295 


45 
46 
47 
48 
49 


60 

51 
52 
53 

54 


04285 
■04293 
■04302 
■04310 
■04319 




.04477 
•04486 
.04495 
.04504 
.04514 


•04805 
•04814 
•04323 
•04832 
•04841 




.05047 
.05057 
.05067 
.05077 
.05087 


.05354 
.05363 
.05373 
.05382 
•05391 




.05657 
.05667 
.05678 
.05688 
05699 


•05932 
•05942 
•05951 
.05961 
•05971 




.06306 

.06317 

06328 

06339 

06350 


50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 


55 
56 
57 
58 
59 


■04327 
•04336 
•04344 
•04353 
■043R1 




.04523 
.04532 
.04541 
.04551 
■04560 


•04850 
•04858 
•04867 
•04876 
04885 




.05097 
.05107 
-05116 
.05126 
.05136 


•05401 
.05410 
.05420 
.05429 
•05439 




.05709 
05720 
05730 
05741 
05751 


•05981 
•05991 
•06001 
•06011 
•06021 




06362 
06373 
06384 
06395 
06407 


60 


■04370 




•04569 


• 04894 




.05146 


•05448 




05782 


.06031 




06418 



780 



TABLE X— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



30° 



31"= 



33° 



33° 



A^ 


Vers. I 


]Xc sec. 


Vers. I 


]x. sec. 


Vers. 


Ex. sec. 


Vers. I 


jX. sec. 


/ 





.06031 


06418 


.06642 


07115 


•07282 


•07853 


•07950 • 


086S6 





1 


.06041 


06429 


.06652 


07126 


•07293 


-07866 


•07961 


08649 


1 
X 


2 


■06051 


06440 


■06663 


07138 


.07303 


.07879 


•07972 - 


08663 


2 


3 


.06061 


06452 


■06673 


07150 


.07314 


-07892 


•07984 


G8676 


3 


4 


06071 
06081 


0646J_ 
'06474 


.06684 


07162 
07174 


.07325 


07904 


•07095 . 


C8690 
08703 


4 


5 


.06694 . 


■07336 


.07917 


■ 0i;006 


5 


fi 


•06091 


06486 


-06705 . 


07186 


■07347 


07930 


■08018 


08717 


6 


7 


.06101 


06497 


.06715 


07199 


■07358 


07943 


. 08029 


08730 


7 


8 


.06111 


06508 


.06726 


07211 


•07389 


-07955 


■ 08041 


08744 


8 


9 


06121 


06520 
06531 


.06736 
06747 


07223 
07235 


■07380 


-07968 
•07981 


■08052 


08757 


9 


10 


■06131 


■07391 


■08064 


08771 


10 


11 


.06141 


06542 


.06757 


07247 


■07402 


.07994 


■08075 


08784 


11 


12 


■06151 


06554 


•06768 • 


07259 


•07413 


-08006 


.08086 


08798 


12 


13 


■06161 


06565 


.06778 


07271 


•07424 


•08019 


■08098 


08811 


13 


14 


■06171 


06577 


■06789 


07283 


•07435 


•08032 


■08109 
.08121 


08825 
08839 


14 


15 


■06181 


06588 


•06799 


07295 


•07446 


-08045 


15 


16 


■06191 


06600 


■06810 


07307 


•07457 


.08058 


.08132 


08852 


13 


17 


■06201 


06611 


■06820 


07320 


-07468 


-08071 


•08144 


08866 


17 


18 


■06211 


06622 


■06831 


07332 


•07479 


-08084 


-08155 


08880 


18 


19 


■06221 


06634 


■06841 


07344 


•07490 


•08097 


08167 


08893 


19 


30 


■06231 


06645 


■06852 


07356 


-07501 


•08109 


-08178 


08907 


30 


21 


.06241 


06657 


■06863 


07368 


-07512 


-08122 


.08190 


08921 


21 


22 


.06252 


06668 


.06873 


07380 


•07523 


•08135 


-08201 


08934 


22 


23 


.06262 


06680 


.06884 


07393 


•07534 


.08148 


.08213 


08948 


23 


24 


■06272 


06691 


.06894 


07405 


•07545 
•07556 


08161 


.08225 


08962 
08975 


24 


25 


06282 


06703 


06905 


07417 


-08174 


■08236 


25 


26 


.06292 


06715 


.06916 


07429 


•07568 


.08087 


08248 


08989 


26 


27 


.06302 


06726 


.06926 


07442 


•07579 


-08200 


■08259 


09003 


27 


28 


.06312 


06738 


.06937 


07454 


•07590 


-08213 


.08271 


09017 


28 


29 


.06323 


06749 


.06918 


07486 


•07601 


•08226 


08282 


.09030 


29 


30 


.08333 


06761 


.06958 


07479 


•07612 


-08239 


.08294 


-09044 


30 


31 


.06343 


06773 


•06969 


07491 


07623 


-08252 


.08306 


-09058 


31 


32 


.06353 


06784 


-06980 


07503 


•07634 


-08265 


.08317 


.09072 


32 


33 


.06363 


06796 


•06990 


07516 


•07645 


-08278 


■08329 


•09086 


33 


34 


.06374 


06807 


.07001 


07528 


■07657 


.08291 


■ 08340 


^90.99 
.09113' 


34 


35 


08384 


06819 


.07012 


.07540 


•07668 


•08305 


■08352 


35 


36 


06394 


06831 


.07022 


07553 


•07679 


.08318 


■08364 


09127 


36 


37 


■06404 


06843 


•07033 


07565 


•07690 


.08331 


.08375 


•09141 


37 


38 


■06415 


06854 


.07044 


07578 


07701 


•08344 


•08387 


■09155 


38 


39 


■06425 


06866 
06878 


■07055 
■07065 


07590 


•07713 


•08357 
•08370 


•08399 
•08410 


09169 
09183 


39 


40 


■06435 


07J02 


■07724 


40 


41 


■06445 


06889 


■07076 


07615 


-07735 


•08383 


•08422 


09197 


41 


42 


■06456 


06901 


.07087 


07627 


•07746 


08397 


08434 


09211 


42 


43 


■06466 


06913 


■07098 


07040 


-07757 


•08410 


• 08445 


09224 


48 


44 


■08476 


06925 


■07108 


07652 
07665 


-07769 


•08423 


•08457 


09238 


44 


45 


06486 


06936 


.07119 


07780 


-08438 


08469 


09252 


45 


46 


■06497 


06948 


-07130 


07677 


•07791 


• 08449 


-08481 


09266 


46 


47 


■06507 


06960 


•07141 


07690 


•07802 


-08463 


-08492 


09280 


47 


48 


■06517 


06972 


•07151 


07702 


07814 


08476 


-08504 


0S294 


48 


49 


.06528 


06984 


•07162 


07715 


•07825 


-0848i, 


-08516 


09308 


49 


50 


.06538 


06995 


•07173 


07727 


•07836 


-08503 


-08528 


09323 


50 


51 


■06548 


07007 


•07184 


07740 


07848 


•08518 


•08539 . 


09337 


51 


52 


■06559 


07019 


.07195 


07752 


07859 


.08529 


•08551 . 


09351 


52 


53 


06569 


07031 


.07206 


07765 


•07870 


•08542 


-08563 


09365 


53 


64 


■06580 


07043 


•07216 


07778 


•07881 


-08556 . 


-08575 


09379 


54 


55 


■06590 


07055 


■07227 


07790 


-07893 


-08569 


•08586 


09393 


55 


56 


.06600 


07067 


■07238 


07803 


•07904 


-08582 


08598 


09407 


56 


57 


06611 


07079 


■07249 


07816 


•07915 


-08596 


08610 


09421 


57 


58 


■06621 


07091 


•07260 


07828 


•07927 


•08069 


•08622 


09435 


58 


59 


■06632 


07103 
07115 


■07271 


07841 


07938 


•08623 
• 08636 


08634 


09449 


59 


60 


06642 


■07282 


07853 


•07950 


•08845 • 


09464 


60 



781 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL 
24° 25° 26° 27° 


SECANTS. 


/ 


Vers. 


Ex. sec, 

.09464 
.09478 
.09492 
.09506 
•09520 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


~ I 
1 
2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 




1 

2 
3 

4 


08645 
.08657 
.08669 
.08681 
.08693 


•09369 
•09382 
•09394 
.09406 
09418 




10338 
10353 
10368 
i0383 
10398 


10121 
.10133 
.10146 
.10159 

10172 




11260 
11276 
11292 
11308 
11323 


•10899 
10913 
•10926 
•10939 
.10952 




12233 
1224b 
12266 
12283 
12299 


5 
6 
7 
8 
9 


.08705 
.08717 
. 08728 
.08740 
.08752 

.08764 
.08776 
.08788 
.088C0 
.08812 


•09535 
.09549 
•09563 
.09577 
.09592 


•09431 
• 09443 
•09455 
09468 
•09480 




10413 
10428 
10443 
10458 
10473 


•10184 
•10197 
.10210 
.10223 
•10236 




11339 
11355 
11371 
11387 
11403 


•10965 
•10979 
■10992 
■11005 
•11019 




12316 
12333 
12349 
12366 
12383 


10 

11 
12 
13 
14 


.09606 
.09620 
.09635 
.09649 
.09663 


.09493 
•09505 
•09517 
•09530 
.09542 




10488 
10503 
10518 
10533 
10549 


.10248 
..10261 
•10274 
.10287 
.10300 




11419 
11435 
11451 
11467 
11483 


•11032 
•11045 
•11058 
11072 
-11085 




12400 
12416 
12433 
12450 
12467 


15 
16 
17 
18 

19 


.08824 
.08838 
.08848 
•08860 
08872 


.09678 
.09^92 
.09707 
.09721 
^09735 


09554 
.09567 
.09579 
•09592 
.09604 




10564 
10579 
10594 
10609 
10625 


•10313 
•10326 
•10338 
•10351 
•10364 




11499 
11515 
11531 
11547 
11563 


■11098 
•11112 
•11125 
■11138 
■11152 




12484 
12501 
12518 
125b4 
12551 


20 

21 
22 
23 
24 


•08834 
•08896 
•08908 
•08920 
•08932 


.09750 
.09764 
.09779 
.09793 
.09808 


.09617 
.09629 
•09642 
.09654 
•09666 




10640 
10655 
10670 
10686 
30701 


•10377 
•10390 
-10403 
.10416 
10429 




11579 
11595 
11611 
11627 
11643 


■11165 
•11178 
•11192 
•11205 
-11218 




12568 
12585 
12602 
12619 
12636 


25 
26 
27 
28 
29 


•08944 
•08956 
•08968 
•08980 
.08992 


.09822 
.09837 
.09851 
.09866 
.09880 


.09679 
.09691 
.09704 
.09716 
•09729 




10716 
10731 
10747 
10762 
10777 


• 10442 
.10455 
•10468 
•10481 

• 10494 




11659 
11675 
11691 
11708 
11724 


•11232 
•11245 
•11259 
•11272 
•11285 




12653 
12670 
12687 
•12704 
12721 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•09004 
•09016 
•09028 
.09040 
•09052 


.09895 
.09909 
.09924 
•09939 
•09953 


•09741 
•09754 
•09767 
•09779 
•09792 




1079a 
10808 
10824 
10839 
10854 


•10507 
•10520 
•10533 
•10546 
.10559 




11740 
11756 
11772 
11789 
11805 


•11299 
•11312 
•11326 
•11339 
•11353 




12738 
12755 
12772 
12789 
12807 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.09064 
.09076 
.09089 
•09101 
.09113 


•09968 
•09982 
.09997 
•10012 
•10026 


•09804 

09817 

09829 

•09842 

.09854 

.09867 
.09880 
.09392 
.09905 
•09918 




10870 
10885 
10901 
10916 
10932 


.10572 
.10585 
.10598 
.10611 
•10624 

.10637 
.10650 
.10663 
■10676 
■10689 




11821 
11838 
11854 
11870 
11886 


•11366 
•11380 
•11393 
•11407 
•11420 




12824 
.12841 
•12858 
.1..875 

12892 


35 
36 
37 
38 

39 


40 

41 
42 
43 
44 


•09125 
•09137 
.09149 
.09161 
.09174 


.10041 
.10055 
•10071 
.10085 
.10100 




10947 
10963 
10978 
10994 
11009 




11903 
•11919 

11936 
•11952 
•11968 


•11434 
•11447 
•11461 
•11474 
■11488 




12910 

.12927 

•12944 

12961 

12979 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


-09186 
.09198 
09210 
•09222 
•09234 


.10115 
.10130 
.10144 
.10159 
.10174 


■09930 
•09943 
•09955 
•09968 
•09981 




11025 
11041 
11056 
•11072 
11087 


•10702 
•10715 
•10728 
■10741 
■10755 




•11985 
12001 
12018 
12034 
12051 


■11501 
•11515 
■11528 
■11542 
■11555 




12996 
13013 
13031 
13048 
13065 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.03247 
09259 
09271 
09283 

.09296 


.10189 
.10204 
.10218 
.10233 
.10248 


■09993 
•10006 
• 10019 
•10032 
•10044 




•11103 

.11119 

•11134 

11150 

11166 


■10768 
■10781 

10794 
■10807 

10820 




12067 
12084 
12100 
12117 
12133 


11569 

-11583 

11596 

11610 

■11623 




13083 
13100 
13117 
13135 
13152 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•09308 

•09320 

•09332 

09345 

09357 


.10263 
.10278 
.10293 
.10308 
.10323 


•10057 
•10070 
•10082 
•10095 
10108 

•10121 




.11181 
.11197 
.11213 
.11229 
11244 


.10833 
•10847 
•10860 
•10873 
.10886 




12150 
12166 
12183 
12199 
12218 


•11637 
•11651 
.11664 
•11678 
•11692 




13170 
13187 
13205 
13222 
13240 


55 
56 
57 
58 
59 


60 


•09369 


.10338 




11260 


•10899 




12233 


-11705 




13257 


60 



782 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS* 





28' 


3 


29° 


30° 


31° 




t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.11705 
•11719 
.11733 
.11746 
.11760 




13257 
13275 
13292 
13310 
13327 


.12538 
•12552 
•12566 
•12580 
•12595 




14335 
14354 
14372 
14391 
14409 


•13397 
.13412 
.13427 
•13441 
.13456 


•15470 
.15489 
.15509 
.15528 
.15548 


•14283 
•14298 
.14313 
.14328 
•14343 


•16663 
.16684 
.16704 
.16725 
.16745 




1 
2 
3 

4 


5 
6 
7 
8 
9 


.11774 
•11787 
•11801 
•11815 
. 11828 




13345 
13362 
13380 
13398 

13415 


•12609 
•12623 
•12637 
•12651 
•12665 




14428 
14446 
14465 
14483 
14502 


•13470 
•13485 
•13499 
•13514 
•13529 


.15567 
.15587 
•15606 
•15626 
■15645 


• 14358 
•14373 
•14388 

• 14403 
- 14418 


.16766 
.16786 
.16806 
.16827 
.16848 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•11842 
•11856 
•11870 
-11883 
•11897 




13433 
13451 
13468 
13486 
13504 


•12679 
•12694 
•12708 
•12722 
•12736 




14521 
.14539 
.14558 
.14576 

14595 


•13543 
•13558 
•13573 
•13587 
-13602 


•15665 
•15684 
•15704 
•15724 
•15743 


- 14433 
- 14449 
- 14464 
- 14479 
- 14494 


.16868 
.16889 
-16909 
.16930 
.16950 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


.11911 
.11925 
.11938 
.11952 
•11966 


— 


13521 
13539 
13557 
13575 
13593 

13610 
13628 
13646 
13664 
13682 


•12750 
•12765 
.12779 
.12793 
•12807 

•12822 
.12836 
.12850 
.12864 
•12879 




.14614 
.14632 
.14651 
.14670 
14689 


-13616 
■13631 
-13646 
■13660 
.13675 


•15763 
•15782 
.15802 
.15822 
•15841. 

.15861 
.15881 
.15901 
.15920 
•15940 


•14509 
•14524 
•14539 
•14554 
■14569 


.16971 
.16992 
.17012 
•17033 
■17054 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.11980 
.11994 
.12007 
.12021 
.12035 




.14707 
.14726 
.14745 
.14764 
14782 


-13690 
-13705 
-13719 
-13734 
■13749 


■14584 
•14599 
•14615 
•14630 
•14645 


•17075 
•17095 
.17116 
•17137 
•17158 


20 

21 
22 
23 

24 


25 
26 
27 
28 
29 


• 12049 
•12063 
.12077 
■12091 
12104 




13700 
13718 
13735 
13753 
13771 


•12893 
•12907 
•12921 
•12936 
•12950 




.14801 
•14820 
•14839 
•14858 
14877 


■13763 
•13778 
•13793 
•13808 

•13822 


•15960 
•15980 
•16000 
•16019 
•16039 


•14660 
•14675 
•14690 
•14706 
•14721 


•17178 
.17199 
.17220 
•17241 
•17262 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•12118 
•12132 
•12146 
-12160 
12174 




13789 
13807 
13825 
13843 
13861 


•12964 
.12979 
.12993 
.13007 
.13022 




14896 
14914 
14933 
14952 
14971 


■13837 
•13852 
•13867 
•13881 
■13896 

•13911 
■13926 
•13941 
■13955 
-18970 

13985 

■ 14000 
•140] 5 

■ 14030 
-14044 


•16059 
•16079 
•16099 
•16119 
•16139 


•14736 
•14751 
•14766 
•14782 
•14797 


-17283 
•17304 
•17325 
•17346 
•17367 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•12188 
•12202 
•12216 
•12230 
• 12244 

.12257 
.12271 
.12285 
.12299 
.12313 




13879 
13897 
13916 
13934 
13952 


•13036 
•13051 
.13065 
-13079 
■13094 

■13108 
.13122 
.13137 
.13151 
•13168 




14990 
15009 
15028 
15047 
15066 


•16159 
•16179 
•16199 
•16219 
•16239 


.14812 
•14827 
• 14843 
.14858 
■ 14873 1, 


•17388 
•17409 
•17430 
•17451 
-17472 

-17493 
•17514 
.17535 
.17556 
.17577 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 




13970 
13988 
34006 
14024 
14042 




15085 
1510^ 
15124 
15143 
15162 


.16259 
•16279 
•16299 
•16319 
■16339 


•14888 
•14904 
•14ai9 
•14934 
■14949 


40 

41 
42 
43 

44 


45 
46 
47 
48 
49 


•12327 
.]234i 
.12355 
.12369 
12383 




14061 
14079 
14097 
14115 
14134 


•13180 
13195 
•13209 
•13223 
•13238 




15181 
15200 
15219 
15239 
15258 


14059 

14074 

-14089 

-14104 

14119 


16359 
•16380 
•16490 
•16420 
• 16440 


•14965 
•14980 
•14995 
•15011 
■15026 


.17598 
.17620 
.17641 
.17662 
• 1768-3 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.1239*; 
. 124H 
.12425 
.12439 
•12454 




14152 
14170 
34188 
14207 
14225 


•13252 
•13267 
•13281 
•13296 
-13310 




15277 
15296 
15315 
15335 
15354 


-14134 
-14149 
-14164 
-14179 
•14194 


•16460 
16481 
•16501 
.16521 
■16541 


15041 
•15057 
•15072 
•15087 
■15103 


•17704 
•17726 
•17747 
•17768 
■17790 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•12468 
.12482 
.12496 
.12510 
•12524 




14243 
14262 
14280 
14299 
14317 


.13325 
.13339 
.13354 
.13368 
.13383 




15373 
15393 

15412 
15431 
15451 


•14208 
•14223 
.14238 
.14253 
■14268 


.16562 
.16582 
.16602 
.16623 
■16643 


15118 
•15134 
■15149 
•15164 
■15180 


•17811 
.17832 
.17854 
.1787^ 
•17896 


55 
56 
57 
58 
59 


60 


.12538 




.14335 


•13397 




15470 


•14283 


.16663 


•15195 


•17918 


60 



783 



TABLE X— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



33= 



33° 



34' 



IC-O 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ . 




1 

2 
3 
4 


•15195 
.15211 
.15226 
.15241 
.15257 




.17918 
•17939 
.17961 
.17982 
.18004 


.16133 
•16149 
•16165 
•16181 
•16196 




.19236 
.19259 
•19281 
•19304 
•19327 


.17096 
•17113 
•17129 
•17145 
•17161 




.20622 
.20645 
.20669 
-20693 
.20717 


; 18085 
-18101 
-18118 
.18135 
•18152 

•18168 
•18185 
•18202 
•18218 
•18235 




•22077 
.22102 
-22127 
-22152 
-22177 



1 
2 
3 

4 


5 
6 
7 
8 
9 


.15272 
.15288 
.15303 
.15319 
.15334 




.18025 
.18047 
•18068 
•18090 
•18111 


.16212 
.16228 
-16244 
•16260 
•16276 




.19349 
.19372 
.19394 
.19417 
.19440 


•17178 
•17194 
•17210 
•17227 
•17243 




.20740 
.20764 
-20788 
-20812 
-20836 




-22202 
-22227 
-22252 
-22277 
•22302 


9 
6 
7 
8 

9 


10 

11 
12 
13 
14 


.15350 
.15365 
.15381 
.15396 
•15412 




•18133 
•18155 
•18176 
•18198 
•18220 


•16292 
•16308 
•16324 
•16340 
•16355 




.19463 
.19485 
.19508 
.19531 
.19554 


•17259 
•17276 
.17292 
.17308 
.17325 




-20859 
-20883 
-20907 
-20931 
-20955 


•18252 
•18269 
•18286 
•18302 
•18319 




• 22327 
•22352 
-22377 
. 22402 
. 22428 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•15427 
• 15443 
•15458 
•15474 
•15489 




•18241 
•18263 
•18285 
.18307 
•18328 


•16371 
■16387 
•16403 
• 16419 
•16435 




.19576 
.19599 
.19622 
.19645 
•19668 


•17341 
•17357 
.17374 
•17390 
•17407 




-20979 
-21003 
-21027 
-21051 
-21075 


•18336 
•18353 
•18369 
•18386 
•18403 




.22453 
.22478 
.22503 
.22528 
.22554 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•15505 
•15520 
•15536 

• 15552 

• 15567 




•18350 
•18372 
•18394 
.18416 
•18437 


•16451 
•16467 
•16483 
•16499 
16515 




•19691 
•19713 
.19736 
.19759 
.19782 


•17423 
.17439 
.17456 
•17472 
•17489 




-21099 
-21123 
-21147 
-21171 
.21195 


•18420 
•18437 
•18454 
•18470 
•18487 




.22579 
.22604 
.22629 
.22655 
.22680 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•15583 
•15598 
•15614 
•15630 
•15645 




.18459 
18481 
.18503 
.18525 
.18547 


•16531 
•16547 
•16563 
•16579 
16595 




.19805 

•19828 

19851 

19874 

19897 


•17505 
•17522 
•17538 
•17554 
•17571 




.21220 
.21244 
.21268 
.21292 
.21316 


•185C4 
•18521 
•18538 
•18555 
.18572 

•18588 
•18605 
•18622 
•18639 
•18656 




.22706 
-22731 
-22756 
■22782 
-22807 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•15661 
•15676 
•15692 
•15708 
•15723 




.18569 
.18591 
.18613 
.18635 
.18657 


•16611 

16627 

•16644 

•16660 

16676 




19920 
19944 
19967 
19990 
20013 


•17587 
•17604 
•17620 
•17637 
•17653 




.21341 
.21365 
-21389 
-21414 
-21438 




-22833 
•22858 
•22884 
•22909 

•22935 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•15739 
•15755 
•15770 
•15786 
•15802 




.18679 
.18701 
.18723 
.18745 
•18767 


•16692 
•16708 
•16724 
•16740 
•16756 




20036 
20059 
20083 
20106 
20129 


•17670 
-17686 
•17703 
•17719 
17736 




21462 
21487 
21511 
21535 
21560 


•18673 
•18690 
•18707 
•18724 
18741 




22960 
22986 
23012 
23037 
23063 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•15818 
•15833 
•15849 
•15865 
•15880 




•18790 

•18812 

•18834 

18856 

18878 


•16772 
•16788 
•16805 
•16821 
.16837 




20152 
20176 
20199 
20222 
20246 


•17752 
•17769 
•17786 
•17802 
•17819 




21584 
21609 
21633 
21658 
21682 


•18758 
18775 
•18792 
•18809 
•18826 




23089 
23114 
23140 
23166 
23192 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•15896 
15912 
•15928 
•15943 
•15959 




18901 
18923 
18945 
18967 
18990 


.16853 
.16869 
•16885 
•16902 
•16918 

•16934 
•16950 
•16966 
•16983 
•16999 




20269 
20292 
20316 
20339 
20363 


•17835 
•17852 
•17868 
•17885 
•17902 

•17918 
•17935 
•17952 
•17968 
•17985 

•18001 
•18018 
•18035 
•18051 
•18068 




21707 
21731 
21756 
21781 
21805 


•18843 
•18860 
•18877 
•18894 
•18911 




23217 
23243 
23269 
23295 
23321 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•15975 
•15991 
16006 
•16022 
• 18038 




19012 
19034 
19057 
19079 
19102 




20386 
20410 
20433 
20457 
20480 




21830 
21855 
21879 
21904 
21929 


•18928 
•18945 
•18962 
•18979 
•18996 




23347 
23373 
23399 
23424 
23450 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•16054 
•16070 
•16085 
• 16101 
•16117 




19124 
19146 
19169 
19191 
19214 


•17015 
•17031 
•17047 
•17064 
•17080 




20504 
20527 
20551 
20575 
20593 




21953 
21978 
22003 
22028 
22053 


-19013 
•19030 
•19047 
•19064 
•19081 




23476 
23502 
23529 
23555 
23581 


55 
56 
57 
58 
59 


60 


•16133 




19236 


.17096 




20622 


.18085 




22077 


•19098 




23607 


60 



784 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS 



3G° 



37' 



38' 



39° 



/ 


Vers. 


Ex. sec. 


Vers. 

•20136 
•20154 
•20171 
•20189 
•20207 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


» 




1 

3 
4 


•19098 
•19115 
•19133 
•19150 
.19167 


•23607 
.23633 
•23659 
•23685 
•23711 


•25214 
•25241 
•25269 
-25296 
• 25324 


•21199 
•21217 
•21235 
•21253 
•21271 


.26902 
.26931 
.26960 
•26988 
.27017 

.27046 
.27075 
.27104 
.27133 
■27162 


22285 
•22304 
•22322 
■22340 
■22359 


.28676 
•28706 
•28737 
•28767 
•28797 




1 
2 

3 

4 


5 
6 
7 
8 
9 


•19184 
•19201 
•19218 
•19235 
.19252 


.23738 
.23764 
.23790 
.23816 
.23843 


•20224 
•20242 
•20259 
•20277 
•20294 


•25351 
•25379 
•25406 
.25434 
_^2546^ 

•25489 
•25517 
•25545 
•25572 
•25600 

•25628 
•25656 
•25683 
•25711 
■25739 


•21289 
•21307 
•21324 
•21342 
•21360 


•22377 
•22395 
■22414 
•22432 
■22450 


•28828 
•28858 
•28889 
•28919 
•28950 

•28980 
•29011 
•29042 
•29072 
•29103 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•19270 
•19287 
•19304 
•19321 
•19338 . 


.23869 
.23895 
•23922 
.23948 
.23975 


•20312 
•20329 
•20347 
.20365 
.20382 


•21378 
•21396 
•21414 
■21432 
■21450 


.27191 
.27221 
.27250 
.27279 
■27308 


•22469 
•22487 
•22506 
•22524 
•22542 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•19358 
•19373 
•19390 
•19407 
•19424 


.24001 
.24028 
•24054 
•24081 
•24107 


. 20400 
.20417 
•20435 
•20453 
•20470 


.21468 
.21486 
.21504 
.21522 
.21540 


.27337 
.27366 
.27396 
.27425 
•27454 


•22581 
•22579 
•22598 
•22616 
•22634 


•29133 
•29164 
•29195 
•29226 
.29256 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•19442 
•19459 
•19476 
•19493 
•19511 


•24134 
.24160 
.24187 
•24213 
.24240 

.24287 
•24293 
•24320 
•24347 
•24373 


•20488 
•20506 
•20523 
•20541 
•20559 


•25767 
•25795 
•25823 
•25851 
•25879 


.21558 
■21576 
.21595 
.21613 
.21631 


■27483 
.27513 
•27542 
•27572 
•27601 


■22653 
•22871 
•22690 
•22708 
•22727 


•29287 
•29318 
•29349 
•29380 
•29411 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•19528 

•19545^ 

•19562 

•19580 

•19597 


•20576 
•20594 
•20612 
•20629 
•20647 


•25907 
•25935 
•25963 
•25991 
•26019 


■21649 
.21667 
■21685 
•21703 
•21721 


.27630 
.27660 
•27689 
•27719 
•27748 


•22745 
•22764 
■22782 
•22801 
■22819 


•29442 
•29473 
.29504 
.29535 
.29566 

•29597 
•29628 
•29659 
•29690 
-29721 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•19614 
•19832 * 
•19649 
•19686 
•19884 


• 24400 

• 24427 
•24454 
•24481 
•24508 


•20665 
■20682 
■20700 
•20718 
•20736 


•26047 
•26075 
•26104 
•26132 
•26160 

•26188 
•26216 
•26245 
•26273 
■26301 


•21739 
•21757 
•21775 
.21794 
■21812 


•27778 
•27807 
•27837 
•27867 
•27896 


■22838 
•22856 
•22875 
•22893 
•22912 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•19701 
•19718 
•19736 
•19753 
•19770 


•24534 
•24561 
•24588 
•24615 
• 24642 


.20753 
•20771 
•20789 
•20807 
■20824 


.21830 
•21848 
■21866 
■21884 
■21902 

■21921 
■21939 
•2]957 
■21975 
•21993 


•27926 
•27956 
•27985 
•28015 
28045 

•28075 
•28105 
•28134 
•28164 
•28194 


•22930 
•22949 
■22967 
■22986 
23004 


•29752 
•29784 
•29815 
•29848 
•29877 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•19788 
•19805 
•19822 
•19840 
•19857 


•24669 
•24696 
•24723 
•24750 
•24777 


•20842 
.20860 
•20878 
•20895 
•20913 


■26330 
•26358 
•26387 
•26415 
•28443 


■23023 
■23041 
•23060 
•23079 
•23097 


•29909 
•29940 
•29971 
.30003 
.30034 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.19875 
•19892 
•19909 
•19927 
• 19944 


• 24804 
•24832 

24859 
•24886 

• 24913 


•20931 
•20949 
■20967 
■20985 
•21002 


•26472 
•26500 
•26529 
•26557 
•26586 

•28615 
.26643 
•26672 
•26701 
•26729 


.22012 
.22030 
•22048 
■22066 
•22084 


•28224 
•28254 
•28284 
•28314 
•28344 


•23116 
•23134 
•23153 
23172 
•23190 


.30066 
.30097 
.30129 
.30160 
.3C192 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•19962 
• 19979- 
•19997 
•20014 
•20032 


• 24940 
•24967 

• 249,95 
. 25022 
.25049 


•21020 
•21038 
•21056 
•21074 
■21092 


•22103 
•22121 
•22139 
•22157 
•22176 


•28374 
• 28404 
•28434 
•28464 
•28495 


•23209 
•23228 
■23246 
.23265 
■23283 


.30223 
.30255 
.30287 
•30318 
•30350 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


• 20049 
•20066 
■20084 
•20101 
•20119 


•25077 
•25104 
•25131 
•25159 
.25186 


•21109 
•21127 
•21145 
•21163 
■21181 

•21199 


•26758 
.26787 
•28815 
•26844 
.26873 


.22194 
.22212 
.22231 
.22249 
■22267 

.22285 


•28525 
•28555 
•28585 
•28615 
■28646 


.23302 
•23321 
23339 
•23358 
.23377 


.30382 
•30413 
•30445 
•30477 
•30509 


55 
56 
57 
56 
-59 


60 


•20136 


.25214 


.26902 


•28676 


.23396 


•30541 


60 



785 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS, 



40° 



41° 



43= 



43*' 



/ 


-Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


# ■■ 




1 

2 
3 

4 


•23396 
.23414 
.23433 
.23452 
•23470 


.30541 
.30573 
.30605 
.30636 
.30668 


.24529 
.24548 
•24567 
.24586 
.24605 


.32501 
.32535 
.32568 
.32602 
.32636 


.25686 
.25705 
.25724 
.25744 
.25763 


.34563 
.34599 
.34634 
• .34669 
•34704 


.26865 
.26884 
.26904 
.26924 
.26944 


.36733 
.36770 
.36807 
.36844 
•36881 

.36919 
•36956 
•36993 
.37030 
.37068 




1 
2 
3 

4 


5 
6 
7 
8 
9 


.23489 
.23508 
•23527 
.23545 
.23584 


.30700 
.30732 
•30764 
.30796 
.30829 


.24625 
.24644 
.24663 
.24682 
.24701 


•32669 
.32703 
.32737 
.32770 
_^32804 

.32838 
.32872 
.32905 
.32939 
.32973 


.25783 
.25802 
.25822 
-25841 
•25861 


.34740 
.34775 
.34811 
.34846 
.34882 


.26964 
.26984 
.27004 
.27024 
.27043 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


■23583 
.23602 
•23620 
.23639 
.23658 


.30861 
.30893 
.30925 
.30957 
.30989 


.24720 
.24739 
.24759 
.24778 
.24797 


.25880 
.25900 
.25920 
.25939 
.25959 

.25978 
.25998 
.26017 
.26037 
.26056 

.26076 
.26096 
.26115 
.26135 
.26154 


.34917 
.34953 
.34988 
.35024 
.35060 


27063 
-27083 
.27103 
.27123 
•27143 


.37105 
.37143 
.37180 
.37218 
-37255 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


•23677 
.23696 
•23714 
.23733 
.23752 


.31022 
.31054 
.31086 
.31119 
.31151 


.24816 
.24835 
.24854 
.24874 
.24893 


.33007 
.33041 
.33075 
.33109 
.33143 


.35095 
.35131 
.35167 
.35203 
.35238 


.27163 
.27183 
-27203 
.27223 
.27243 


.37293 
.37330 
•37368 
•37406 
-37443 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.23771 
.23790 
.23808 
.23827 
.23846^ 

.23865 
•23884 
•23903 
•23922 
•23941 


.31183 
.31216 
.31248 
.31281 
.31313 


.24912 
.24931 
.24950 
.24970 
.24989 


.33177 
.33211 
.33245 
.33279 
.33314 

•33348 
•33382 
•33416 
•33451 
.33485 


.35274 
.35310 
.35346 
.35382 
.35418 


.27263 
-27283 
-27303 
.27323 
.27343 


.37481 
.37519 
.37556 
.37594 
-37632 

.37670 
.37708 
.37746 
.37784 
.37822 


30 

21 

22 
23 
24 


25 
26 
27 
28 
29 


.31346 
.31378 
.31411 
.31443 
.31476 


.25008 
.25027 
.25047 
.25066 
.25085 

.25104 
.25124 
.25143 
.25162 
.25182 

.25201 
-25220 
.25240 
.25259 
.25278 


.26174 
-26194 
.26213 
.26233 
.26253 


.35454 
.35490 
.35526 
.35562 
.35598 


-27363 
.27383 
-27403 
.27423 
. 27443 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•23959 
•23978 
•23997 
•24016 
•24035 


.31509 
.31541 
.31574 
.31607 
.31640 


•33519 
.33554 
.33588 
.33622 
.33657 

.33691 
.33726 
.33760 
.33795 
.33830 


.26272 
.26292 
.26312 
.26331 
.26351 


.35634 
.35670 
.35707 
.35743 
.35779 


.27463 
-27483 
-27503 
-27523 
-27543 


.37860 
.37898 
.37936 
.37974 
.38012 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•24054 
•24073 
•24092 
•24111 
•24130 


.31672 
.31705 
.31738 
.31771 
.31804 


.26371 
.26390 
.26410 
.26430 
.26449 


.35815 
.35852 
.35888 
•35924 
.35961 


-27563 
-27583 
-27603 
-27623 
-27643 


.38051 
•38089 
.38127 
•38165 
.38204 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•24149 
.24168 
.24187 
.24206 
•24225 


.31837 
.31870 
.31903 
.31936 
.31969 


.25297 
.25317 
.25336 
.25356 
.25375 


.33864 
.33899 
.33934 
.33968 
.34003 


.26469 
.26489 
-26509 
.26528 
.26548 

.26568 
.26588 
.26607 
.26627 
.26647 


.35997 
.36034 
.36070 
.36107 
.36143 


.27663 
.27683 
.27703 
.27723 
.27743 


•38242 
•38280 
.38319 
.38357 
.38396 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•24244 
•24262 
•24281 
•24300 
•24320 


.32002 
.32035 
.32068 
.32101 
.32134 


.25394 
.25414 
.25433 
.25452 
.25472 


.34038 
.34073 
•34108 
.34142 
.34177 


.36180 
.36217 
.36253 
.36290 
.36327 


.27764 
.27784 
.27804 
.27824 
.27844 


.38434 
.38473 
.38512 
.38550 
.38589 

.38628 
.38666 
.38705 
•38744 
.38783 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•24339 
•24358 
•24377 
•24396 
.24415 


.32168 
.32201 
.32234 
.32267 
.32301 

.32334 
.32368 
.32401 
.32434 
•32468 


.25491 
.25511 
.25530 
.25549 
.25569 

.25508 
.25608 
.25627 
.25647 
.25666 


•34212 
.34247 
.34282 
.34317 
.34352 

.34387 

.34423 
.34458 
.34493 
.34528 

.34563 


.26667 
.26686 
.26706 
.26726 
.26746 


.36363 
.36400 
.36437 
.36474 
.36511 


.27864 
.27884 
.27905 
.■.»7925 
.37945 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


. 24434 
.24453 
.24472 
.24491 
•24510 


.26766 
.26785 
.26805 
.26825 
.26845 


.36548 
.36585 
.36622 
.36659 
-36696 


-27965 
-27985 
.J?8005 
.P.8026 
-28046 


.38822 
.38860 
.38899 
.38938 
.38977 


55 
56 
57 
58 
59 


60 


.24529 


.32501 


.25686 


.26865 


.36733 


.P8066 


.39016 


60 



786 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 





44° 


45° 


46° 


47** 




« 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 





.28066 


.39016 


29289 


•41421 


.30534 


•43956 


.31800 


.46628 


(I 


1 


.28086 


•39055 


.29310 


•41463 


■30655 


•43999 


.31821 


•46674 


1 
2 
3 
4 

5 
6 
7 
8 
g 


2 


.28106 


.39095 


29330 


.41504 


.30576 


.44042 


•31843 


•46719 


3 


.28127 


.39134 


•29351 


.41545 


.30597 


.44086 


•31864 


•46765 


4 


.28147 


.39173 


•29372 


.41586 


.30618 


•44129 


31885 


• ■^6811 
•46857 


5 


.28167 


.39212 


•29392 


.41627 


•30639 


.44173 


.31907 


6 


.28187 


.39251 


.29413 


.41669 


•30660 


.44217 


•31928 


•46903 


7 


.28208 


.39291 


•29433 


.41710 


•30681 


.44260 


•31949 


.46949 


8 


.28228 


.39330 


•29454 


.41752 


•30702 


•44304 


•31971 


.46995 


9 


.28248 


.39369 


.29475 


.41793 


.30723 


.44347 


.31992 


'57041 


10 


.28268 


•39409 


.29495 


•41835 


.30744 


.44391 


.32013 


.47087 


io 


11 


.28289 


.39448 


.29516 


•41876 


.30765 


•44435 


.32035 


•47134 


]i 


12 


.28309 


.39487 


•29537 


•41918 


.30786 


.44479 


•32056 


•47180 


12 


13 


.28329 


•39527 


•29557 


.41959 


•30807 


.44523 


•32077 


•47226 


13 


14 


•28350 


.39566 
.39606 


•29578 


.42001 


-30828 


•44567 


•32099 


.47272 


14 


15 


.28370 


.29599 


.42042 


•30849 


.44610 


•32120 


•47319 


ir> 


16 


.28390 


.39646 


.29619 


.42084 


-30870 


•44654 


•32141 


•47365 


ifi 


17 


•28410 


.39685 


.29640 


.42126 


.30891 


.44698 


■32163 


•47411 


17 


18 


•28431 


.39725 


.29661 


.42168 


.30912 


•44742 


•32184 


•47458 


18 


19 


•28451 


•39764 


•29681 


•42210 
•42251 


•30933 
.30954 


•44787 
•44831 


•32205 


.■«V504 


19 


20 


•28471 


.39804 


•29702 


•32227 


•47551 


30 


21 


.28492 


.39844 


.29723 


.42293 


•30975 


•44875 


•32248 


•47598 


21 


22 


.28512 


.39884 


.29743 


.42335 


•30996 


.44919 


•32270 


•47644 


22 


23 


•28532 


.39924 


.29764 


.42377 


.31017 


.44963 


•32291 


.47691 


23 


24 


28553 


•39963 


•29785 


.42419 


•31038 


• 45007 


•32312 


•47738 


24 


25 


•28573 


.40003 


.29805 


•42461 


•31059 


.45052 


•32334 


.47784 


25 


26 


•28593 


•40043 


.29826 


.42503 


•31080 


.45096 


•32355 


.47831 


26 


27 


•28614 


•40083 


.29847 


.42545 


•31101 


.45141 


•32377 


.47878 


27 


28 


•28634 


.40123 


.29868 


.42587 


•31122 


.45185 


.32398 


.47925 


28 


29 


•28655 


.40163 


•29888 


.42630 


•31143 


.45229 


•32420 


.47972 


2S 


30 


•28675 


.40203 


•29909 


.42672 


-31165 


.45274 


-32441 


.48019 


3C 


31 


•28695 


.40243 


•29930 


.42714 


-31186 


.45319 


•32462 


.48066 


31 


32 


•28716 


.40283 


•29951 


.42756 


•31207 


.45363 


•32484 


.48113 


32 


33 


.28736 


.4C?2A 


.29971 


.42799 


•31228 


.45408 


•32505 


.48160 


33 


34 


.28757 


.40364 


•29992 


.42841 


.31249 
•31270 


•45452 


•32527 


•48207 


34 


35 


.28777 


.40404 


•30013 


.42883 


.45497 


•32548 


•48254 


35 


36 


•28797 


.40444 


•30034 


.42926 


•31291 


.45542 


•32570 


•48301 


36 


37 


.28818 


.40485 


•30054 


.42968 


•31312 


.45587 


■32591 


•48349 


37 


38 


.28838 


.40525 


•30075 


.43011 


•31334 


.45631 


•32613 


•48396 


38 


39 


.28859 


.405P5 


•30096 


.43053 


■31355 


•45676 


,32634 


•48443 


39 


40 


•28879 


.40606 


•30117 


.43096 


•31376 


.45721 


•32656 


•48491 


40 


41 


•28900 


. i0646 


•30138 


.43139 


•31397 


.45766 


.32677 


.48538 


41 


42 


.28920 


.40687 


•30158 


.43181 


.31418 


.45811 


•32699 


-48586 


42 


44 


.28941 


.40727 


.30179 


.43224 


•31439 


.45856 


-32720 


•48633 


43 


44 


.28961 


.40768 


.30200 


.43267 


.31461 


.45901 


•32742 


.48681 


44 


45 


.28981 


.40808 


.30221 


.43310 


•31482 


.45946 


•32763 


.48728 


45 


46 


.29002 


.40849 


•30242 


.43352 


•31503 


.45992 


•32785 


.48776 


46 


47 


.29022 


.40890 


•30263 


.43395 


•31524 


.46037 


•32806 


.48824 


47 


48 


.29043 


.40930 


•30283 


.43438 


.31545 


-46082 


•32828 


.48871 


48 


49 


-29063 


.40971 


•30304 


.43481 


•31567 


-46127 


•32849 


•48919 


49 


50 


.29084 


.41012 


•30325 


•43524 


•31588 


.46173 


•32871 


•48967 


50 


51 


.29104 


.41053 


•30346 


•43567 


.31609 


•46218 


•32893 


.49015 


bl 


52 


.29125 


.41093 


30367 


•43610 


31630 


•46263 


.32914 


•49063 


52 


53 


•29145 


.41134 


30388 


•43653 


.yi651 


•46309 


•32936 


•49111 


53 


54 


.29166 


.41175 


30409 


.43696 


.31673 


.46354 


-32957 


.49159 


64 


55 


.29187 


.41216 


30430 


.43739 


.31694 


.46400 


•32979 


•49207 


55 


56 


.29207 


.41257 


30451 


•43783 


•31715 


•46445 


•33C01 


.49255 


66 


57 


.29228 


.41298 


30471 


.43826 


•31736 


•46491 


33022 


.49303 


67 


58 


29248 


.41339 


30492 


.43869 


.31758 


.46537 


33044 


.49S51 


68 


59 


29269 


.41380 


30513 


.A3912 


-31779 


.46582 . 


^^(^85 


• '3P399 


69 


60 


.29289 


.41421 


30534 


•43956 


31800 


.46628 


33087 


.49448 


60 



787 



TABLE X— NATURAL VJ^RSED SINES AND EXTEp.NAL SECANT% 





48° 


49° 


50° 


61' 




i 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. spp. 


/ 





•33087 


•49448 


.34394 


.52425 


•35721 


.55572 


.37068 


.58902 





1 


.33109 


.49496 


.34416 


.52476 


•35744 


.55626 


.37091 


.58959 


i 


2 


.33130 


•49544 


.34438 


.52527 


•35766 


.55680 


•37113 


.59016 


2 


3 


.33152 


.49593 


.34460 


•52579 


•35788 


.55734 


•37136 


.59073 


3 


4 


.33173 


.49641 


.34482 


.52630 


-35810 


.55789 


.37158 


.59130 


4 


5 


.33195 


.49690 


.34504 


.52681 


•35833 


.55843 


.37181 


•59188 


5 


6 


.33217 


.49738 


•34526 


.52732 


•35855 


.55897 


•37204 


.59245 


8 


7 


.33238 


.49787 


.34548 


.52784 


•35877 


.55951 


.37226 


.59302 


7 


8 


.33260 


.49835 


.34570 


.52835 


•35900 


.56005 


.37249 


.59360 


8 


9 


-33282 


.49884 


-34592 


.52886 


•35922 


.56060 


.37272 


■59418 


9 


10 


.33303 


.49933 


.34614 


.52938 


•35944 


.56114 


.37294 


.59475 


IQ 


11 


.33325 


.49981 


.34636 


.52989 


•35967 


.56169 


.37317 


.59533 


11 


12 


.33347 


.50030 


.34658 


.53041 


.35989 


.56223 


.37340 


.59590 


12 


13 


.33368 


.50079 


.34680 


.53092 


•36011 


.56278 


.37362 


.59648 


13 


14 


.33390 


.50128 


.34702 


.53144 


.36034 


.56332 


.37385 


.59706 


14 


15 


.33412 


.50177 


.34724 


.53196 


•36056 


.56387 


•37408 


.59764 


15 


16 


.33434 


.50228 


.34746 


.53247 


.36078 


.56442 


•37430 


.59822 


16 


17 


.33455 


.50275 


•34788 


.53299 


.36101 


.56497 


•37453 


.59880 


17 


18 


.33477 


.50324 


•34790 


.53351 


.36123 


.56551 


•37476 


.59938 


1^ 


19 


.33499 


-50373 
.50422 


.34812 
.34834 


.53403 


•36146 
•3G168 


.56606 


•37498 


.59996 


19 


20 


.33520 


.53455 


.56661 


•37521 


.60054 


30 


21 


.33542 


.50471 


.34853 


.53507 


•36190 


.56716 


•37544 


.60112 


21 


22 


.33584 


.50521 


.34873 


.53559 


.36213 


.56771 


•37567 


.60171 


22 


23 


.33588 


.50570 


.34900 


•53611 


.36235 


.56826 


•37589 


.60229 


23 


24 


.33607 


.50819 


.34923 


.53663 


.36258 


.56881 


-37612 


.60287 


24 


25 


.33629 


.50669 


.34945 


.53715 


^36280 


.56937 


•37635 


.60346 


25 


26 


.33851 


.50718 


.34967 


.53768 


.36302 


.56992 


.37658 


.60404 


26 


27 


.33873 


.50767 


.34989 


.53820 


.36325 


.57047 


.37680 


.60463 


27 


28 


.33694 


.50817 


.35011 


.53372 


.36347 


.57103 


•37703 


•60521 


28 


29 


.33716 


.50866 
.50916 


.35033 


.53924 


.36370 


.57158 


.37726 


.60580 


29 


30 


.33738 


.35055 


.53977 


•36392 


.57213 


.37749 


• 60639 


30 


31 


.33760 


.50966 


.35077 


.54029 


•36415 


.57269 


•37771 


•60698 


31 


32 


.33782 


.51015 


.35099 


.54082 


•36437 


.57324 


.37794 


•60756 


32 


33 


.33803 


.51085 


.35122 


.54134 


•36460 


.57380 


.37817 


.60815 


33 


34 


.33825 


.51115 


.35144 


.54187 


-36482 


.57436 
.57491 


.37340 


.60874 


34 


35 


■33847 


.51165 


.35168 


. 54240 


-36504 


.37862 


.60933 


35 


36 


-33369 


.51215 


•35188 


.54292 


•36527 


.57547 


.37885 


.60992 


36 


37 


•33891 


-51265 


•35210 


.54345 


-36549 


.57603 


.37908 


.61051 


37 


38 


.33912 


.51314 


.35232 


.54398 


.36572 


.57659 


.37931 


.61111 


38 


39 


.33931 


•51384 


.35254 


.54451 


.36594 


.57715 


.37954 


.61170 


39 


40 


•33958 


.51415 


.35277 


.54504 


•36617 


.57771 


•37976 


.61229 


40 


41 


•33978 


.51485 


.35299 


.54557 


•36639 


.57827 


•37999 


.61288 


41 


42 


-34000 


.51515 


.35321 


.54610 


•36662 


.57883 


•38022 


.61348 


42 


43 


.34022 


.51585 


.35343 


.54663 


•36684 


.57939 


•38045 


.61407 


43 


44 


-34044 
-34065 


.51615 


.35365 


.54716 


•36707 


.57995 


■38068 


-61467 


44 


45 


.51685 


•35388 


•54769 


•36729 


•58051 


•38091 


.61526 


45 


46 


■34087 


.51716 


.35410 


.54822 


•36752 


.58108 


•38113 


.61586 


46 


47 


-34109 


.51768 - 


.35432 


.54876 


•36775 


.58164 


•38138 


.61646 


47 


48 


-34131 


.51817 


■35454 


.54929 


•36797 


.58221 


•38159 


.61705 


48 


49 


•34153 
-34175 


.51837 


•35476 


.54982 


•36820 


.58277 


•38182 


.61765 


49 


50 


.51918 


•35499 


.55036 


.36842 


•58333 


.38205 


.61825 


50 


bi 


■34197 


.51988 


•35521 


.55089 


.36865 


.58390 


•38228 


.61885 


51 


52 


-34219 


.52019 


•35543 


.55143 


.36887 


.58447 


•38251 


.61945 


52 


53 


•34241 


.52069 


•35565 


.55196 


.36910 


.58503 


•38274 


.62005 


53 


54 


•34262 


.52120 


•35588 


.55250 


■36932 
.36955 


.58560 


•38296 


.62065 
.62125 


54 


55 


•34284 


.52171 


•35610 


.55303 


.58617 


•38319 


55 


56 


•34306 


.52222 


•35632 


.55357 


.36978 


.58674 


•38342 


•62185 


56 


5V 


■34328 


.52273 


.35654 


.55411 


■37000 


.58731 


.38365 


•62246 


57 


58 


■34350 


.52323 


•35677 


.55465 


■37023 


•58788 


.38388 


•62306 


58 


59 


-34372 


.52374 


•35699 


.55518 


-37045 


.58845 


.38411 
•38434 


•62366 


59 


60 


•34394 


.52425 


.35721 


.55572 


.37068 


.58902 


.62427 


60 



788 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS 
53° 53° 54° 55° 



» 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


* 




1 

2 
3 

4 


.38434 
.38457 
.38480 
.38503 
•38526 

.38549 
-38571 
.38594 
.38617 
•38640 


• 62427 
•62487 
•62548 
•62609 
■62669 


•39819 

•39842 

•39865 

•39888 

.3991L 

•39935 

•39958 

•39981 

•40005 

•40028 


•66164 
.66228 
.66292 
•66357 
•66421 


.41221 
•41245 
•41269 
.41292 
•41316 


.70130 
.70198 
•70267 
•70335 

• 70403 
•70472 
•70540 
•70609 
•70677 

• 70746 


•42642 
•42668 
•42690 
•42714 
.42738 


• 74345 
•74417 

• 74490 
•74562 
•74635 




1 
2 
3 
4 


5 
6 
7 
8 
9 


•62730 
•62791 
.62852 
•62913 
•62974 


•66486 
•66550 
•66615 
•66679 

•66744 


•41339 
.41363 
.41386 
•41410 
•41433 


•42762 
•42785 
•42809 
•42833 
.42857 


• 74708 
•74781 
•74854 
•74927 
■75000 


5 
6 
7 
8 
9 


to 

11 
12 
13 
14 


.38663 
.38686 
•38709 
.38732 
•38755 


•63035 .40051 
.63096 .40074 
•63157 .40098 
•63218 .40121 
.63279 .40144 


•68809 
•66873 
•66938 
.67003 
•87068 


•41457 
•41481 
•41504 
•41528 
■41551 


70815 
.70884 
.70953 
•71022 
•71091 


•42881 
•42305 
•42929 
•42953 
.42976 


•75073 
•75146 
•75219 
•75293 
•75366 

• 75440 
•75513 
•75587 
•75861 
.75734 


10 

11 
12 

13 
14 


15 
16 
17 
18 


•38778 
.38801 
.38824 
•38847 
•38870 


.63341 
.63402 
• 63464 
.63525 
•63587 


•40188 
•40191 
•40214 
•40237 
.40261 

•40284 
•40307 
•40331 
•40354 
•40378 

•40401 
•40424 
•40448 
•40471 
.40494 


•67133 
.67199 
.67264 
.67329 
.67394 

•67460 
.67525 
.67591 
•67656 
•67722 

•67788 
•67853 
•67919 
•67985 
.68051 


•41575 
•41599 
.41622 
.41646 
•41670 


.71160 
.71229 
.71298 
.71368 

•71437 


.43000 
•43024 
• 43048 
43072 
.43096 


15 
16 
17 
18 

19 


30 

21 
22 
23 
24 


•38893 
•38916 
•38939 
•38962 
.38985 

.39009 
.39032 
.39055 
.39078 
•39101 


•63648 
.63710 
•63772 
•63834 
.63895 

•63957 
•64019 
•64081 
• 64144 
•64206 


•41693 
•41717 
•41740 
•41764 
•41788 


.71506 
.71576 
.71646 
.71715 
•71785 


•43120 
-43144 
.43163 
.43152 
43216 


•75808 
•75882 
•75956 
•76031 
•76105 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•41811 
.41835 
.41859 
.41882 
.41906 


.71855 
.71925 
.71995 
.72065 
•72135 


.43240 
.43284 
.43287 
.43311 
.43335 


•76179 
•76253 
•76328 

• 76402 

• 76477 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.39124 
.39147 
.39170 
.39193 
•39216 


•64268 
•64330 
•64393 
•64455 
•64518 


•40518 
•40541 
•40565 
•40588 
•40611 


•68117 
•68183 
•68250 
•68316 
.68382 


.41930 
.41953 
.41977 
.42001 
• 42024 


•72205 
•72275 
. 72346 
•72416 
•72487 


•43359 
.43383 
. 43407 
•43431 
•43455 


•76552 
•76626 
•76701 
•76776 
•76851 


30 

31 
33 
33 
34 


35 
36 
37 
S8 
39 


•39239 
.39262 
.39286 
.39309 
•39332 


•64580 
. 64643 
.64705 
.64768 
.64831 


-40635 
•40658 
•40682 
•40705 
•40728 


•68449 
.68515 
•68582 
•68648 
•68715 

•68782 
•68848 
•68915 
•68982 
•69049 


• 42048 
•42072 
•42096 
•42119 
.42143 


•72557 
.72628 
•72698 
.72769 
.72840 


•43479 
•43503 
•43527 
•43551 
.43575 


•76926 
•77001 
•77077 
•77152 
•77227 


3§ 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.39355 
.39378 
.39401 
.39424 
.39447 


• 64894 
•64957 
•65020 
.65083 
•65146 


•40752 
•40775 
•40799 
•40822 
•40846 

•40869 
•40893 
•40916 
•40939 
.40963 


•42167 
.42191 
. 42214 
.42238 
•42262 


•72911 
•72982 
•73053 
.73124 
•73195 


•43599 
•43623 
•43647 
•43671 
•43695 


•77303 
•77378 
•77454 
•77530 
•77606 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•39471 
•39494 
•39517 
•39540 
•39563 


•65209 
•65272 
.65336 
.65399 
•65462 


•69116 
•69183 
.69250 
.69318 
•69385 


•42285 
.42309 
.42333 
.42357 
.42381 


.73267 
.73338 
• 73409 
.73481 
•73552 


•43720 
•43744 
•43768 
•43792 
.43816 


•77681 
•77757 
•77833 
•77910 
•77986 


45 
46 
47 

48 
49 


50 

51 
52 
53 
54 


•39586 
•39610 
•39633 
•39656 
•39679 


•65528 
.65539 
•65653 
• 6571? 
•65780 


.40986 
•41010 
.41033 
•41057 
•41080 


•69452 
.69520 
•69587 
.69655 
.69723 


•42404 
•42428 
•42452 
•42476 
•42499 


.73624 
•73696 
•73768 
•73840 
•73911 


.43840 
.43864 
•43888 
•43912 
.43936 


•78062 
.78138 
•78215 
•78291 
•78368 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


•39702 
•39726 
•39749 
•39772 
•39795 


•65844 • 41104 
.65908 .41127 
.65972 .41151 
.66036 ^41174 
•66100 •41198 


•6P790 
.69858 
.69926 
•69994 
•70062 

.70130 


•42523 
•42547 
•42571 
•42595 
•42619 

•42642 


•73983 

•74056 
•74128 
.74200 
- 74272 

.74345 


.43960 
•43984 
•44008 
•44032 
.44057 


• 78445 
.78521 
•78598 
•78675 
.78752 


55 
56 
57 
58 
59 


60 


• 39819 


.66164 .41221 


•44081 


.78829 


60 



789 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



56° 



57' 



58' 



59*= 



/ 


Vers. 


Ex. sec. 


Vers, 

■45536 
■45560 
•45585 
•45609 
■45634 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


•44081 
•44105 
•44129 
•44153 
. 44177 




78829 
78906 
78984 
79061 
79138 




83608 
83690 
83773 
83855 
83938 


•47008 
•47033 
•47057 
•47082 
•47107 

•47131 
•47156 
•47181 
•47206 
•47230 




88708 

88796 

.88884 

.88972 

89060 


■48496 
■48521 
.48546 
■48571 
■48596 


.94160 
.94254 
.94349 
.94443 
•94537 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.44201 
•44225 
•44250 
•44274 
.44298 




79216 
79293 
79371 
79449 
79527 


45658 
■45683 
■45707 
■45731 
■45756 




84020 
84103 
84186 
84269 
84352 




89148 
89237 
89325 
89414 
89503 


■48621 
■48646 
■48671 
■48696 
■48721 


-94632 
•94726 
.94821 
.94916 
.95011 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•44322 
• 44346 
•44370 
•44395 
■44419 




79604 
79682 
79761 
79839 
79917 


■45780 

■45805 
■45829 
■45854 
•45878 




84435 
84518 
84601 
84685 
84768 


•47255 
•47280 
•47304 
•47329 
•47354 




89591 
89680 
89769 
89858 
89948 


■48746 
■48771 
.48796 
•48821 
•48846 


.95106 
-95201 
-95296 
.95392 
■95487 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


■ 44443 
■44467 
■44491 
•44516 
. 44540 




79995 
80074 
80152 
80231 
80309 


■45903 
■45927 
■45951 
■45976 
■46000 




84852 
84935 
85019 
85103 
85187 


•47379 
•47403 
■47428 
•47453 
•47478 


— 


90037 
90126 
90216 
90305 
90305 

90485 
90575 
90685 
90755 
90845 


•48871 
•48896 

•48921 
•48946 
.48f)71 


-95583 
•95678 
-95774 
-95870 
.95966 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


■44584 
■44588 
■44612 
■44637 
■44661 




80388 
80467 
80546 
80625 
80704 


■46025 
■46049 
■46074 
■46098 
■46123 




85271 
85355 
85439 
85523 
85603 


•47502 
■47527 
■47552 
■47577 
■47601 


•48996 
•49021 
•49046 
•49071 
•49096 


.9G062 
.96158 
-96255 
-96351 
-96448 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


■44685 
■44709 
■44734 
■44758 
■44782 




80783 
80862 
80942 
81021 
81101 


■46147 
■46172 
■48196 
■46221 
■46246 




85692 
85777 
85861 
85946 
86031 


■47626 
■47651 
•47676 
•47701 
•47725 




90935 
91026 
91116 
91207 
91297 


■49121 
-49146 
•49171 
•49196 
•49221 


-96544 
.96641 
•96738 
•96835 
•96932 


25 
26 
27 
28 
29- 


30 

31 
32 
33 
34 


■44806 
■44831 
■44855 
■44879 
•44903 




81180 
81260 
81340 
81419 
81499 


■46270 
■46295 
■46319 
•46344 
•46368 


•86116 
.86201 
.86286 
•86371 
•86457 


•47750 
•47775 
• 47S00 ■ 
.47825 
■47849 




91388 
91479 
91570 
91661 
91752 


•49246 
■49271 
•49296 
49321 
.49346 


.97029 
•97127 
.97224 
.97322 
•97420 


30 

3] 
32 
33 
34 


35 
36 
37 
38 
39 


•44928 
■44952 
■44976 
■45001 
■45025 




81579 
81659 
81740 
81820 
81900 


•46393 
•46417 
•46442 
•46466 
■46491 


•86542 
•86627 
•86713 
•86799 
•86885 


•47874 
•47899 
•47924 
•47949 
•47974 




91844 
91935 
92027 
92118 
92210 


•49372 
•49397 
•49422 
•49447 
•49472 


.97517 
•97615 
•97713 
•97811 
-97910 


35 
38 
37 
38 
39 


40 

41 
42 
43 
44 


■45049 
■45073 
■45098 
■45122 
■45146 




81981 
82061 
82142 
82222 
82303 


■46516 
■46540 
•46565 
•46589 
•46614 


•86970 
.87056 
•87142 
•87229 
•87315 


•47998 
•48023 
•48048 
•48073 
• 48;D98 




92302 
92394 
92486 
92578 
92670 


•49497 
•49522 
•49547 
•49572 
.49597 


-98008 
•98107 
.98205 
•98304 
•98403 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


■45171 
■45195 
■45219 
■45244 
45268 




82384 
82465 
82546 
82627 
82709 


•46G39 
•46663 
•46688 
■46712 
■46737 


•87401 
•87488 
-87574 
•87661 
•87748 


•48123 
•48148 
•48172 
•48197 
•48222 




92762 
92855 
92947 
93040 
93133 


.49623 
.49648 
.49673 
■49698 
■49723 


•98502 
.98601 
•98700 
.98799 
-98899 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


■45292 
■45317 
■45341 
■45365 
■45390 




82790 
82871 
82953 
83034 
83116 


■46762 
■46786 
■46811 
■46836 
■46860 


•87834 
•87921 
•88008 
•88095 
•88183 


•48247 
•48272 
•48297 
•48322 
• 4,8347 




93226 
93319 
93412 
93505 
93598 


.49748 
.49773 
-49799 
•49824 
■49849 


-98998 
-99098 
.99198 
.99298 
-99398 


50 

51 
52 
53: 
54 


55 
56 
57 
58 
59 


•45414 
•45439 
■45463 
■45487 
■45512 




83198 
83280 
83362 
83444 
83526 


■46885 
■48909 
■46934 
•46959 
•46983 


•88270 
•88357 
•88445 
•88532 
•88620 


.48372 
•48396 
•48421 
•48446 
48471 




93692 
93785 
93879 
93973 
94066 


■49874 
•49899 
•49924 
•49950 
•49975 


.99498 
.99598 
.99698 
.99799 
-99899 


55: 
56 
57 
58 
59 


60 


■45536 




83608 


.47008 


•88708 


.48496 


L 


94160 


. 50000 


1. 00000 


60 



790 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 





60° 




61° 


63'' 


63° 




t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 





•50000 


1^00000 


•51519 


1.06267 


.53053 


1.13005 


-54601 


1.20269 





1 


•50025 




•00101 


•51544 


1.06375 


•53079 


1-13122 


-54627 


1-20395 


1 


2 


•50050 




•00202 


■51570 


1.06483 


•53104 


1.13239 


-54653 


1-20521 


2 


3 


•50076 




•00303 


.51595 


1.06592 


•53130 


1.13356 


-54679 


1-20647 


3 


4 


•50101 




•00404 


.51621 


1.06701 
1-06809 


•53156 


1.13473 


■54705 


1-20773 
1.20900 


4 


5 


•50126 




•00505 


.51646 


•53181 


1.13590 


-54731 


5 


6 


•50151 




•00607 


•51672 


1.06918 


•53207 


1.13707 


-54757 


1-21C26 


6 


7 


•50176 




•00708 


•51697 


1.07027 


•53233 


1-13825 


- 54782 


1-21153 


7 


8 


•50202 




•00810 


•51723 


1-07137 


•53258 


1-13942 


-54808 


1.21280 


8 


9 


•50227 




00912 


•51748 


1^07246 
1^07356 


•53284 


1-14060 


■54834 


1.21407 


9 


10 


•50252 




01014 


•51774 


•53310 


1-14178 


-54860 


1.21535 


10 


11 


•50277 




01116 


•51799 


1-07465 


•53336 


1-14296 


-54886 


1.21662 


11 


12 


•50303 




01218 


.51825 


1-07575 


.53361 


1 . 14414 


-54912 


1-21780 


12 


13 


.50328 




01320 


.51850 


1-07685 


.53387 


1.14533 


-54938 


1.21918 


13 


14 


•50353 




01422 


.51876 


1-07795 
1.07905 


.53413 


1-14651 


• 54964 

• 54990 


1-22045 


14 


15 


•50378 




01525 


.51901 


.53439 


1.14770 


1-22174 


15 


16 


• 50404 




01628 


■51927 


1-08015 


.53464 


1.14889 


•55016 


1-22302 


16 


17 


.50429 




01730 


■51952 


1.08126 


-53490 


1.15008 


• 55042 


1^22430 


17 


18 


•50454 




01833 


■51978 


1-08236 


-53516 


1.15127 


■55068 


1-22559 


18 


19 


• 50479 




01936 


■52003 


1.08347 
1.08458 


.53542 


] .15246 


■55094 


l-5;688 


19 


20 


•50505 




02039 


.52029 


.53567 


1-15366 


■55120 


1^22817 


20 


21 


.50530 




02143 


.52054 


1-08569 


.53593 


1-15485 


■55146 


1^22946 


21 


22 


.50555 




02246 


.52080 


1-08680 


.53619 


1-15605 


■55172 


1-23075 


22 


23 


•50581 




02349 


.52105 


1-08791 


.53645 


1-15725 


■55198 


1.23205 


23 


24 


•50606 




02453 


.52131 


1-08903 


■53670 


1-15845 


■55224 


1 .23334 


24 


25 


•50631 




02557 


.52156 


1.09014 


.53696 


1-15965 


•55250 


1-23464 


25 


26 


•50656 




02661 


.52182 


1.09126 


.53722 


1-16085 


-55276 


1-23594 


26 


27 


.5G682 




02765 


.52207 


1.09238 


.53748 


1-16206 


-65302 


1-23724 


27 


28 


•50707 


i 


02869 


.52233 


1.09350 


.53774 


1.16326 


■55328 


1-23855 


28 


29 


.50732 




02973 


.52259 


1-09462 
1.09574 


.53799 


1-16447 
1-16568 


-55354 


1-23985 
1.24116 


29 


30 


.50758 




03077 


•52284 


.53825 


■55380 


30 


81 


.50783 




03182 


.52310 


1.09686 


.53851 


1.16689 


-55406 


1 . 24247 


31 


32 


.50808 




03286 


.52335 


1.09799 


.53877 


1-16810 


-55432 


1.24378 


32 


33 


.50834 




03391 


.52361 


1.09911 


.53903 


1-16932 


-55458 


1.24509 


33 


34 


•50859 




03496 


.52386 


1.10024 
1-10137 


.53928 


1-17053 


•55484 


1.24640 


34 


35 


•50884 




03601 


■52412 


.53954 


1.17175 


.55510 


1-24772 


35 


36 


•50910 




03706 


.52438 


1-10250 


-53980 


1-17297 


.55536 


1-24903 


36 


37 


•50935 




03811 


■52463 


1.10363 


- 54006 


1-17419 


55563 


1-25035 


37 


38 


•50960 




03916 


■52489 


1.10477 


-54032 


1.17541 


•55589 


1. 25167 


38 


39 


.50986 




04022 


■52514 
■52540 


1.10590 


■ F4058 


1-17663 


.fE615 


1 .25300 


39 


40 


.51011 




04128 


1.10704 


-54083 


1-17786 


•55641 


1.25432 


40 


41 


.51036 




04233 


.52566 


1.10817 


■54109 


1-17909 


•55667 


1.25565 


41 


42 


.51062 




04339 


.52591 


1.10931 


- 54135 


1-18031 


■55693 


1.25697 


42 


43 


.51087 




04445 


■52617 


1.11045 


•54161 


1-18154 


•55719 


1.25830 


43 


44 


.51113 




04551 


■52642 


1.11159 


-54187 


1-18277 


•55745 


1.25963 


44 


45 


.51138 




04658 


■52668 


1.11274 


-54213 


1-18401 


•55771 


1.26097 


45 


46 


.51163 




04764 


■52694 


1-11388 


-54238 


1-18524 


•55797 


1.26230 


46 


47 


.51189 




04870 


■52719 


1-11503 


-54264 


1-18648 


•55823 


1.26364 


47 


48 


.51214 




04977 


■52745 


1.11617 


-54290 


1-18772 


-55849 


1.26498 


48 


49 


.51239 




05084 


■52771 


1-11732 


.54316 


1-18895 


-55876 


1.26632 


49 


50 


.51265 




05191 


■52796 


1-11847 


.54342 


1-19019 


-55902 


1.26766 


50 


51 


5129"0 




05298 


•52322 


1-11963 


.54368 


1-19144 


.55928 


1.269C0 


bl 


52 


.51316 




05405 


■52848 


1-12078 


.54394 


1-19268 


55954 


1-27035 


52 


53 


.51341 




05512 


■52873 


1-12193 


. 54420 


1.19393 


55980 


1.27169 


53 


54 


.51366 




05619 


■52899 


1.12309 


■ 54446 


1-19517 


56006 


1.27304 


54 


55 


.51392 




05727 


.52924 


1.12425 


■54471 


1-19642 


56032 


1.27439 


55 


56 


.51417 




05835 


•52950 


1.12540 


■54497 


1-19767 


56058 


1.27574 


56 


57 


.51443 




05942 


.52976 


1.12657 


■54523 


1.19892 


56084 


1.27710 


57 


58 


.51488 




06050 


.53001 


!• 12773 


.54549 


1.20018 - 


56111 


1.27845 


58 


59 


.51494 




06158 


■53027 


3.12889 


54575 


1.20143 ■ 


56137 


1.27981 


5t 


60 


•51519 


!• 


06267 


.53053 1-13005 | 


.54601 


1-20269 - 


56163 


1-28117 


60 



791 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 66° 67° 



> 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 

2 
3 
4 


.56163 ; 
.56189 : 
.56215 ; 
.56241 ; 
.56267 J 

•56294 ] 
•56320 ] 
•56346 ] 
•56372 ] 
•56398 ] 


L. 28117 
L. 28253 
L. 28390 
L. 28526 
L. 28663 


.57738 
.57765 
.57791 
.57817 
.57844 


1-36620 
1.36768 
1.36916 
1^ 37064 
1^ 37212 


-59326 
-59353 
.59379 
.59406 
-59433 




45859 

46020 

46181 

.46342 

-46504 


-60927 
.60954 
.60980 
.61007 
.61034 




55930 
56106 
56282 
56458 
56634 




1 

2 
3 

4 


5 
6 
7 
8 
9 


L. 28800 
L. 28937 
L. 29074 
[•29211 
[•29349 


.57870 
•57896 
•57923 
•57949 
-57976 


1^37361 
1^37509 
1-37658 
1-37808 
1-37957 

1-38107 
1-38256 
1-38406 
1-38556 
1-38707 


-59459 
-59486 
-59512 
.59539 
-59566 




-46665 

-46827 

-46989 

47152 

47314 


.61061 
.61088 
-61114 
-61141 
-61168 




56811 
56988 
57165 
57342 
57520 


5 
6 
7 
8 
9 


lo 

11 
12 
13 
14 


•56425 ] 
•56451 ] 
•56477 ] 
•56503 ] 
•56529 ] 


[•29487 
[.29625 
[-29763 
[•29901 
[•30040 


-58002 
-58028 
-58055 
-58081 
-58108 


-59592 
-59619 
-59845 
-59672 
-59699 

-59725 
-59752 
-59779 
-59805 
-59832 




47477 

47640 

47804 

-47967 

-48131 


-61195 
-61222 
-61248 
-61275 
-61302 




57698 
-57876 

58054 
-58233 
-58412 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•56555 ] 
•56582 ] 
•56608 ] 
•56634 ] 
.56660 ] 


[•30179 
[•30318 
[•30457 
[-30596 
.-30735 


-58134 
-58160 
-58187 
-58213 
58240 


1-38857 
1-39008 
l^39159 
1-39311 
1-39462 


— 


48295 
48459 
48624 
48789 
48954 

49119 
49284 
49450 
49616 
49782 

49948 
50115 
50282 
50449 
50617 


-61329 
-61356 
-61383 
-61409 
-61436 




-58591 

-58771 

58950 

59130 

59311 


15 
16 
17 
18 
19 


30 

21 
§2 
23 
24 


•56687 ] 
•56713 ] 
•56739 ] 
•56765 ] 
•56791 ] 


.-30875 
[-31015 
.-31155 
.-31295 
.-31436 

..31576 
.-31717 
.-31858- 
.-31999 
-32140 


-58266 
58293 : 
58319 : 
58345 ; 

-58372 ] 

-58398 ] 

-58425 ] 

58451 ] 

58478 ] 

58504 ] 


1-39614 
1-39766 
1.39918 
1.40070 
1-40222 


■59859 
■59885 
■59912 
59938 
■59965 

■59992 
■60018 
-60045 
-60072 
-60098 


.61463 
.61490 
.61517 
.61544 
-61570 




59491 
59672 
59853 
60035 
60217 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•56818 ] 
•56844 ] 
•56870 ] 
•56896 ] 
•56923 ] 


1-40375 
L. 40528 
[•40681 
[•40835 
[-40988 


-61597 
-61624 
-61651 
-61678 
-61705 




60399 
60581 
60763 
60946 
61129 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•56949 ] 
•56975 ] 
-57001 ] 
•57028 ] 
•57054 1 


-32282 
.-32424 
-32566 
-32708 
-32850 


58531 ] 
58557 ] 
58584 ] 
58610 ] 
58837 ] 


[-41142 
[-41296 
[•41450 
[•41605 
[•41760 

[-41914 
[•42070 
[-42225 
[-42380 
[-42536 

[-42692 
1-42848 
1. 43005 
L43162 
1-43318 


-60125 
-60152 
60178 
■60205 
- 60232 




50784 
50952 
51120 
51289 
51457 


-61732 
61759 
-61785 
-61812 
-61839 




61313 
61496 
61680 
61864 
62049 


30 

31 
3L' 
33 
34 


35 
36 
37 
38 
39 


•57080 1 
•57106 1 
•57133 1 
•57159 1 
•57185 ] 


•32993 
•33135 
-33278 

-33422 
-33585 


58363 ] 
■58690 ] 
■58716 ] 

58743 ] 
■58769 ] 


-60259 
-60285 
-60312 
-60339 
- 60365 

-60392 
60419 

■ 60445 

■ 60472 
-60499 

-60526 
-60552 
.60579 
•60606 
-60633 


L 


51626 
51795 
51965 
52134 
52304 

52474 
52645 
52815 
52986 
53157 


-61866 
-61893 
-61920 
-61947 
-61974 

-62001 
-62027 
-62054 
-62081 
-62108 




62234 
62419 
62604 
62790 
62976 

63162 
63348 
63535 
63722 
63909 


35 
36 
37 
38 
39 


40 

42 

II 


•57212 1 
•57238 ] 
•57264 ] 
■57291 ] 
•57317 ] 


-33708 

-33852 

-33996 

.-34140 

.-34284 


■58796 ] 
■58822 ] 
■58849 ; 
58875 ; 
■53902 : 

■58928 : 
-58955 
-58981 
- 59008 
-59034 

-59061 
•59087 
•59114 
-59140 
•59167 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•57343 ] 
•57369 ] 
•57396 ] 

•57422 ] 
•57448 ] 


.-34429 
•34573 
[•34718 
[•34883 
1^35009 


1 - 43476 
1-43633 
1-43790 
1-43948 
1-44106 




58329 
53500 
53672 
53845 
54017 


-62135 
-62162 
-62189 
-62216 
-62243 

-62270 
-62297 
-62324 
-62351 
-62378 




64097 
64285 
64473 
64662 
64851 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•57475 ] 
•57501 ] 
•57527 ] 
•57554 ] 
•57580 ] 


[-35154 
[.35300 
[-35446 
[•35592 
[-35738 

L-35885 
L-36031 
1-36178 
1-36325 
1-36473 


1-44264 
1-44423 
1.44582 
1^44741 
1-44900 


-60659 
-60886 
-60713 

- 60740 
-60766 

.60793 
-60820 
-60847 
•60873 

- 60900 




54190 
54363 
54536 
54709 
54883 




65040 
65229 
65419 
65609 
65799 


50 

51 
52 
55 
54 


55 
56 
57 
58 
59 


•57806 : 

.57633 

•57659 

•57685 

•57712 


-59194 
-59220 
-59247 
-59273 
■59300 

■59326 


1-45059 
1-45219 
1-45378 
1.45539 
1-45699 




55057 
55231 
55405 
55580 
55755 


-62405 
-62431 
-62458 
■62485 
■62512 




65989 
66180 
66371 
66563 
66755 


55 

58 
59 


60 


•57738 


1-36620 


1-45859 


■60927 


1 


55930 


-62539 


1 


66947 


60 



792 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTE 





68° 


69° 


70° 




71° 






1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


# 




1 

2 
3 

4* 


•62539 
.62566 
.62593 
.62620 
.62647 


1.66947 
1.67139 
1.67332 
1-67525 
1.67718 


-64163 
-64190 
-64218 
- 64245 
-64272 


1-79043 
1-79254 
1-79466 
1-79679 
1-79891 

1-80104 
1-80318 
1.80531 
1.80746 
1.80960 

1.81175 
1.81390 
1.81605 
1.81821 
1-82037 


65798 

65825 

65853 

-65880 

.65907 




92380 
92614 
92849 
93083 
93318 


. 67443 
.67471 
.67498 
.67526 
.67553 


2. 
2. 
1. 
2. 
2. 


07155 
07415 
07675 
07936 
08197 



1 

2 
3 

4 


5 
6 
7 
8 
9 


.62674 
.62701 
.62728 
.62755 
■62782 


1.67911 
1.68105 
1.68299 
1.68494 
1.68689 


-64299 
-64326 
-64353 
-64381 
■ 64408 

- 64435 

- 64462 
•64489 
•64517 
-64544 


-65935 
-65962 
-65989 
■66017 
■ 66044 




93554 
93790 
94026 
94263 
94500 


.67581 
.67608 
.67636 
-67663 
-67691 


2- 
2- 
2. 
2. 
2- 


08459 
08721 
08983 
09246 
09510 


5 
6 
7 
8 
9 


10 

11 
12 
13 

14 


•62809 
.62836 
.62863 
.62890 
•62917 


1.68884 
1.69079 
1.69275 
1.69471 
1.69667 


■66071 
■66099 
■66126 
■66154 
■66181 




94737 
94975 
95213 
95452 
95691 


-67718 
-67746 
-67773 
-67801 
-67829 


2. 
2- 
2. 
2- 
2- 


09774 
10038 
10303 
10568 
10834 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


• 62944 
•62971 
•62998 
•63025 
•63052 


1.69864 
1.70061 
1.70258 
1.70455 
1.70653 


-64571 
-64598 
-64625 
-64653 
-64680 

- 64707 

- 64734 
-64761 

- 64789 
■64816 


1-82254 
1-82471 
1-82688 
1-82906 
1-83124 

1-83342 
1-83561 
1-83780 
1-83999 
1-84219 


.66208 
.66236 
■66263 
■66290 
■66318 

■66345 
■66373 
■66400 
■66427 
■66455 




95931 
96171 
96411 
96652 
96893 

97135 
97377 
97619 
97862 
98106 


.67856 
.67884 
.67911 
.67939 
.67986 


2. 
2. 
2. 
2- 
2. 


11101 
11367 
11835 
11903 
12171 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•63079 
•63106 
•63133 
•63161 
•63188 


1.70851 
1^ 71050 
1-71249 
1.71448 
1.71647 


.67994 
.68021 
-68049 
-68077 
-68104 


2. 

2- 

2- 

2 

2 


12440 
]2709 
12979 
13249 
13520 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•63215 
•63242 
63269 
.63296 
•63323 


1.71847 
1-72047 
1.72247 
1 . 72448 
1.72649 


- 64843 

- 64870 
.64893 
-64925 
-64962 


1.84439 
1.84659 
1-84880 
1-85102 
1-85323 


-66482 
-66510 
-66537 
-66564 
66592 




98349 
98594 
98838 
99083 
99329 


-68132 
-68159 
-68187 
-68214 
-68242 


2 
2 
2 
2 
2 


13791 
14063 
14335 
14608 
14881 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


63350 
•63377 
• 63404 
•63431 
.63458 


1.72850 
1.73052 
1.73254 
1.73456 
1.73659 


-64979 
-65007 
-65034 
-65061 
-65088 


1-85545 
1-85767 
1-85990 
1.86213 
1.86437 


■66619 
■66647 
■66674 
■66702 
■66729 


2 
2 
2 


-99574 

.99821 

0G067 

00315 

00562 


-68270 
- 68397 

68325 
■68352 

68380 

■ 68408 
.68435 
■68463 
■68490 
■68518 

■68546 
■68573 
■68601 
■68628 
.68656 


2 
2 
2 
2 
2 


15155 
15429 
15704 
15979 
16255 


30 

ai 

32 
33 
34 


35 
36 
37 
38 
39 


.63485 
•63512 
•63539 
•63566 
.63594 


1-73862 
1-74065 
1-74269 
1 . 74473 
1-74677 


-65116 
•65143 

65170 
■65197 

65225 


1.86661 
1.86885 
1.87109 
1.87334 
1-87560 


■66756 
■66784 
■66811 
■66839 
■66866 


2 
2 
2 
2 
2, 

2* 
2 
2 
2 
2. 


00810 
01059 
01308 
01557 
01807 

-02057 
-02308 
-02559 
-02810 
-03062 


2 
2 
2 
2 
2_ 

2 
2 
2 
2 


16531 
16808 
17085 
17363 
17641 

17920 
18199 
18479 
18759 
19040 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•63621 
•63648 
•63675 
•63702 
•63729 


1.74881 
1.75086 
1-75292 
1-75497 
1-75703 


-65252 
-65279 
-65306 
-65334 
.65361 


1-87785 
1.88011 
1.88238 
1.88465 
1-88692 


■66894 
■66921 
■66949 
■66976 
■67003 

-67031 
-67058 
•67086 
.67113 
.67141 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•63756 
•63783 
•63810 
•63838 
63865 


1-75909 
1-76116 
1-76323 
1-76530 
1-76737 


.65388 
.65416 
. 65443 
.65470 
-65497 


1.88920 
1.89148 
1.89376 
1.89605 
1-89884 


2 
2 
2 
2 
2 


-03315 
-03568 
-03821 
-04075 
-04329 


.68684 
■68711 
■68739 
■68767 
.68794 


2 
2 
2 
2 
2 


19322 
-19604 
-19886 
■20169 
■20453 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•63892 
.63919 

• 63946 
•63973 

• 64000 

•64027 
•64055 
•64082 
•64109 
■64136 


1-76945 
1-77154 
1-77362 
1-77571 
1-77780 


-65525 
.65552 
.65579 
-65607 
-65634 


1.90063 
1.90293 
1-90524 
1-90754 
1.90986 

1-91217 
1-91449 
1.91681 
1-91914 
1-92147 


-67168 
-67196 
.67223 
.67251 
.67278 


2 
2 
2 
2 
2 


-04584 
-C4839 
-05094 
.05350 
-05607 


.68822 
. 68849 
.68877 
.68905 
■68932 


2 
2 
2 
2 
2 


■20737 
■21021 
■21306 
■21592 
■21878 


50 

51 
52 
53 
54 


55 
56 
57 
58 

59 


1-77990 
1-78200 
1-78410 
1-78621 
1-78832 


-65661 
-65689 
-65716 
-65743 
-65771 


.67306 
.67333 
.67361 
.67388 
-67416 


2 
2 
2 
2 
2 


-05864 
-06121 
-06379 
-06637 
-06896 


■68960 
■68988 
.69015 
. 69043 
■69071 


2 
2 
2 
2 
2 


-22165 
■22452 
■22740 
.23028 
.23317 


55 
56 
57 
58 
59 


60 


•64i63 


1-79043 


-65798 


1-92380 


- 67443 


2 


.07155 


.69098 


2 


-23607 


60 



793 



/ 



TABLE X.— NATURAL VEIlJSED STNES AND EXTERNAL SECANTSL 



73' 



73' 



74° 



75= 






Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


• 
Vers. 


Ex. sec. 


* 




1 

2 
3 
4 


•69098 
.69126 
.69154 
.69181 
.69209 


2.23607 
2.23897 
2.24187 
2.24478 
2.24770 


.70763 
.70791 
.70818 
. 70846 
.70874 


2.42030 
2.42356 
2.42683 
2.43010 
2.43337 


.72438 
.72464 
.72492 
.72520 
.72548 


2 
2 
2 
2 
2 


.62796 
.63164 
.63533 
.63903 
.64274 


-74118 
-74146 
.74174 
. 74202 
.74231 


2 
2 
2 
2 
2_ 

2 
2 
2 
2 
2 


.86370 
.86790 
.87211 
.87633 
.88056 

.88479 
.88904 
.89330 
.89756 
.90184 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.69237 
.69264 
.69292 
.69320 
■69347 


2.25062 
2.25355 
2.25648 
2^25942 
2.26237 


.70902 
.70930 
•70958 
•70985 
.71013 


2.43666 
2.43995 
2.44324 
2.44655 
2.44986 


.72576 
. 72604 
. 72632 
.72660 
.72688 


2 
2 
2 
2 
2 


.64645 
.65018 
.65391 
.65765 
.66140 


. 74259 
.74287 
.74315 
. 74343 
.74371 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•69375 
69403 
•69430 
.69458 
•69486 


2.26531 
2.26827 
2.27123 
2.27420 
2.27717 


.71041 
.71069 
.71097 
.71125 
•71153 


2-45317 
2-45650 
2.45983 
2.46316 
2.46651 


.72716 
.72744 
.72772 
.72800 
.72828 


2 
2 
2 
2 
2 


.66515 
.66892 
-67269 
-67647 
-68025 


.74399 
. 74427 
•74455 
. 74484 
•74512 


2 
2 
2 
2 
2 


.90613 
.91042 
.91473 
.91904 
.92337 


10 

11 
12 
13 

14 


15 
16 
17 
18 
19 


.69514 
.69541 
•69569 
69597 
•69624 


2.28015 
2.28313 
2.28612 
2.28912 
2-29212 


•71180 
•71208 
.71236 
.71264 
.71292 


2.46986 
2.47321 
2.47658 
2.47995 
2.48333 


.72856 
.72884 
.72912 
.72940 
.72968 

•72996 
•73024 
.73052 
.73080 
.73108 


2 
2 
2 
2 

2 


68405 
68785 
69167 
59549 
69931 


• 74540 
•74568 
•74596 
.74824 
.74652 

.74680 

• 74709 
•74737 
•74765 
. 74793 


2 
2 
2 
2 
2 


92770 
93204 
93640 
94076 
94514 


15 
18 
17 
18 
19 


20 

21 
22 
23 
24 


•69652 
.69680 
.69708 
.69735 
.69763 


2.29512 
2.29814 
2.30115 
2-30418 
2-30721 


.71320 
.71348 
.71375 
.71403 
.71431 


2.48671 
2-49010 
2.49350 
2.49691 
2.50032 


2 

2. 

2. 

2. 

2. 


70315 
70700 
71085 
71471 
71858 


I 

2 
2 
2 


94952 
95392 
95832 
96274 
96716 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.69791 
.69818 
•69846 
•69874 
•69902 


2.31024 
2.31328 
2.31633 
2.31939 
2.32244 


.71459 
.71487 
.71515 
.71543 
.71571 


2. 50374 
2.50716 
2.51060 
2.51404 
2.51748 


.73136 
.73164 
.73192 
.73220 
• 73248 


2. 
2- 
2- 
2- 
2- 


72246 
72635 
73024 
73414 
73806 


.74821 
. 74849 
.74878 
. 74906 
. 74934 


2 
2 
2 
2 
2. 


97160 
97604 
98050 
98497 
98944 


23 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•69929 
•69957 
.69985 
.70013 
. 70040 


2.32551 
2.32858 
2.33166 
2.33474 
2.33783 


.71598 
.71626 
.71654 
.71682 
•71710 


2.52094 
2.52440 
2.52787 
2.53134 
2.53482 


•73276 
-73304 
•73332 
.73360 
.73333 


2. 

2. 
2. 
2. 
2. 


74198 
74591 
74984 
75379 
75775 


.74962 
.74990 
.75018 
.75047 
.75075 


2. 
2. 
3^ 
3^ 
3^ 


99393 
99843 
00293 
00745 
01198 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.70068 
.70096 
.70124 
•70151 
70179 


2.34092 
2.34403 
2.34713 
2.35025 
2.35336 


•71738 
•71766 
•71794 
.71822 
.71850 

.71877 
.71905 
•71933 
.71961 
.71989 

.72017 
.72045 
.72073 
.72101 
.72129 


2.53831 
2.54181 
2.54531 
2.54883 

2.55235 

2.55587 
2.55940 
2.56294 
2.56649 
2.57005 


.73416 
. 73444 
• 73472 
•73500 
.73529 


2. 

2. 
2. 
2- 
2. 


73171 
76568 
76966 
77365 
77765 


.75103 
.75131 
.75159 
•75187 
•75216 


3. 
3. 
3. 
3. 

3. 


01652 
02107 
02563 
03020 
03479 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•70207 
•70235 
•70263 
• 70290 
70318 


2.35649 
2.35962 
2.36276 
2.36590 
2.36905 


.73557 
.73585 
.73613 
.73641 
.73669 


2. 
2. 
2. 
2. 
2. 


78168 
78588 
78970 
79374 
79778 


•75244 
•75272 
•75300 
•75328 
.75356 


3. 
3. 
3. 
3. 
3. 


03938 
04398 
04860 
05322 
05786 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•70346 
•70374 
• 70401 
. 70429 
.70457 


2.37221 
2.37537 
2.37854 
2.38171 
2.38489 


2.57361 
2.57718 
2.58076 
2 . 58434 
2.58794 


.73697 
.73725 
.73753 
.73781 
.73809 


2. 
2. 
2. 
2. 
2. 


80183 
80589 
80996 
81404 
81813 


.75385 
.75413 
.75441 
.75469 
.75497 


3. 
3. 
3. 
3. 

3. 


06251 
06717 
07184 
07652 
08121 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.70485 
•70513 
•70540 
•70568 
.70596 


2.38808 
2^39128 
2 •39448 
2.39768 
2.40089 


.72157 
.72185 
.72213 
.72241 
.72269 


2.59154 
2.59514 
2.59876 
2.60238 
2.60601 


.73837 
.73865 
.73893 
.73921 
.73950 


2. 
2. 
2. 
2. 

2. 


82223 
82633 
83045 
83457 
83871 


.75526 
.75554 
.75582 
■75610 
•75639 


3- 
3^ 
3^ 
3^ 
3^ 


08591 
09063 
09535 
10009 
10484 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•70624 
.70652 
.70679 
.70707 
.70735 


2.40411 
2.40734 
2.41057 
2.41381 
2.41705 


.72296 
•72324 
.72352 
.72380 
.72408 


2.60965 
2.61330 
2.61695 
2.62061 
2-62428 


-73978 
. 74006 
. 74034 
. 74062 
•74090 


2. 
2. 
2. 
2. 
2 


84285 
84700 
85116 
85533 
85951 


•75667 
•75695 
•75723 
75751 
.75780 


3. 
3. 
3. 
3. 
3. 


10960 
11437 
11915 
12394 
12875 


55 
56 
57 
58 
59 


60 


•70763 


2.42030 


.72436 


2.62796 


•74118 


2 


86370 


.75808 


3. 


13357 


60 



794 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
76° 77° 78° 79° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers, 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


•75808 
75836 
•75864 
•75892 
•75921 

•75949 
•75977 
•76005 
•76034 
•76062 


3 
3 
3 
3 
3 


13357 
13839 
14323 
14809 
15295 


■77505 
•77533 
•77562 
•77590 
•77618 


3. 
3- 
3^ 
3^ 
3^ 


44541 
45102 
45664 
46228 
46793 


•79209 a 
•79237 S 
•79266 S 
• 79294 !a 
•79323 S 


[•80973 
[•81633 
.•82294 
[•82956 
[•83621 


•80919 
•80948 
•80976 
•81005 
■81C33 


4 
4 
4 
4 
4 


24C84 
24870 
25658 
26448 
27241 




1 
2 
3 
4 


5 
6 
7 
8 
9 


3 

3 

3 

3^ 

3. 


15782 
16271 
16761 
17252 
17744 


•77647 
•77675 
•77703 
•77732 
■77760 


3^ 
3. 
3^ 
3^ 
3^ 
3- 
3- 
3^ 
3^ 
3^ 


47360 
47928 
48498 
49069 
49642 

50216 
50791 
51368 
51947 
52527 


•79351 c 
•79380 J 
•79408 : 
•79437 i 
•79465 ; 


[•84288 
{•84956 
J. 85627 
J .86299 
3 •86973 


•81062 
•81090 
•81119 
•81148 
•81176 


4 
4 
4 
4 
4 


28036 
28833 
29634 
30436 
31241 


5 
6 
7 
8 
9 


10 

11 
12 
13 

14 


•76090 
•76118 
•76147 
•76175 
•76203 


3^ 
3^ 
3^ 
3^ 
3^ 


18238 
18733 
19228 
19725 
20224 


•77788 
•77817 
•77845 
•77874 
•77902 


•79493 . 
•79522 
•79550 
•79579 
79607 


3-87649 
3 •88327 
3.89007 
3-89689 
3-90373 


•81205 
•81233 
•81262 
•81290 
•81319 


4 
4 
4 
4 
4 


•32049 
•32859 
•33671 
.34486 
.35304 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•76231 
•76260 
•76288 
•76316 
•76344 


3^ 
3^ 
3- 
3^ 
3^ 


20723 
21224 
21726 
22229 
22734 


• 77930 
•77959 
•77987 
•78015 
. 78044 


3 

3 

3 

3 

3. 

3 

3 

3 

3 

3 


53109 
53692 
54277 
54863 
55451 

. 56041 
.56632 
.57224 
.57819 
•58414 


•79636 
•79664 
.79693 
.79721 
•79750 


3^91058 
3^91746 
3-92436 
393128 
3-93821 


•81348 
■81376 
■81405 
•81433 
■8]462 


4 
4 
4 
4 
4 


.36124 
.36947 
•37772 
•38600 
•39430 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•76373 
•76401 . 
•76429 
•76458 
•76486 


3^ 

3 

3 

3 

3 


23239 
23746 
24255 
24764 
25275 


• 78072 
•78101 
•78129 
•78157 
•78186 

.78214 
. 78242 
•78271' 
•78299 
•78328 


• 79778 
•79807 
•79835 
•79864 
■79892 


394517 
3-95215 
3-95914 
3-96616 
3-97320 


•81491 
.81519 
.81548 
.81576 
•81605 


4 
4 
4 
4 
4 


•40263 
•41099 
•41937 
•42778 
43622 


30 

2a 

23 
24 


25 
26 
27 
28 
29 


•76514 
•76542 
-76571 
•76599 
•76627 


3 
3 
3 
3 
3 


25787 

26300 

•26814 

•27330 

27847 


3 
3 
3 
3 
3 


•59012 
•59611 
•60211 
•60813 
■61417 


•79921 
•79949 
•79978 
•80006 
80035 


3-98025 
398733 
3-99443 
4-00155 
^■C0869 


•81633 
•81662 
•81691 
•81719 
■81748 


4 
4 
4 
4 
4 


44468 
45317 
46169 
47023 
47881 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•76655 
•76684 
•76712 
• 76740 
76769 


3 
3 
3 
3 
3_ 

3 
3 
3 

1 


.28366 
.28885 
.29406 
.29929 
30452 

30977 
31503 
32031 
32560 
33090 


•78356 
•78384 
•78413 

• 78441 

• 78470 


3 
3 
3 
3 
3 


■62023 
•62630 
•63238 
•63849 
. 64461 


•80063 
•80092 
•80120 
•80149 
-80177 


4-01585 
4-02303 
4-03024 
4-03746 
4-04471 


■81776 
•81805 
•81834 
•81862 
•81891 


4 
4 
4 
4 
4 


48740 
49603 
50468 
•51337 
52208 


30 

31 
32 
33 
34 


35 
36 
87 
38 

39 


•76797 
•76825 
•76854 
•76882 
•76910 


• 78498 
•78526 
•78555 
•78583 
•78612 

.78640 
•78669 
•78697 
•78725 
•78754 


3 
3 
3 
3 
3 


•65074 
•65690 
•66307 
•66925 
■67545 


•80206 
.80234 
.80263 
.80291 
•80320 ' 

•80348 ' 
•80377 ' 
•80405 ' 
•80434 ' 
•80462 ' 


4-05197 
4-05926 
406657 
4.07390 
1.08125 


•81919 
•81948 
.81977 
.82005 
•82034 

.82063 
.82091 
.82120 
.82148 
.82177 


4 
4 
4 
4 
4 


•53081 

•53958 

54837 

55720 

56605 


35 
36 
37 
38 

39 


40 

41 
42 
43 
44 


.76938 
•76967 
•76995 
•77023 
•77052 


3. 
3. 
3. 
3. 
3^ 


33622 
34154 
34689 
35224 
35761 


3 
3 
3 
3 
3 


68167 
68791 
69417 
70C44 
70673 


i. 08863 
1-09602 
1-10344 
1-11088 
1-11835 


4 
4 
4 
4 


57493 
58383 
59277 
60174 
61073 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•77080 
•77108 
•77137 
•77165 
•77193 


3^ 
3^ 
3^ 
3- 
3^ 


36299 
36839 
37380 
37923 
38466 


.78782 
.78811 
.78839 
•78868 
•78896 

•78924 
•78953 
•78981 
.79010 
•79038 


3 

3. 

3. 

3. 

3. 


71303 
71935 
72569 
73205 
73843 


•80491 ' 
•80520 ' 
•80548 ' 
•80577 ' 
•80605 ' 


1-12583 
1.13334 
1.14087 
1-14842 
1-15599 


.82206 
.82234 
.82263 
■82292 
■82320 


4 
4 
4 
4 
4. 


61976 
62881 
63790 
64701 
65616 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.77222 
•77250 
.77278 
•77307 
•77335 


3^ 

3^ 

3^ 

3 

3 


39012 
39558 
40106 
40656 
41206 


3. 
3. 
3. 
3. 
3. 


74482 
75123 
75766 
76411 
77057 


•80634 ' 
•80662 ' 
.80691 ^ 
.80719 ^ 
•80748 ^ 

•80776 4 
•80805 4 
•80833 4 
.80862 4 
•80891 4 


1-16359 
1-17121 
1-17886 
1-18652 
1-19421 

t- 20193 
t-20966 
t- 21742 
t-22521 
t- 23301 


■82349 
•82377 
•82406 
.82435 
•82463 

.82492 
.82521 
.82549 
.82578 
•82607 


4 
4 
4 
4 
4 


66533 
67454 
68377 
69304 
70234 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.77363 
.77392 
•77420 
• 77448 
•77477 


3 
3 
3 
3 
3 


41759 
42312 
42867 
43424 
43982 


•79067 
•79095 
•79123 
.79152 
•79180 


3. 
3^ 
3. 
3. 

L 
3. 


77705 
78355 
79007 
79661 
80316 

80973 


4 
4 
4 
4 
4. 


71166 
72102 
73041 
73983 
74929 


55 
56 
57 
58 
-19 


60 


•77505 


3 


44541 


.79209 


•80919 4 


t. 24084 


.82655 


4. 


75877 


60 



795 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



80' 



sr 



83' 



83° 



* 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sete. 


f 




1 

2 
3 

4 


•82685 
.82664 
.82692 
.82721 
.82750 


4.75877 
4.76829 
4.77784 
4.78742 
4.79703 


.84357 
.84385 
.84414 
. 84443 
.84471 


5 
5 
5 
5 
5 


39245 
40422 
41602 
42787 
43977 


.86083 
.86112 
.86140 
.86169 
.86198 


6 
6 
6 
6 
6 


18530 
20020 
21517 
23019 
24529 


.87813 
.87842 
•87871 
•87900 
•87929 


7.20551 
7.22500 
7.24457 
7.26425 
7.28402 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.82778 
.82807 
•82836 
.82864 
.82893 


4-80667 
4-81635 
4-82606 
4-83581 
4.84558 


.84500 
.84529 
.84558 
84586 
.84615 


5 
5 
5 
5 
5 


45171 
46369 
47572 
48779 
49991 


•86227 
-86256 
.86284 
.88313 
•86342 


6 

6 
6 
6 
6 


26044 
27563 
29095 
30630 
32171 


•87957 
•87988 
•88015 
• 88044 
•88073 


7.30388 
7.32384 
7.34390 
7.36405 
7. 38431 


5 
6 
7 
8 
9 


10 

11 
12 
13 
U 


.82922 
82950 
.82979 
.83008 
.83036 


4-85539 
4-86524 
4-87511 
4.88502 
4.89497 


.84644 
.84673 
.84701 
.84730 

.84759 

.84788 
.84816 
.84845 
.84874 
.84903 

.84931 
•84960 
.84989 
-85018 
-85046 


5 
5 
5 
5 
5 


51208 
52429 
53655 
54886 
56121 


•86371 
•86400 
•86428 
•88457 
•86486 


6 
6 
6 
8 
6 


33719 
35274 
36835 
38403 
39978 


•88102 
•88131 
•88160 
•88188 
•88217 


7.40466 
7.42511 
7.44566 
7.46632 
7. 48707 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.83065 
.83094 
.83122 
.83151 
.83180. 


4.90495 
4-91496 
4-92501 
4-93509 
4-94521 

4.95536 
4-96555 
4-97577 
4-98603 
4-99633 


5 
5 

5 
5 
5 


57361 
58606 
59855 
61110 
62369 


•86515 
•86544 
•86573 
•88601 
•86630 


6 
6 
6 
6 
6 


41560 
43148 
44743 
46348 
47955 


•88246 
•88275 
•88304 
•88333 
•88362 


7.50793 
7.52889 
7.54996 
7.57113 
7.59241 


15 
16 
17 
18 
19 


30 

21 
22 
S3 
24 


83208 
.83237 
.83266 
.83294 
.83323 


5 
5 
5 
5 
5 


63633 
64902 
66176 
67454 
68738 


•86659 
.88683 
.86717 
.86748 
•86774 


6 
6 
6 

6 
6 


49571 
51194 
52825 
54462 
56107 


•88391 
•88420 
•88448 
•88477 
•88506 


7.61379 
7.63528 
7.65688 
7.67859 
7.70041 


30 

21 

2| 

24 


25 
26 
27 
28 
29 


.83352 
.83380 
.83409 
.83438 
.83467 


5-00686 
5-01703 
5-02743 
5-03787 
5-04834 


-85075 
-85104 
.85133 
85162 
.85190 


5 

5 
5. 
5 
5 


70027 
71321 
72620 
73924 
75233 


•86803 
•80832 
.88861 
.86890 
.86919 


6 
6 
6 
6 
6_ 

6 
6 
6 
6 
6 


57759 
.59418 
61085 
62759 
64441 


•88535 
•88564 
•88593 
•88622 
•88651 

•88680 
•88709 
•88737 
•88766 
•88795 


7.72234 
7.74438 
7.76653 
7-78880 
7-81118 


25 
26 
27 

2J! 

29 


30 

31 
32 
33 
34 


.83495 
.83524 
.83553 
.83581 
83610 


5-05886 
5.06941 
5-08000 
5.09062 
5.10129 


-85219 
-85248 
-85277 
-85305 
-85334 


5 
5 
5 
5 
5 


76547 
77866 
79191 
80521 
81856 


.86947 
.86976 
.87005 
.87034 
.87063 

.87092 
.87120 
.87149 
.87178 

.87207 

.87236 
•87265 
•87294 
-87322 
.87351 


66130 
67826 
69530 
71242 
72962 


7-83367 
7-85628 
7-87901 
7-90186 
7-92482 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


83639 
.83667 
.83696 
.83725 
.83754 


5.11199 
5.12273 
5-13350 
5-14432 
5.15517 


■85363 
.85392 
.85420 
.85449 
.85478 


5 

5 
5 
5 

L 

5 

5 

i 

5 


83196 
84542 
85893 
87250 
88612 

89979 
91352 
92731 
94115 
95505 


6 
6 
6 
6 

L 

6 
6 
6 
6 


74689 
76424 
78167 
79918 
81677 


•88824 
•88853 
•88882 
•88911 
• 88940 


7-94791 
7-97111 
7-99444 
8-01788 
8-04146 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.83782 
.83811 
.83840 
.83868 
•83897 


5.16607 
5.17700 
5.18797 
5.1989n 
5.21004 


.85507 
.85536 
.85564 
.85593 
.85622 


83443 
85218 
87001 
88792 
90592 


•88969 
•88998 
•89027 
•89055 
•89084 


8.06515 
8.08897 
8.11292 
8.13699 
8.16120 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.83926 
-83954 
.83983 
.84012 
.84041 


5-22113 
5-23226 
5-24343 
5-25464 
5-26590 


.85651 
85680 
.85708 
.85737 
.85766 


5 
5 
5 
6 
6 


96900 
98301 
99708 
01120 
02538 


.87380 
.87409 
•87438 
•87467 
•87496 


6 
6 
6 
6 
6 


92400 
94216 
96040 
97873 
99714 


•89113 
•89142 
•89171 
•89200 
•89229 


8.18553 
8.20999 
8-23459 
8.25931 
8.28417 


45 
46 
47 
48 
4d 


50 

51 
52 
53 
54 


.84069 
.84098 
.84127 
.84155 
.84184 


5.27719 
5-28853 
5-29991 
5-31133 
5-32279 


.85795 
.85823 
.85852 
.85881 
.85910 

.85939 
.85967 
.85996 
.86025 
.86054 

.86083 


6 
6 
6 
6 
6 


03962 
05392 
06828 
08269 
09717 


.87524 
.87553 
.87582 
.87611 
•87640 


7 
7 
7 
7 
7 


01565 
03423 
05291 
07167 
09052 


•89258 
•89287 
•89316 
•89345 
•89374 


8.30917 
8.33430 
8.35957 
8.38497 
8.41052 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.84213 
. 84242 
.84270 
•84299 
.84328 


5-33429 
5-34584 
5.35743 
5.36906 
5-38073 


6 
6 
6 
6 


11171 
.12630 
.14096 
.15568 
.17046 


•87669 
•87698 
•87726 
•87755 
•87784 


7 
7 
7 
7 
7 


10946 
12849 
14760 
16681 
18612 


.89403 
.89431 
.89460 
•89489 
.89518 


8.43620 
8.46203 
8.48800 
8.51411 
8.54037 


55 
56 
57 
58 
59 


60 


.84357 


5.39245 


.18530 


.87813 


7 


20551 


.89547 


8.56677 


60 



796 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

84° 85° 86° 



/ 


Vers. 
.89547 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 





8.56677 


•91284 


10.47371 


93024 


13.33559 





1 


-89576 


8.59332 


.91313 


10.51199 


•93053 


13-39547 


1 


2 


•89605 


8 


. 62002 


•91342 


10.55052 


.93082 


13.45586 


2 


3 


•89634 


8 


.6468? 


•91371 


10 


•58932 


•93111 


13.51676 


3 


4 


.89663 


8 


.67387 


•91400 


10 


•62837 


.93140 


13.57817 


4 


5 


89692 


8 


•70103 


91429 


10 


66769 


•93169 


13.64011 


5 


6 


89721 


8 


.72833 


91458 


10 


•70728 


•93198 


13.70258 


6 


7 


89750 


8 


75579 


91487 


10 


•74714 


•93227 


13.76558 


7 


8 


89779 


8 


78341 


91516 


10 


•78727 


•93257 


13.82913 


8 


9 


89808 


8 


81119 


91545 


10 


.82768 


•93286 


13.89323 


9 


10 


89836 


8 


83912 


91574 


10 


.86837 


.93315 


13.95788 


10 


11 


89865 


8 


86722 


91603 


10 


90934 


.93344 


14^02310 


11 


12 


89894 


8 


89547 


91632 


10 


95060 


.93373 


14^08890 


12 


13 


89923 


8 


92389 


91661 


10 


99214 


93402 


14^15527 


13 


14 


89952 


8 


95248 


91690 


11 


03397 


93431 


14^22223 


14 


15 


89981 


8 


98123 


91719 




07610 


93460 


14.28979 


15 


16 


90010 


9 


01015 


91748 


11 


11852 


93489 


14.35795 


16 


17 


90039 


9 


03923 


91777 


11 


16125 


93518 


14.42672 


17 


18 


90068 


9 


06849 


91806 


11 


20427 


93547 


14-49611 


18 


19 


90097 


9 


09792 


91835 


11 


24761 


93576 


14-56614 


19 


30 


90126 


9 


12752 


91864 


11 


29125 


93605 


14-63679 


20 


21 


90155 


9 


15730 


91893 


11 


33521 


93634 


• 14-70810 


21 


22 


90184 


9 


18725 


91922 


11 


37948 


93663 


14-78005 


22 


23 


90213 


9. 


21739 


91951 


11 


42408 


93692 


14.85268 


23 


24 


90242 


9 


24770 


91980 


11 


46900 


93721 


14-92597 


24 


25 


90271 


9. 


27819 


92009 


ii 


51424 


93750 


14-99995 


25 


26 


90300 


9. 


30887 


92038 


11 


55982 


93779 


15-07462 


26 


27 


90329 


9. 


33973 


92067 


11 


60572 


93808 


15-14999 


27 


28 


90358 


9. 


37077 


92096 


11 


65197 


93837 


15-22607 


28 


29 


90386 


9. 


40201 


92125 


11 


69856 


93866 


15-30287 


29 


30 


90415 


43343 


92154 


u 


74550 


93895 


15-38041 


30 


31 


90444 


9 


46505 


92183 


11 


79278 


93924 


15-45869 


31 


32 


90473 


9 


49685 


92212 


11 


84042 


93953 


15.53772 


32 


33 


90502 


o . 


52886 


92241 


11 


88841 


93982 


15.61751 


33 


34 


90531 


9 


56106 


92270 


u 


93677 


94011 


15.69808 


34 


35 


90560 


9. 


59346 


92299 


11 


98549 


94040 


15.77944 


35 


36 


90589 


9. 


62605 


92328 


12 


03458 


94069 


15.86159 


36 


37 


90618 


9. 


65885 


92357 


12 


08404 


94098 


15.94456 


37 


38 


90647 


9. 


69186 


92386 


12 


13388 


94127 


16.02835 


38 


39 


90676 


9. 


72507 


92415 
92444 


12 
12. 


18411 


94156 


16.11297 


39 


40 


90705 


9. 


75849 


23472 


94186 


16-19843 


40 


41 


90734 


9. 


79212 


92473 


12 


28572 


94215 


16.28476 


41 


42 


90763 


9. 


82596 


92502 


12 


33712 


94244 


16.37196 


42 


43 


90792 


9. 


86001 


92531 


12 


38891 


94273 


16.46005 


43 


44 


90821 


9. 


89428 


92560 


12 


44112 


94302 


16.54903 


44 


45 


90850 


9. 


92877 


92589 


12 


49373 


94331 


16.63893 


45 


46 


90879 


9. 


96348 


92618 


12. 


54676 


94360 


16-72975 


46 


47 


90908 


9. 


99841 


92647 


12 


60021 


94389 


16-82152 


47 


48 


90937 


10. 


03356 


92876 


12. 


65408 


94418 


16-91424 


48 


49 


90966 


10. 


06894 


92705 


12 


70838 


94447 


17.00794 


49 


50 


90995 


10. 


10455 


92734 


12. 


76312 


94476 


17.10262 


50 


51 


91024 


10. 


14039 


92763 


12 


81829 


94505 


17-19830 


bl 


52 


91053 


10. 


17646 


92792 


12 


87391 


94534 


17-29501 


52 


53 


91082 


10. 


21277 


92821 


12 


92999 


94563 


17.39274 


53 


54 


91111 


10 


24932 


92850 


12 


98651 


94592 


17.49153 


54 


55 


91140 


10 


28610 


92879 


13 


04350 


94621 


17.59139 


55 


56 


91169 


10 


32313 


92908 


13 


10096 


94650 


17.69233 


56 


57 


91197 


10 


36040 


92937 


13 


15889 


94679 


17.79438 


57 


58 


91226 


10 


39792 


92966 


13 


21730 


94708 


17.89755 


b8 


59 


91255 


10 


43569 


92995 
93024 


13 


27620 


94737 


18.00185 


59 


60 


91284 


10 


47371 1 


13 


33559 


94766 


18.10732 


60 



797 



TABLE X.— NATURAL VERSED SINES AND EXTE^INAL SECANTS. 





87° 


88° 




89° 




1 


Vera. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 





.94766 


18.10732 


.96510 


27.65371 


98255 


P,S. 29869 





1 


.94795 


18.21397 


.96539 


27 


89440 


98284 


57.26976 


1 


2 


.94825 


18.32182 


.96568 


28 


13917 


98313 


58.27431 


2 


3 


.94854 


18.43088 


.96597 


28 


38812 


98342 


59.31411 


3 


4 


.94883 


18.54119 


.96626 


28 


64137 


98371 


60.39105 


4 


5 


.94912 


18.65275 


.96655 


28 


89903 


98400 


61.50715 


5 


6 


.94941 


18.76560 


.96684 


29 


16120 


98429 


62.66460 


6 


7 


.94970 


18.87976 


.96714 


29 


42802 


98458 


63-86572 


7 


8 


.94999 


18.99524 


.96743 


29 


69960 


98487 


65.11304 


8 


9 


.95C2B 


19.11208 


.96772 


29 


97607 


98517 


66.40927 


9 


10 


.95057 


19.23028 


.96801 


30 


25758 


98546 


67.75736 


10 


11 


.95086 


19.34989 


.96830 


30 


54425 


98575 


69.16047 


11 


12 


.95115 


19.47093 


.96859 


30 


83623 


.98604 


70.62205 


12 


13 


.95144 


19 59341 


.96888 


31 


13366 


98633 


72-14583 


13 


14 


.95173 


19.71737 


.96917 


31 


43671 


98662 


73.73586 


14 


15 


.95202 


19.84283 


.96946 


31 


74554 


98691 


75.39655 


15 


16 


.95231 


19-96982 


.96975 


32 


06030 


98720 


77.13-274 


16 


17 


.95260 


20.09838 


.97004 


32 


38118 


98749 


78.94968 


17 


18 


.95289 


20.22852 


.97033 


32 


70835 


98778 


80-85315 


18 


J9- 


.95318 
.95347 


20.36027 
20.49368 


-97062 


33 


04199 


98807 


82-84947 


19 


20 


.97092 


33 


38232 


98836 


84-94561 


20 


21 


.95377 


20.62876 


.97121 


33 


72952 


S8866 


87.14924 


21 


22 


.95406 


20.76555 


.97150 


34 


08380 


98895 


89-46886 


22 


23 


.95435 


20.90409 


.97179 


34 


44539 


98924 


91-91387 


23 


24 


.95464 


21.04440 


.97208 


34 


81452 


98953 


94-49471 


24 


25 


.95493 


21.18653 


.97237 


35 


19141 


98982 


97-22303 


25 


26 


.95522 


21.33050 


.97266 


35 


57633 


99011 


100.1119 


26 


27 


.95551 


21.47635 


.97295 


35 


96953 


99040 


103.1757 


27 


28 


.95580 


21.62413 


.97324 


36 


37127 


99069 


106-4311 


28 


29 


.95609 


21-77386 


.97353 


36 


78185 


99098 


109-8966 


29 


80 


.95638 


21-92559 


.97382 


37 


20155 


99127 


113-5930 


30 


31 


.95667 


22-07935 


.97411 


37 


63068 


99156 


117-5444 


31 


32 


.95696 


22-23520 


.97440 


38 


06957 


99186 


121.7780 


S2 


33 


.95725 


22-39316 


.97470 


38 


51855 


99215 


126-3253 


33 


34 


•95754 


22-55328 


.97499 


38 


97797 


99244 


131-2223 


34 


35 


.95783 


22.71563 


.97528 


39 


44820 


99278 


136-5111 


35 


36 


.95812 


22.88022 
23.04712 


.97557 


39 


92963 


99302 


142-2406 


36 


37 


.95842 


.97586 


40 


42266 


99331 


148-4684 


37 


38 


.95871 


23.21637 1 


.97615 


40 


92772 


99360 


155-2623 


38 


39 


.95900 


23. 38802 


97644 


41 


44525 


99389 


162-7033 


39 


40 


■95929 


23.56212 


.97673 


41 


97571 


9941J. 


170-8883 


40 


41 


.95958 


23-73873 


.97702 


42 


51961 


994^ 


179-9350 


41 


42 


.95987 


23-91790 


.97731 


43 


07746 


99476 


189-9868 


42 


43 


.96016 


24-09969 


.97760 


43 


64980 


99505 


201.2212 


43 


44 


.96045 


24-28414 


.97789 


44 


23720 


99535 


213-8600 


44 


45 


.96074 


24-47134 


.97819 


44 


84026 


99564 


228.1839 


45 


46 


.96103 


24-66132 


.97848 


45 


459S3 


99593 


244-5540 


46 


47 


.96132 


24.85417 


.97877 


46 


09596 


99622 


263.4427 


47 


48 


.96161 


25.04994 


.97906 


46 


74997 


99651 


285.4705 


48 


49 


.96190 


25.24869 


.97935 


47 


42241 


9S680 


311-5230 


49 


50 


.96219 


25.45051 


.97964 


48 


11406 


.99709 


342.7752 


%0 


51 


.96243 


25.65546 


•979^3 


48 


82576 


.99738 


380-9723 


51 


52 


.9627? 


25.86360 


.98^22 


49 


55840 


.99767 


428.7187 


52 


53 


.96307 


2ft. 97503 


.98051 


50 


31290 


.99796 


490.1070 


-'33 


54 


.96336 


26 ^-rsgci 


-98080 


51 


09027 


99825 


571-9581 


54 


55 


.96365 


26.N>0804 


.98109 


51 


89156 


.99855 


686.5496 


55 


56 


.96394 


26.V2978 


.98138 


52 


71790 


.99884 


858.4369 


56 


57 


.96423 


26.95513 


.98168 


53 


57046 


.99913 


1144.916 


57 


58 


.96452 


27.18417 


.98197 


54 


45053 


.99942 


1717.874 


53 


i9- 


,96481 
.86510 


27.41700 


.98226 


55 


35946 


99971 


3^36-747 


59 


•50 


27.65371 


.98255 


56 


29869 1 


00000 


Infinite 


6Q 



798 



TABLE XI.— REDUCTION OF BAROMETER READING TO 32° F. 















[nches. 












Temp. 
























O 
r:!hr. 


26-0 


26-5 


27-0 


27.5 


28.0 


28.5 


29.0 


29.5 


30.0 


30.5 


310 


45 


-.039 


-.039 


-.040 


-.041 


-.042 


^ =042 


-.043 


-.044 


-■045 


-■045 


-■046 


46 


.041 


.042 


.043 


.043 


■ 044 


■ 045 


.046 


•046 


■047 


■048 


■049 


47 


•043 


•044 


.045 


.046 


■047 


.048 


.048 


•049 


■050 


■051 


■ 052 


48 


.046 


-047 


.047 


.048 


■ 049 


.050 


.051 


•052 


■053 


■053 


■ 054 


49 


•048 


•049 


.050 


.051 


■052 


•052 


.054 


.054 


■055 


■056 


.057 


50 


.050 


•051 


.052 


.053 


■054 


.055 


.056 


.057 


.058 


■059 


.060 


51 


.053 


.054 


.055 


.056 


■057 


.058 


■059 


.060 


.061 


■062 


■063 


52 


.055 


.056 


•057 


•058 


■059 


■060 


■061 


•062 


■064 


■065 


■ 066 


53 


•057 


.058 


.060 


.061 


■062 


■ 063 


■064 


•065 


■066 


■ 067 


■068 


54 


.060 


.061 


•062 


■063 


■064 


■ 065 


.067 


.068 


■069 


■070 


■071 


55 


.062 


.063 


.064 


.065 


■066 


■ 068 


.069 


.070 


■071 


■073 


■074 


56 


■064 


•065 


■ 067 


.068 


■ 069 


■070 


•072 


.073 


■074 


■075 


• 077 


57 


.067 


.068 


■ 069 


.070 


■ 072 


.073 


.075 


.076 


■077 


■078 


■080 


58 


•069 


.070 


•071 


.073 


■ 074 


■ 076 


■077 


•078 


■080 


■081 


■082 


59 


.072 


.073 


.074 


.075 


■ 077 


.078 


.080 


•081 


.083 


■084 


■ 085 


60 


.074 


.076 


.077 


.078 


■ 079 


.081 


.082 


•084 


.085 


■ 086 


■088 


61 


.076 


.077 


.079 


.080 


■082 


.083 


.085 


■ ^086 


.088 


■ 089 


■ 091 


62 


.079 


.080 


.082 


.083 


■ 085 


■086 


.088 


• 089 


.091 


.092 


■094 


63 


•081 


.082 


•084 


.085 


■087 


■088 


■090 


•091 


.093 


■ 095 


■096 


64 


.083 


.085 


.086 


.088 


■090 


.091 


.093 


• 094 


.096 


■ 097 


■099 


65 


•086 


.087 


.089 


.090 


■092 


.093 


.095 


■ 097 


■099 


■ 100 


■ 102 


66 


.088 


.089 


.091 


.093 


■095 


.096 


.098 


•099 


■ 101 


■ 103 


■ 105 


67 


.090 


.092 


•094 


.095 


■097 


.099 


.101 


• 102 


■ 104 


■ 106 


■ 108 


68 


.093 


.094 


.096 


.098 


.100 


-101 


.103 


• 105 


.107 


.108 


• 110 


69 


.095 


.097 


.099 


.100 


.102 


.104 


.106 


• 107 


■ 110 


■ 111 


.113 


70 


.097 


.099 


.101 


.103 


.105 


.106 


.109 


• 110 


■ 112 


■ 114 


.116 


71 


.100 


.101 


.103 


.105 


.107 


.109 


■ 111 


■ 113 


.115 


■ 117 


.119 


72 


.102 


.104 


.106 


.108 


.110 


.112 


.114 


• 116 


.118 


■ 120 


• 122 


73 


.104 


.106 


• 108 


• 110 


.112 


• 114 


.116 


• 118 


.120 


■ 122 


.124 


74 


.107 


.109 


.111 


.113 


.115 


• 117 


.119 


■ 121 


.123 


.125 


.127 


75 


.109 


.111 


.113 


.115 


.117 


.119 


.122 


• 124 


.126 


.128 


.130 


76 


.111 


.113 


.116 


.118 


.120 


.122 


.124 


• 126 


.128 


.130 


.133 


77 


• 114 


.1.16 


■ 118 


.120 


.122 


.124 


.127 


• 129 


.131 


.133 


.136 


78 


.116 


.118 


.120 


.122 


.125 


.127 


■ 129 


-131 


.134 


.136 


.138 


79 


• 118 


.120 


•123 


.125 


• 127 


.129 


.132 


• 134 


.137 


.139 


..141 


80 


.121 


.123 


.125 


.127 


.130 


.132 


.135 


.137 


.139 


.141 


.144 


81 


.123 


.125 


.128 


■ 130 


.132 


.134 


■ 137 


.139 


■ 142 


.144 


• 147 


82 


• 125 


.128 


.130 


.132 


.135 


.137 


.140 


.142 


■ 145 


■ 147 


.149 


83 


.128 


• 130 


• 133 


.135 


• 138 


.140 


.142 


.145 


■ 147 


■ 149 


.152 


84 


.130 


• 132 


.135 


.138 


.140 


.142 


.145 


.147 


■ 150 


.152 


155 


85 


.132 


.134 


.137 


.140 


• 143 


.145 


.148 


.150 


.153 


.155 


.158 


86 


• 135 


■ 137 


• 140 


.142 


.145 


.148 


• 150 


.153 


.155 


.158 


.161 


87 


• 137 


.139 


■ 142 


.144 


• 148 


.150 


.153 


.155 


.158 


.161 


• .163 


88 


.139 


.142 


.145 


.147 


.150 


.152 


.155 


.158 


.161 


■ 163 


.166 


89 


rl42 


.144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.166 


.169 


90 


.144 


.147 


.150 


.153 


.155 


.158 


• 161 


.164 


.166 


.169 


.172 


91 


-.146 


-.149 


-.152 


-.155 


-.158 


-.160 


-.163 


-.166 


-.169 


-■172 


-.175 



799 



TABLE XII.— BAROMETRIC ELEVATIONS.* 



B 




A 


Inches. 


Feet. 


20 


11,047 


20 


1 


10,911 


20 


2 


10,776 


20 


3 


10,642 


20 


4 


10,508 


20 


5 


10.375 


20 


6 


10.242 


20 


7 


10,110 


20 


8 


9,979 


20 


9 


9,848 


21 





9,718 


21 


1 


9,589 


21 


2 


9,460 


21 


3 


9,332 


21 


4 


9. 204 


21 


5 


9,077 


21 


6 


8,951 


21 


7 


8,825 


21 


8 


8,700 


21 


9 


8,575 


22 





8,451 


22 


1 


8,327 


22 


2 


8,204 


22 


3 


8,082 


22 


4 


7,960 


22 


5 


7,838 


22 


6 


7,717 


22 


7 


7,597 


22 


8 


7,477 


22 


9 


7358 


23 





7,239 


23 


1 


7,121 


23 


2 


7,004 


23 


3 


6,887 


23 


4 


6,770 


23 


5 


6,554 


23 


6 


6,538 


23 


7 


6,423 



Diff. for 
• 01. 



Feet. 



-13 


6 


13 


5 


13 


4 


13 


4 


13 


3 


13 


3 


13 


2 


13 


1 


13 


1 


13 





12 


9 


12 


9 


12 


8 


12 


8 


12 


7 


12 


6 


12 


6 


12 


5 


12 


5 


12 


4 


12 


4 


12 


3 


12 


2 


12 


2 


12 


2 


12 


1 


12 





12 







9 




9 




8 




7 




7 




7 




6 




6 


~11 


5 



B 



Inches. 



23 
23 
23 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
27 
27 
27 
27 
27 



A 



Feet. 

6,423 
6,308 
6,194 
6,080 
5,967 
5-854 
5,741 
5,629 
5,518 
5-407 
5,296 
5,186 
5,077 
4,968 
4,859 
4,751 
4,643 
4,535 
4,428 
4,321 
4,215 
4109 
4,004 
3 899 
3,794 
3 690 
3,586 
3,483 
3,380 
3 277 
3,175 
3,073 
2 972 
2-871 
2 770 
2,670 
2,570 
2.470 



Diff. for 
■ 01. 



Feet. 



-11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 



10 



B 



Inches. 



27 
27 
27 
27 
27 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
31 



A 



Feet. 

2,470 

2,371 

2,272 

2,173 

2,075 

1,977 

1,880 

1-783 

] 686 

1,589 

1,493 

1,397 

1,302 

1.207 

1,112 

1,018 

924 

830 

738 

643 

550 

458 

366 

274 

182 

91 



-91 

181 

271 

361 

451 

540 

629 

717 

805 

-893 



Diff. for 
.01. 



Feet. 



-9 


9 


9 


9 


9 


9 


9 


8 


9 


8 


9 


7 


9 


7 


9 


7 


9 


7 


9 


6 


9 


6 


9 


5 


9 


5 


9 


5 


9 


4 


9 


4 


9 


4 


9 


4 


9 


3 


9 


3 


9 


2 


9 


2 


9 


2 


9 


2 


9 


1 


9 


1 


9 


1 


9 





9 





9 





9 





8 


9 


8 


9 


8 


8 


8 


8 


-8 


8 



* Compiled from Report of U. S. C. & G. Survey for 1881. App. 10 Table XI. 



TABLE XIII.— COEFFICIENTS FOR CORRECTIONS FOR TEMPERATURE 

AND HUMIDITY.* 



t-\-t' 



0° 
10 
20 
30' 
40 
50 
60 



.1024 
.0915 
■ 0806 
.0698 
• 0592 
.0483 
.0380 



Diff. for 
1°. 



10 


9 


10 


9 i 


10 


8 


10 


6 


10 


6 : 


10 


6 ! 



i + t' 


C 


60° 


-.0380 


70 


.0273 


80 


.0166 


90 


--0058 


100 


+ .0049 


110 


.0156 


120 


+ .0262 



Diff. for 
1°. 



10 
10 
10 
10 
10 



7 
8 
7 
7 
10.6 



t+t' 


C 


120° 


+ .0262 


130 


.0368 


140 


.0472 


150 


.0575 


160 


.0677 


170 


.0779 


180 


+ .0879 



Diff. for 
1°. 



10 
10 
10 
10 
10 
10 



.6 
.4 
-3 
.2 
.2 
• 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10, Tables I, IV. 

800 



TABLE XIV. — USEFUL TRIGONOMETRICAL FORMULA. 



9 

10 

11 

12 



sin a 



1 tan a / i — cos 2a 



/ 1 — cos 



coseca Vl + tan-'a V ^ Vl + cot2a 

= cos a tan a = v 1 — cos^ a = 2 sin ^a cos ^ a 

1 + cos a 2 tan ^a ^ , 

= — T = -;— — — :n — = veis a cot ^a. 

cot ia 1 + tan" ^a 

1 cot a 1 

cos a = 



see a Vl + cot2 a Vl + tan2a 
= 1 — vers a = sin a cot a = V 1— sin^ a = 2 cos^ ^a — 1 



tan a = 



sin a cot i^a — 1 = cos^ ^a — sin^ ^a = 1 — 2 sin^ Ja. 
1 sin a sec a 1 



cot a cos a cosec a Vcosec^ a — 1 
= vers 2a cosec 2a = cot a — 2 cot 2a=sin asec a 

sin 2a , , o ^ o 

= -— = exsec a cot ia = 3xsec 2a cot 2a. 

1 + cos 2a 

, 1 cos a sin 2a 1 + cos 2a 

cot a = •= = = ■ — - 

tan a sm a 1 — cos 2a sin 2a 

= V cosec^ a — 1 = cot ^a — cosec a. 
5 vers a = 1 — cos a =sin a tan ^a = 2 sin^ ^a = cos a exsec a„ 

Q exsec a = sec a — 1 = tan a tan Ja = vers a sec a. 

'versa sin a vers a cos ^a 



2 cos ^a sin a 



cos 



, ^ / 1 + cos a sin a sin a sin ^a 

•j^a =,4/ = = — 

^22 sin ^a vers a 



tan ^a = vers a cosec a = cosec a — cot a = • 



1 +sec a 



f.r.1- Xr, _ 1 + cos a tan a 

cot ta = -. =cosec a + cot a = - 



sm a exsec a cosec a— cot a 

vers i-a = 1 - V^(l + cosa). ■ 

exsec ^a = — — i. 

V^(l + cosa) 



801 



TABLE XIV. — USEFUL TRIGONOMETRICAL FORMULiB. 



sin 2a 


„ . ^ taxi c* 


1 + tan2 a 


cos 2a 


= cos2a— sin2a=l — 2sin2a=2 cos^a — 




1 — tan2 a 




l + tan2a* 


tan 2a 


2 tan a 
1 — tan^a' 



cut 2a =^ cot a-i tana =^^^^^^^=i^=*^B^ 

2 cot a 2 tan a 

vers 2a = 2 sin2 a = 1 - cos 2a = 2 sin a cos a tan a. 

exsec 2a=i^B_2£L ^■ 2tan2a _. 2 sin2 g 

cot a 1 - tan2 a 1-2 sin2 a ' 

sin (a ± 6) =sin a cos & ± cos a sin &. 

cos (a ± 6) = cos a cos 6 T sin a sin 6. 

sin a + sin & =3 sin i(a + 6) cos i(a-&). 

sin a- sin & =2 sin i(a-6) oos Ka + 6). 

cos a + cos 6 = 2 cos i(a + 6) cos ^(a — 6). 

cos a — cos 6= —2 sin i{a + b) sin J(a — 6). 



Call the sides of any triangle A, B, C, and the opposite angles a, ht 
andc. CaHs = iiA+B + C). 

tan ^(a — &) = . , p tan i(a + &)= . , ^ cot ic 

A. -+■ H J± +Jj 

^^ cosl^a + &} ^ BinUa + b) 

cosiia — b) ^sini(a — 6) 



amia = 4/ jtt^ . 






cosia = 4/ ^^ -. 



2(s-B)(s-C) 



versa = - ^^ 



Area =x^«(«-A)(3-^)(«-C)=^^ ^^^ ^^^^" . 



802 



TABLE XV.— USEFUL FORMULA AND CONSTANTS. 





Logarithm. 


Circumference of a circle (radius — r)= 2irr. 






Area of a circle =irr2. 






Area of sector (length of arc = —^Ir. 






•• •• " (angle of arc =a°) =^,7rr2. 






Area of segment (chord = c, mid. ord. =m) = |cm (approx.). 




Volume of a cone or pyramid = area of base X k height. 




Area of a circle to radius 1 








Circumference of a circle to diameter 1 


= 7r = 


3.1415927 


0.497 1499 


Surface of a sphere to diameter 1 


■ 






Volume of a sphere to radius 1 = 4ir -r- 3 = 


4.1887902 


. 622 0886 




degrees = 


57.2957795 


1 . 758 1226 


Arc equal to radius expressed in 


minutes = 


3437.7467708 


3 . 536 2739 




seconds = 


206264.8062471 


5.314 4251 


Length of arc of 1°, radius unity 


0.01745329 


8.2418774 


Sine of one second = . 000004S4S 

Weight of one cubic foot of w 
39°.8 F., barom. 30") 


1 




4 . 685 5749 


ater at maximum density (therm. 
62.379 


1.795 0384 


Weight of one cubic foot of water at ordinary temperature (therm. 
62° F.) fi2 . .S21 


1 . 794 6349 


Acceleration due to gravity at latitude of N 
square second 


ew York in feet per 
32.15945 


1 . 507 3086 


1 yard (U. S. standard) -3937 meter = 


0.914402 m. 


9.961 1371 


1 foot = 


0.304801 m. 


9.484 0158 


1 inch = 


0.025400 m. 


8.404 8346 


1 meter = 


3.28083 feet 


0.515 9842 


= 


39 . 3700 inches 


1 . 595 1654 


1 pound (avoirdupois) = 


0.453592 kilogr. 


9.656 6659 


1 kilogram = 


2 . 20462 pounds 


1.343 3341 


1 bushel (U. S. standard) = 


2150.420 cu. in. 




= 


1.244 cu. ft. 




1 gallon (U. S. standard) => 


231. cu. in. 








= 


0.1337 cu. ft. 





803 





TABLE XVi. — SQUARES, CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cubie Roots. 


Reciprocals. 


1 
2 
3 
4 
5 


1 

4 

9 

16 

25 


1 
8 

27 
64 

125 


1.0000000 
1.4142136 
1.7320508 
2.0000000 
2.2360680 


1.0000000 
1.2599210 
1.4422496 
1.5874011 
1.7099759 


1.000000000 
-500000000 
.333333333 
-250000000 , 
.200000000 : 


6 
7 
8 
9 
10 


36 
49 
64 
81 
100 


216 
343 
512 
729 
1000 


2.4494897 
2.6457513 
2.8284271 
3.0000000 
3-1622777 


1.8171206 
1.9129312 
2.0000000 
2.0800837 

2.1544347 


.166666667 
.142857143 
-125000000 
.111111111 
.100000000 


11 
12 
13 
14 
15 


121 
144 
169 
196 

2.25 


1331 
1728 
2197 
2744 
3375 


3.3166248 
3.4641016 
3.6055513 
3.7416574 
3-8729833 


2.2239801 
2.2894286 
2.3513347 
2.4101422 
2-4662121 


.090909091 
-083333333 
.076923077 
.071428571 
-066666667 


16 
17 
18 
19 
30 


256 
289 
324 
361 
400 


4096 
4913 
5832 
6859 
8000 


4.0000000 
4.1231056 
4-2426407 
4.3588989 
4-4721360 


2.5198421 
2.5712816 
2-6207414 
2-6684016 
2-7144177 

2-7589243 
2-8020393 
2-8438670 
2-8844991 
2-9240177 


-062500001 
.058823529 
.055555556 
.052631579 
-050000000 


21 
22 
23 
U 
25 


441 
484 
529 
576 
625 


9261 
10648 
12167 
13824 
15625 


4-5825757 
4.6904158 
4.7958315 
4.8989795 
5-0000000 


.04V619048 
.045454545 
.043478261 
.041666667 
.040000000 


26 
27 
28 
29 
30 


676 
729 
784 
841 
900 


17576 
19683 
21952 
24389 
27000 


5-0990195 
5.1961524 
5.2915028 
5-3851648 
5-4772256 


2-9624960 
3-0000000 
3-0365889 
3-0723168 
3-1072325 


.038461538 
.037037037 
.035714286 
.034482759 
-033333333 


31 
32 
33 
34 
35 


961 
1024 
1089 
1156 
1225 


29791 
32768 
35937 
39304 
42875 


5-5677644 
5.6568542 
5.7445626 
5.8309519 
5-9160798 


3-1413806 
3.1748021 
3.2075343 
3.2396118 
3-2710663 


.032258065 
.031250000 
.030303030 
-029411765 
.028571429 


36 
37 
38 
39 
40 


1296 
1369 
1444 
1521 
1600 


46656 
50653 
54872 
59319 
64000 


6-0000000 
6-0827625 
6-1644140 
6-2449980 
6-3245553 


3.3019272 
3.3322218 
3-3619754 
3.3912114 
3-4199519 


.027777778 
-027027027 
-026315789 
.025641026 
.025000000 


41 
43 
43 
44 
. 45 


1681 
1764 
1849 
1936 
2025 


68921 
74088 
79507 
85184 
91125 


6-4031242 
6.4807407 
6.5574385 
6.6332496 
6.7082039 


3.4482172 
3.4760266 
3.5033981 
3.5303483 
3-5568933 


-024390244 
.023809524 
.023255814 
.022727273 
-022222222 


46 
47 
48 
49 
50 


2116 
2209 
23b4 
24D1 
2500 


97336 
103823 
110592 
117649 
125000 


6.7823300 
6-8556546 
6.9282032 
7.0000000 
7-0710678 


3-5830479 
3.6088261 
3.6342411 
3.6593057 
3.6840314 


.021739130 
.021276600 
.020833333 
.020408163 
-020000000 


51 
52 
53 
54 
55 


2601 
2704 
2809 
2916 
3025 


132651 
140608 
148877 
157464 
166375 


7.1414284 
7.2111026 
7.2801099 
7.3484692 
7-4161985 


3.7084298 
3-7325111 
3-7562858 
3-7797631 
3.8029525 


.019607843 
-019230769 
.018867925 
.018518519 
.018181818 


56 
57 
58 
59 
60 


3136 
3249 
3364 
3481 
3600 


175616 
185193 
195112 
205379 
216000 


7.4833148 
7.5498344 
7.6157731 
7.6811457 
7.7459667 


3.8258624 
3.8485011 
3.8708766 
3.8929965 
3.9148676 


.017857143 
.017543860 
.017241379 
.016949153 
.016666667 



804 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. . 


Cubes. 


Square Roots. 


Cube Roots. 


Reciproisals. 


61 


3721 


226981 


7-8102497 


3-9364972 


.016393443 


62 


3844 


238328 


7-8740079 


3-9578915 


.016129032 


63 


3969 


250047 


7-9372539 


3-9790571 


.015873016 


64 


4096 


262144 


8 - 0000000 


4-0000000 


.015625000 


65 


4225 
4356 


274625 


8-0622577 


4-0207256 


.015384615 


66 


287496 


8-1240384 


4-0412401 


.015151515 


67 


4489 


300763 


8-1853528 


4-0615480 


.014925373 


68 


4624 


314432 


8-2462113 


4.0816551 


.014705882 


69 


4761 


328509 


8-3066239 


4-1015661 


.014492754 


70 


4900 


343000 


8-3666003 


4-1212853 


-014285714 


71 


5041 


357911 


8-4261498 


4-1408178 


.014084507 


72 


5184 


• 373248 


8-4852S14 


4-1601676 


.013888889 


73 


5329 


389017 


8-5440037 


4-1793390 


.013698630 


74 


5476 


405224 


8-6023253 


4-1983364 


.013513514 


75 


5625 


421875 


8-6602540 


4-2171633 


.01^333333 


76 


5776 


438976 


8-7177979 


4-2358236 


.013157895 


77 


5929 


456533 


8-7749644 


4-2543210 


.012987013 


78 


6084 


474552 


8-8317609 


4-2726586 


.012820513 


79 


6241 


493039 


8-8881944 


4-2908404 


.012658228 


80 


6400 


512000 
531441 


8-9442719 


4-3088695 


-012500000 


81 


6561 


9-0000000 


4-3267487 


.012345679 


82 


6724 


551368 


9-0553851 


4-3444815 


.012195122 - 


83 


6889 


571787 


9-1104336 


4-3620707 


.012048193 


84 


7056 


592704 


9-1651514 


4-3795191 


.011904762 


85 


7225 


614125 


9-2195445 


4-3968296 


.011764706 


86 


7396 


636056 


9-2736185 


4-4140049 


.011627907 


87 


7569 


658503 


9-3273791 


4-4310476 


.011494253 


88 


7744 


681472 


9-3808315 


4-4479602 


.011363636 


89 


7921 


704969 


9-4339811 


4-4647451 


.011235955 


90 


8100 


729000 


9-4868330 


4-4814047 


.011111111 


91 


8281 


753571 


9-5393920 


4-4979414 


.010989011 


92 


8464 


778688 


9-5916630 


4-5143574 


.010869565 


93 


8649 


804357 


9-6436508 


4-5306549 


.010752688 


94 


8836 


830584 


9-6953597 


4-5468359 


.010638298 


95 


9025 


857375 


9-7467943 


4-5629026 


.010526316 


96 


9216 


884736 


9-7979590 


4-5788570 


.010416667 


97 


9409 


912673 


9-8488578 


4-5947009 


.010309278 


98 


9604 


• 941192 


9-8994949 


4-6104363 


.010204082 


99 


9801 


970299 


9-9498744 


4-6260650 


.010101010 


100 


10000 


1000000 


10-0000000 


4-6415888 


-010000000 


101 


10201 


1030301 


10-0498756 


4-6570095 


.009900990 


102 


10404 


1061208 


10-0995049 


4-6723287 


.009803922 


103 


10609 


1092727 


10.1488918 


4-6875482 


.009708738 


104 


10816 


1124864 


10.1980390 


4-7026694 


.009615385 


105 


11025 


1157626 


10-2469508 


4-7176940 


-009523810 


106 


11236 


1191016 


10-2956301 


4-7326235 


.009433962 


107 


11449 


1225043 


10-3440804 


4-7474594 


.009345794 


108 


11664 


1259712 


10-3923048 


4-7622032 


.009259259 


109 


11881 


1295029 


10-4403065 


4-7768562 


.009174312 


110 


12100 


1331000 


10-4880885 


4-7914199 


.009090909 


ill 


12321 


1367631 


10-5356538 


4-8058955 


.009009009 


112 


12544 


1404928 


10-5830052 


4-8202845 


-C08928571 


113 


12769 


1442897 


10-6301458 


4-8345881 


.008849558 


114 


12996 


1481544 


10-6770783 


4.8488076 


.008771930 


115 


13225 


1520875 


10-7238053 


4-8629442 


-008695652 


116 


13456 


1560896 


10-7703296 


4-8769990 


.008620690 


117 


13689 


1601613 


10-8166538 


4-8909732 


.008547009 


118 


13924 


1643032 


10-8627805 


4.9048681 


.008474576 


119 


14161 


1685159 


10-9087121 


4.9186847 


.008403361 


130 


14400 


1728000 


10-9544512 


4.9324242 


.008333333 



805 





TABLE XVI. SQUARES, 


CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 
122 
123 
124 
125 


14641 
14884 . 
15129 
15376 
15625 


1771561 
1815848 
1860867 
1906624 
1953125 


11 
11 
11 
11 
11 


0000000 
0453610 
09U5365 
1355287 
.iyu33i;9 


4 
4 
4 
4 
5 


9460874 
9596757 
9731898 
9856310 
OOOOOUO 




008264463 
008196721 
008130081 
008064516 
008000000 ; 


126 
127 
128 
129 
130 


15876 
16129 
16384 
16641 
16900 


2000376 
2048383 
2097152 
2148689 
2197000 


11 
11 
11 
11 
11 


2249722 

2694277 

3137085 

•3578167 

.4017543 


5 
5 
5 
5 

5 


0132979 

0265257 

.0396842 

.0527743 

.0657970 




.007936508 i 
•007874016 ' 
•007812500 i 
•007751938 
.007692308 ; 


131 
132 
133 
134 
135 


17161 
17424 
17689 
17956 
18225 


2248091 
2299968 
2352637 
2406104 
2460375 


11 
11 
11 
11 
11 


4455231 
4891253 
5325626 
5758369 
6189500 


5 
5 
5 
5 
5 


.0787531 
.0916434 
•1044687 
•1172299 
1299278 




•007633588 
•007575758 
•007518797 
•007462687 
•007407407 


136 
137 
138 
139 
140 


18496 
18769 
19044 
19321 
19600 


2515456 
2571353 • 
2828072 
2685619 
2744000 


11 
11 
11 
11 
11 


6619038 
7046999 
7473401 
7898261 
8321596 


5 
5 
5 
5 
5 


1425 
1551 
1676 
1801 
1924 


632 
367 
493 
015 
941 

279 
034 
215 
828 
879 




•007352941 

•007299270 

007246377 

007194245 

007142857 


141 
142 
143 
144 
145 


19881 
20164 
20449 
20736 
21025 


2803221 
2863288 
2924207 
2985984 
3048625 


11 
11 
11 
12 
12 


8743421 
9163753 
9582607 
0000000 
0415946 


5 
5 
5 
5 
5 


2048 
2171 
2293 
2414 
2535 




007092199 
007042254 
006993007 
006944444 
006896552 


146 
147 
148 
149 
150 


21316 
21609 
21904 
22201 
22500 


3112136 
3176523 
3241792 
3307949 
3375000 


12 

12- 

12- 

12. 

12 


0830460 
1243557 
1655251 
2065556 
2474487 


5 
5 
5 
5 
5 


2656374 
2776321 
2895725 
3014592 
3132928 




•0C6849315 
006802721 
006756757 
006711409 
006666667 


151 
152 
153 
154 
155 


22801 
23104 
23409 
23716 
24025 


3442951 
3511808 
3581577 
3652264 
3723875 


12 
12 
12 
12 
12 


2882057 
3288280 
3693169 
4096736 
4498996 


5 
5 
5 
5 
5 


3250740 
3368033 
3484812 
3601084 
3716854 




006622517 
006578947 
006535948 
006493506 
006451613 


156 
157 
158 
159 
160 


24336 
24649 
24964 
25281 
25600 


3796416 
3869893 
3944312 
4019679 
4096000 


12 
12 
12 
12 
12 


4899960 
5299641 
5698051 
6095202 
6491]nR 


5 
5 
5 
5 
5 


3832126 
3946907 
4061202 
4175015 
4288352 




006410256 
006369427 
006329114 
006289308 
006250000 


161 
162 
163 
164 
165 


25921 
26244 
26569 
26896 
27225 


4173281 
4251528 
4330747 
4410944 
4492125 


12 
12 
12 
12 
12 


6885775 
7279221 
7671453 
8062485 
8452326 


5 
5 
5 
5 
5 


4401218 
4513618 
4625556 
4737037 
4848066 




006211180 
006172840 
006134969 
006097561 
006060606 


166 
167 
168 
169 
170 


27556 
27889 
28224 
28561 
28900 


4574296 
4657463 
4741632 
4826809 
4913000 


12 
12 
12 
13 
13 


8840987 
9228480 
9614814 
0000000 
0384048 


5 
5 
5 
5 
5 


4958647 
5068784 
5178484 
5287748 
5396583 




006024096 
005988024 
005952381 
005917160 
005882353 


171 
172 
173 
174 
175 


29241 
29584 
29929 
30278 
30625 


5000211 
5088448 
5177717 
5268024 
5359375 


13 
13 
13 
13 
13 


0766968 
1148770 
1529464 
1909060 
2287566 


5 
5 
5 
5 
5 


5504991 
5612978 
5720546 
5827702 
5934447 




005847953 
005813953 
005780347 
005747126 
005714286 


176 
177 
178 
179 
180 


30976 
31329 
31684 
32041 
32400 


5451776 
5545233 
■5639752 
5735339 
5832000 


13 
13 
13 
13 
13 


2664992 . 

3041347 

3416641 

3790882 

4164079 


5 
5 
5 
5 
5 


6040787 
6146724 
6252263 
6357408 
6462162 




005681818 
005649718 
005617978 
005586592 
005555556 



806 





CUBE EOOTS 


, AND RECIPROCALS. ' 




1 

No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


181 


32761 


5929741 


13-4536240 


5-6566528 


•005524862 
•005494505 
.005464481 
•005434783 
-005405405 


182 


33124 


6028568 


13-4907376 


5.6670511 


183 


33489 


6128487 


13-5277493 


5.6774114 


184 


33856 


6229504 


13-5646600 


5.6877340 


185 


34225 


6331625 


13-6014705 


5-6980192 


186 


34596 


6434856 


13-6381817 


5-7082675 


•005376344 
.005347594 
.005319149 


187 


34969 


6539203 


13-6747943 


5-7184791 


188 


35344 


6644672 


13-7113092 


5-7286543 


189 


35721 


6751269 


13-7477271 


5-7387936 


■005291005 
-0052631 58 


190 


36100 


6859000 


13-7840488 


5-7488971 


191 


38481 


6967871 


13-8202750 


5. 7589652 


-005235602 


192 


36864 


7077888 


13-8564065 


5-7689982 


.005208333 


193 


37249 


7189057 


13.8924440 


5-7789966 


• .005181347 


194 


37636 


7301384 


13-9283883 


5-7889604 


.005154639 


195 


38025 


7414875 


13-9642400 


5-7988900 


-005128205 


196 


38416 


7529536 


14-0000000 


5-8087857 


.005102041 


197 


38809 


7645373 


14-0356688 


5-8186479 


.005076142 


198 


39204 


7762392 


14-0712473 


5-8284767 


.005050505 


199 


39601 


7880599 


14-1067360 


5-8382725 


.005025126 


200 


40000 


8000000 


14.1421356 


5.8480355 


-005000000 


201 


40401 


8120601 


14-1774469 


5-8577660 


.004975124 


202 


40804 


8242408 


14-2126704 


5-8674643 


-004950495 


203 


41209 


8365427 


14-2478068 


5-8771307 


•004926108 


204 


41616 


8489664 


14-2828569 


58867653 


.004901961 


205 


42025 


8615125 


14-3178211 


5-8963685 


-004878049 


206 


42436 


8741816 


14-3527001 


5-9059406 


.004854369 


207 


42849 


8869743 


14-3874946 


5-9154817 


.004830918 


208 


43264 


8998912 


14-4222051 


5-9249921 


.004807692 


209 


43681 


9129329 


14-4568323 


5-9344721 


•004784689 


210 


44100 


9261000 


144913767 


5-9439220 


-004761905 


211 


44521 


9393931 


14-5258390. 


5-9533418 


-004739336 


212 


44944 


9528128 


14-5602198 


5-9627320 


•004716981 


213 


45369 


9663597 


14. 5945195 


5-9720926 


.004694836 


214 


45796 


9800344 


14-6287388 


5-9814240 


.004672897 


215 


46225 


9938375 


14-6628783 
14-6969385 


59907264 


-004651163 


216 


46656 


10077696 


6-0000000 


.004629630 


217 


47089 


10218313 


14-7309199 


6.0092450 


.004608295 


218 


47524 


10360232 


14-7648231 


6-0184617 


.004587156 


219 


47961 


10503459 


14-7986486 


6-0276502 


.004566210 


220 


48Ann 


10648000 


14.R,^2R970 


R 03fi8107 


-004545455 


221 


48841 


10793861 


14-8660687 


6-0459435 


-004524887 


222 


49284 


10941048 


14-8996644 


6. 0550489 


-004504505 


223 


49729 


11089567 • 


14-9331845 


6-C641270 


.004484305 


224 


50178 


11239424 


14-9666295 


60731779 


.004464286 


225 


50625 


11390625 


15-0000000 


6.0822020 


- 004444444 


226 


51076 


11543176 


15-0332964 


6-0911994 


-004424779 


227 


51529 


11697083 


15-0665192 


6-1001702 


-004405286 


228 


51984 


11852352 


15-0Q96689 


6-1091147 


.004385965 


229 


52441 


12008989 


15-1327460 


6-1180332 


-004366812 


230 


52900 


12167000 


15-1657509 


6-1269257 


.004347826 


231 


53361 


12326391 


15-1986842 


6-1357924 


-004329004 


232 


53824 


12487168 


15-2315462 


6-1446337 


-004310345 


233 


54289 


12649337 


15-2643375 


6-1534495 


-004291845 


234 


54756 


12812904 


15-2970585 


6-1622401 


-004273504 


235 


55225 


12977875 


15-3297097 


6-1710058 


-004255319 


236 


55696 


13144256 


15.3622915 


6-1797466 


.004237288 


237 


56169 


13312053 


15-3948043 


6-1884628 


.004219409 


238 


56644 


13481272 


15-4272486 


6-1971544 


.004201681 


239 


57121 


13651919 


15-4596248 


6-2058218 


.004184100 


24:0 


57600 


13824000 


15.4919334 


6-2144650 


.004166667 



807 





TABLE XVI. SQUARES, 


CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


CubeSo 


Square Roots. 


Cube Roots. 


Reciprocals. 


241 


58081 


13997521 


15.5241747 


6.2230843 


•004149378' 


242. 


58564 


14172488 


15. 


5563492 


6. 


2316797 


.004132231 


243 


59049 


14348907 


15. 


5884573 


6. 


2402515 


.004115226 


244 


59536 


14526784 


15. 


6204994 


6. 


2487998 


.004098361 


245 


60025 


14706125 


15. 


6524758 


6. 


2573248 


.004081633, 


246 


60516 


14886936 


15. 


6843871 


6. 


2658266 


.004065041] 


247 


61009 


15069223 


15. 


7162336 


6. 


2743054 


.004048583 


248 


61504 


15252992 


15. 


7480157 


6 


2827613 


.004032258. 


249 


62001 


15438249 


15. 


7797338 


6 


2911946 


.004016064 


350 


62500 


15625000 


15. 


8113883 


6 


2996053 


.004000000 


251 


63001 


15813251 


15. 


8429795 


6 


3079935 


.003984064 


252 


63504 


16003008 


15. 


8745079 


6 


3163596 


.003908254 


253 


64009 


16194277 


15 


9059737 


6 


3247035 


.003952569 


254 


64516 ■ 


16387064 


15 


9373775 


6 


3330256 


.003937008 


255 


65025 


16581375 


15 


9687194 


6 


3413257 


.003921569 


256 


65536 


16777216 


16 


0000000 


6 


3496042 


.003906250 


257 


66049 


16974593 


16 


0312195 


6 


3578611 


•003891051 


258 


66564 


17173512 


16 


0623784 


6 


3630968 


.003875969 


259 


67081 


17373979 


16 


0934769 


6 


3743111 


.003861004 


260 


67800 


17576000 


16 


1245155 


6 


38;:^5043 


.003846154 ' 


261 


68121 


17779581 


16 


1554944 


6 


3906765 


.003831418 


262 


68644 


17984728 


16 


1864141 


6 


39G8279 


.003816794 


263 


69169 


18191447 


16 


2172747 


6 


40G9585 


.003802281 


264 


69696 


18399744 


16 


2480768 


6 


4150687 


.003787879 


265 


70225 


18609625 


16 


2788206 


6 


4231583 


.003773585 


266 


70756 


18821096 


16 


3095064 


6 


4312276 


.003759398 


267 


71209 


19034163 


16 


3401346 


6 


4392767 


.003745318 


268 


71824 


19248832 


16 


3707055 


6 


4473057 


.003731343 


269 


72361 


19465109 


16 


4012195 


6 


4553148 


.003717472 


370 


72900 


19683000 


16 


4316767 


6 


4633041 


.003703704 


271 


73441 


19902511 


16 


4620776 


6 


4712736 


.003690037 


272 


73984 


20123648 


16 


4924225 


6 


4792236 


.003676471 


273 


74529 


20346417 


16 


5227116 


6 


4871541 


.003663004 


274 


75076 


20570824 


16 


5529454 


6 


4950653 


.003649635 


275 


75625 


20796875 


16 


5831240 


6 


5029572 


-003636364 


i276 


76176 


21024576 


16 


6132477 


6 


5108300 


.003623188 


an 


76729 


21253933 


16 


6433170 


6 


5186839 


.003610108 


278 


77284 


21484952 


16 


6733320 


6 


-5265189 


.003597122 


279 


77841 


21717639 


16 


7032931 


6 


•5343351 


.003584229 


380 


78400 


21952000 


16 


7332005 


6 


•5421326 


•003571429 


281 


78961 


22188041 


16 


7630546 


6 


.5499116 


.003558719 


282 


79524 


22425768 


16 


7928556 


6 


.5576722 


.003546099 


283 


80089 


22665187 


16 


•8226088 


6 


.5654144 


.003533569 


284 


80656 


22906304 


16 


8522995 


6 


.5731385 


.003521127 


285 


81225 


23149125 


16 


•8819430 


6 


•5808443 


.003508772 


286 


81796 


23393656 


16 


•9115345 


6 


•5885323 


.003496503 


287 


82369 


23639903 


16 


•9410743 


6 


•5962023 


.003484321 


288 


82944 


23887872 


16 


•9705627 


6 


•6038545 


.003472222 


289 


83521 


24137569 


17 


•0000000 


6 


•6114890 


.003460208 


390 


84100 


24389000 


17 


•0293864 


6 


•6191060 


.003448276 


291 


84881 


24642171 


17 


•0587221 


6 


•6267054 


.003436426 


292 


85264 


24897088 


17 


•0880075 


6 


.6342874 


.003424658 


293 


85849 


25153757 


17 


•1172428 


6 


.6418522 


.003412969 


294 


86436 


25412184 


17 


•1464282 


6 


•6493998 


.003401361 


295 


87025 


25672375 


17 


•1755640 


6 


•6569302 


•003389831 


296 


87616 


25934336 


17 


.2046505 


6 


•6644437 


.003378378 


297 


88209 


26198073 


17 


•2336879 


6 


•6719403 


.003367003 


298 


88804 


26463592 


17 


•2626765 


6 


•6794200 


.003355705 


299 


89401 


26730899 


17 


.2916165 


6 


•6868831 


.003344482 


300 


90000 


27000000 


17.3205081 


6.6943295 


.003333333 



808 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Ciibeso 


Square Roots. 


Cube Roots. I 


leeiprocals. 


301 


90601 


27270901 


17.3493516 


6.7017593 


003322259 


302 


91204 


27543603 


17 


3781472 


6 


7091729 


003311258 


303 


91809 


27818127 


17 


4068952 


6 


7165700 


003300330 


304 


92416 


28094464 


17. 


4355958 


6 


7239508 


003289474 


305 


93025 


28372625 


17 


4642492 


6 


7313155 


003278689 


306 


93636' 


28652616 


17. 


4928557 


6 


7386641 


003267974 


307 


94249 


28934443 


17. 


5214155 


6 


7459967 


003257329 


308 


94864 


29218112 


17 


5499288 


6 


7533134 


003246753 


309 


95481 


29503629 


17. 


5783958 


6 


7606143 


003236246 


310 


96100 


29791000 


17. 


6068169 


6 


7678995 


003225806 


311 


96721 


30080231 


17. 


6351921 


6 


7751690 


003215434 


312 


97344 


30371328 


17. 


6635217 


6 


7824229 


003205128 


313 


97969 


30664297 


17. 


6918060 


6 


7896613 


003194888 


314 


98596 


30959144 


17. 


7200451 


6 


7968844 


003184713 


315 


99225 


31255875 


17. 


7482393 


6 


8040921 


003174603 


316 


99856 


• 31554496 


17. 


7763888 


6 


8112847 


003164557 


317 


100489 


31855013 


17. 


8044938 


6 


8184620 


003154574 


318 


101124 


32157432 


17. 


8325545 


6 


8256242 


003144654 


319 


101761 


32461759 


17 


8605711 


6 


8327714 


003134796 


_ 330 


102400 ' 


32768000 


17 


8885438 


6^ 

8 


8399037 


003125000 _ 


321 


103041 


33076161 


17 


9164729 


8470213 


003115285 


322 


103684 


33386248 


17 


9443584 


6 


8541240 


003105590 


323 


104329 


33698267 


17 


9722008 


6 


8612120 


003095975 


324 


104976 


34012224 


18 


0000000 


6 


8682855 


003086420 


325 


105625 


34328125 


18 


0277564 


6 


8753443 


003076923 


326 


106276 


34645976 


18 


0554701 


6 


8823888 


003067485 


327 


106929 


34965783 


18 


0831413 


6 


8894188 


003058104 


328 


107584 


35287552 


18 


1107703 


6 


8964345 


003048780 


329 


108241 


35611289 


18 


1383571 


6 


9034359 


003039514 


330 


108900 


35937000 


18 


1659021 


6 


9104232 


003030303 


331 


109561 


36264691 


18 


1934054 


6 


9173964 


003021148 


332 


110224 


36594368 


18 


2208672 


6 


9243556 


003012048 


333 


110889 


36926037 


18 


2482876 


6 


9313008 


003003003 


334 


111556 


37259704 


18 


2756669 


6 


9382321 


002994012 


335 


112225 


37595375 


18 


3030052 


6 


9451496 


002985075 


336 


112896 


37933056 


18 


3303028 


6 


9520533 


002976190 


337 


113569 


38272753 


18 


3575598 


6 


9589434 


002967359 


338 


114244 


38614472 


18 


•3347763 


6 


9658198 


002958580 


339 


114921 


38958219 


18 


4119526 


6 


9726826 


002949853 


340 


115600 


39304000 


18 


4390889 


6 


9795321 


002941176 


341 


116281 


39651821 


18 


.4661853 


6 


9863681 


002932551 


342 


116964 


40001688 


18 


.4932420 


6 


9931906 


002923977 


343 


117649 


40353607 


18 


.5202592 


7 


0000000 


002915452 


344 


118336 


40707534 


18 


.5472370 


7 


0067962 


002906977 


345 


119025 


41063625 


18 


5741756 


7 


0135791 


002898551 


346 


119716 


41421736 


18 


•6010752 


7 


0203490 


002890173 


347 


120409 


41781923 


18 


•6279360 


7 


0271058 


002881844 


348 


121104 


42144192 


18 


•6547581 


7 


0338497 


002873563 


349 


121801 


42508549 


18 


•6815417 


7 


0405806 


002865330 


350 


122500 


42875000 


18 


7082869 


7 


0472987 


002857143 


351 


123201 


43243551 


18 


•7349940 


7 


0540041 


002849003 


352 


123904 


43614208 


18 


•7616630 


7 


0606967 


002840909 


353 


124609 


43986977 


18 


•7882942 


7 


0673767 


002832861 


354 


125316 


44361864 


18 


•8148877 


7 


0740440 


002824859 


355 


126025 


44738875 


18 


•8414437 


7 


0806988 


002816901 


356 


126736 


45118016 


18 


.8679623 


7 


0873411 


002808989 


357 


127449 


45499293 


18 


.8944438 


7 


0939709 


002801120 


358 


128164 


45882712 


18 


•9208879 


7 


1005885 


002793296 


359 


128881 


46268279 


18 


.9472953 


7 


10719.?7 


002785515 


360 


129600 


46656000 


18.9736660 


7.1137Co6 


002777778 



809 





TABLE XVI. SQUARES, 


CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


361 
362 
363 
364 
365 


130321 
131044 
131769 
132496 
133225 


47045881 
47437928 
47832147 
48228544 
48627125 


19 
19 
19 
19 
19 


0000000 
0262976 
0525589 
0787840 
1049732 


7.1203674 
7.1269360 
7.1334925 
7.1400370 
7.1465695 


.002770083 i 

.002762431 

.002754821 

.002747253 

.002739726 


366 
387 
368 
369 
370 


133956 
134689 
135424 
136161 
136900 


49027896 
49430863 
49836032 
50243409 
50653000 


19 
19 
19 
19 
19 


1311265 
1572441 
1833261 
2093727 
2353841 


7.1530901 
7.1595988 
7.1660957 
7.1725809 
7.1790544 


.002732240 

.002724796: 

.002717391 

.002710027 

.002702703 


371 
372 
373 
374 
375 


137641 
138384 
139129 
139876 
140625 


51064811 
51478848 
51895117 
52313624 
52734375 


19 
19 
19 
19 
19 


2613603 
2873015 
3132079 
3390796 
3649187 


7.1855162 
7.1919663 
7.1984050 
7.2048322 
7.2112479 


.002695418 
.002688172 
.002680965 
.002673797 
.002666667 


376 
377 
. 378 
379 
380 


141376 
142129 
142884 
143641 
144400 


53157376 
53582633 
54010152 
54439939 
• 54872000 

55306341 
55742968 
56181887 
58623104 
57066625 


19 

19. 

19. 

19. 

19 


3907194 
4164878 
4422221 
4679223 
4935887 


7.2178522 
7.2240450 
7.2304288 
7.2367972 
7.2431565 


.002659574 
.002652520 
.002645503 
.002838522 
.002631579 


381 
382 
383 
384 
385 


145161 
145924 
146689 
147456 
148225 


19 

19. 

19. 

19. 

19 


5192213 
5448203 
5703858 
5959179 
6214169 


7.2495045 
7.2558415 
7.2821875 
7.2684824 
7.2747864 


.002824672 
.002817801 
.002810966 
.002604167 
.002597403 


386 
387 
388 
389 
390 


148996 
149769 
150544 
151321 
152100 


57512456 
57960603 
58411072 
58863869 
59319000 


19. 

19 

19. 

19 

19 


6468827 
6723156 
6977156 
7230829 
7484177 


7.2810794 
7.2873617 
7.2936330 
7.2998936 
7.3061436 


.002590674 
.002583979 
.002577320 
.002570694 
.002564103 


391 
392 
393 
394 
395 


152881 
153664 
154449 
155236 
156025 


59776471 
60236288 
60698457 
61182984 
61629875- 


19. 

19 

19 

19 

19 


7737199 
7989899 
8242276 
8494332 
8746069 


7.3123828 
7.3186114 
7.3248295 
7.3310369 
7.3372339 


.002557545 
.002551020 
.002544529 
.002538071 
.002531646 


396 
397 
398 
399 
400 


156816 
157609 
158404 
159201 
160000 


62099136 
62570773 
63044792 
63521199 
64000000 


19 
19 
19 
19 
20 

20 
20 
20 
20 
20 


8997487 
9248588 
9499373 
9749844 
0000000 


7.3434205 
7.3495966 
7.3557624 
7.3619178 
7.3880830 


.002525253 
.002518892 
. .002512563 
.002506266 
.002500000 


401 
402 
403 
404 
405 


160801 
161604 
162409 
163218 
164025 


64481201 
64964808 
65450827 
65939264 
66430125 


0249844 
0499377 
0748599 
0997512 
1246118 


7.3741979 
7.3803227 
7.3864373 
7.3925418 
7.3986363 


.002493786 
.002487562 
.002481390 
.002475248 
.002489136 


406 
407 
408 
409 
410 


164836 
165649 
166464 
167281 
168100 


66923416 
67419143 
67917312 
68417929 
68921000 


20 
20 
20 
20 
20 


1494417 

1742410 

.1990099 

.2237484 

.248.4567 


7.4047206 
7.4107950 
7.4168595 
7.4229142 
7.4289589 


.002463054 
.002457002 
.002450980 
.002444988 
.002439024 


411 
412 
413 
414 

415 


168921 
169744 
170569 
171396 

172225 


69426531 
69934528 
70444997 
70957944 
71473375 


20 
20 
20 
20 
20 


.2731349 
.2977831 
.3224014 
.3469899 
.3715488 


7.4349938 
7.4410189 
7.4470342 
7.4530399 
7.4590359 


.002433090 
.002427184 
.002421308 
.002415459 
.002409839 


416 
417 
418 
419 
430 


173056 
173889 
174724 
175561 
176400 


7199129b 
72511713 
73034632 
73560059 
74088000 


20 
20 
20 
20 
20 


.3960781 
.4205779 
.4450483 
.4694895 
.4939015 


7.4850223 
7.4709991 
7.4769864 
7.4829242 
7.4888724 


.002403846 
.002398082 
.002392344 
.002386635 
.00238C'5?.a 



810 



CUBE HOOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 
422 
423 
424 
425 


177241 
178084 
178929 
179776 
180625 


74618461 
75151448 
75686967 
76225024 
76765625 


20.5182845 
20.5426386 
20.5669638 
20-5912603 
20-6155281 


7.4948113 
7.5007406 
7.5066607 
7.5125715 
7.5184730 


-C02375297 
.002369668 
.002364066 
.002358491 
.C02352941 


426 
427 
428 
429 
430 


181476 
182329 
183184 
184041 
184900 


77308776 
77854483 
78402752 
78953589 
79507000 


20-6397674 
20.6639783 
20.6881609 
20.7123152 
20.7364414 


7-5243652 
7-5302482 
7.5361221 
7-5419867 
7.5478423 


.002347418 
.002341920 
.C02336449 
.002331002 
.002325581 


431 
432 
433 
434 
435 


185761 
186624 
187489 
188356 
189225 


80062991 
80621568 
81182737 
81746504 
82312875 


20-7605395 
20.7846097 
20.8086520 
20.8326667 
20-8566536 


7.5536888 
7.5595263 
7.5653548 
7.5711743 
7-5769849 


-C02320186 
-002314815 
-002309469 
.002304147 
.002298851 , 


436 
437 
438 
439 
440 


190096 
190969 
191844 
192721 
193600 


82881856 
83453453 
84027672 
84604519 
85184000 


20.8806130 
20.9045450 
20.9284495 
20.9523268 
20.9761770 


7.5827865 
7-5885793 
7-5943633 
7.6001385 
7-6059049 


-CC2293578 
-C0M88330 
-C02283105 
■C02277904 
-002272727^ 


441 
442 
443 
444 
445 


194481 
195364 
196249 
197136 
198025 


85766121 
86350888 
86938307 
87528384 
88121125 


21.0000000 
21.0237960 
21.0475652 
21.0713075 
21-0950231 


7-6116626 
7-6174116 
7-6231519 
7-6288837 
7-6346067 


-002267574 
-002262443 
.002257336 
.002252252 
-002247191 


446 
447 
448 
449 
450 


198916 
199809 
200704 
201601 
202500 


88716536 
89314623 
89915392 
90518849 
91125000 


21.1187121 
21.1423745 
21.1660105 
21.1896201 
21-2132034 


7-6403213 
7.6460272 
7.6517247 
7.6574138 
7- 6630943 


-002242152 
.002237136 
.002232143 
.002227171 
-002222222 


451 
452 
453 
454 
455 


203401 
204304 
205209 
206116 
207025 


91733851 
92345408 
92959677 
93576664 
94196375 


21.2367606 
21.2602916 
21.2837967 
21.3072758 
21-3307290 


7-6687665 
7-6744303 
7.68G0857 
7-6857328 
7. 6913717 


-002217295 
-002212389 
-002207506 
-002202643 
-002197802 


456 
457 
458 
459 
. 460 


20793C 
208849 
209764 
210681 
211600 


94818816 
95443993 
96071912 
96702579 
97336000 


21.3541565 
21.3775583 
21.4009346 
21.4242853 
21-4476106 


7-6970023 
7-7026246 
7-7082388 
7.7138448 
7. 7194426 


-002192982 
-002188184 
.002183406 
.002178649 
-002173913 


461 
462 
463 
464 
465 


212521 
213444 
214369 
215296 
216225 


97972181 
98611128 
99252847 
99897344 
100544625 


21.4709106 
21.4941853 
21.5174348 
21.5406592 
21.5638587 


7-7250325 
7-7306141 
7-7361877 
7-7417532 
7-74731C9 


.002169197 
.002164502 
.002159827 
.002155172 
-002150538 


466 
467 
468 
469 
470 


217156 
218089 
219024 
219961 
220900 


101194696 
101847563 
102503232 
103161709 
103823000 


21-5870331 
21.6101828. 
21.6333077 
21.6564078 
21.6794834 


7-7528606 
7-7584023 
7.7639361 
7.7694620 
7-7749801 


-002145923 
.002141328 
.002136752 
.002132196 
-002127660 


471 
472 
473 
474 
475 


221841 
222784 
223729 
224676 
225625 


104487111 
105154048 
105823817 
106496424 
107171875 


21.7025344 
21.7255610 
21.7435632 
21-7715411 
21-7944947 


7-7804904 
7.7859928 
7.7914875 
7-7969745 
7.8024538 


-002123142 
-GC2118644 
-G02114165 
.002109705 
-002105263 


476 
477 
478 
479 
480 


226576 
227529 
228484 
229441 
230400 


107850176 
108531333 
109215352 
109902239 
110592000 


21-8174242 
21-8403297 
21.8632111 
21.8860686 
21.9089023 


7.8079254 
7.8133892 
7-8188456 
7-8242942 
7.8297353 


-002100840 
.002096436 
-002092050 
.002087683 
.002083333 



811 





TABLE XVI. SQUARES, 


CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals, 


48X 

482 
483 
484 
485 


231361 
232324 
233289 
234256 
235225 

236196 
237169 
238144 
239121 
240100 


111284641 
111980168 
112678587 
113379904 
114084125 


21 

21 
21 

22 
22 


.9317122 
.9544984 
.9772610 
.0000000 
•0227155 


7.8351688 
7.8405949 
7.8460134 
7.8514244 
7.8568281 


.002079002 
.002074689 
.002070393 
•002066116 1 
.002061856 • 


486 
487 
488 
489 
490 


114791256 
115501303 
116214272 
1169301^9 
117649000 


22 
22 
22 
22 
22 


.0454077 
.0680765 
.0907220 
.1133444 
.1359436 


7.8622242 
7.8676130 
7.8729944 
7.8783684 
7.8837352 


.002057613 
.002053388 
.002049180 
■ .002044990 
.002040816 


491 
492 
493 
494 
495 


241081 
242064 
243049 
244036 
245025 


118370771 
11^095488 
119823157 
120553784 
121287375 


22 
22 
22 
22 
22 


.1585198 

.1810730 

.2036033 

2261108 

2485955 


7-8890946 
7.8944468 
7.8997917 
7.9051234 
7.9104599 


.002036660 
.002032520 
.002028398 
.002024291 
. 00202020^ 


496 
497 
498 
499 
500 


246016 
• 247009 
248004 
249001 
25000Q 


122023936 
122763473 
123505992 
124251499 
125000000 


22 
22 
22 
22 
22 


2710575 
2934968 
3159136 
3383079 
3606798 


7.9157832 
7.9210994 
7.9264085 
7.9317104 
7-9370053 


-002016129 
.002Q12072 
.002008032 
. 002004008 
.002000000 


501 
502 
503 
504 
505 


251001 
252004 
253009 
254016 
255025 


125751501 
126506008 
127263527 
128024064 
128787825 


22 

22. 

22. 

22. 

22 


3830293 
4053505 
4276615 
4499443 
4722051 


7-9422931 
7.9475739 
7.9528477 
7-9581144 
7.9633743 


-001996008 
•001992032 
.001988072 
.001984127 
-001980198 


506 
507 
508 
509 
510 


256036 
257049 
258064 
259081 
280100 


129554216 
130323843 
131096512 
131872229 
132651000 


22- 
22. 
22. 
22. 
22 


4944438 
5166605 
5388553 
5610283 
5831796 


7.9886271 
7.9738731 
7.97911^2 
7-9843444 
7.9895697 


-001976285 
-001972387 
-001968504 
-00196463?^ 
•001960784 


511 
512 
5X3 
514 
515 


261121 
262144 
263169 
264196 
265225 

266256 
267289 
268324 
269361 
270400 


133432831 
134217728 
135005697 
135796744 
136590875 


22. 
22. 
22. 
22. 
22 


6053091 
6274170 
6495033 
6715681 
6936114 


7.9947883 
8.0000000 
8.0052049 
8.0104032 
8.0155946 


-001956947 
-001953125 
.001949318 
.001945525 
-001941748 


516 
517 
518 
519 
5^0 


137388096 
138188413 
138991832 
139798359 
140608000 


22- 

22. 

22. 

22 

22 


7156334 
7376340 
7596134 
7815715 
8035085 


8.0207794 
8.0259574 
8.0311287 
8.0362935 
8-0414515 

8-0466030 
8.0517479 
8-0568862 
8.0620180 
8.0671432 


.001937984 
.001934236 
.001930502 
.001926782 
•001923077 , 


521 
522 
523 
524 
525 


271441 
272484 
273529 
274576 
275625 


141420761 
142236648 
143055667 
143877824 
144703125 


22. 

22. 

22 

22 

22. 


8254244 
8473193 
8691933 
8910463 
9128785 


.001919386 
•001915709 
.001912048 
.001908397 
-001904762 


526 
527 
528 
529 
580 


276676 
277729 
278784 
279841 
280900 


145531576 
146363183 
147197952 
148035889 
148£ 77000 


22. 
22. 
22. 
23. 
23. 


9346899 
9564806 
9782506 
0000000 
0217289 


8.0722620 
8.0773743 
8-0824800 
8.0875794 
8.0926723 


.001901141 
.001897533 
.001893939 
.001890359 
-001886792 


531 
532 
533 
534 
535 


281961 
283024 
284089 
285156 
286225 


149721291 
150568768 
151419437 
152273304 
153130375 


23. 
23. 
23. 
23. 
23. 


0434372 
0651252 
0867928 
1084400 
1300670 


8.0977589 
8.1028390 
8.1079128 
8.1129803 
8.1180414 


.001883239 
.001879699 
.001876173 
.001872659 
.001869159 


536 
537 
538 
539 
640 


287296 
288369 
289444 
290521 
291600 


153990656 
154854153 
155720872 
156590819 
157464000 


23. 
23. 
23. 
23. 
23. 


1516738 
1732605 
1948270 
2163735 
2370001 


8.1230962 
8-1281447 
8.1331870 
8.1382230 ^ 
8-1432529 


.001865672 
.001862197 
.001858736 
.001855288 
.001851852 



812 





CUBE ROOTS, 


AND RECIPROCALS. 




' No. 

541 

1 542 
543 
544 
545 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Rec'procals. 


292681 
293764 
294849 
295936 
297025 


158340421 
159220088 
160103007 
160989184 
161878625 

162771336 
163667323 
164566592 
165469149 
166375000 


23.2594067 
23.2808935 
23.3023604 
23-3238076 
23-3452351 


8.1482765 
8-1532939 
8-1583051 
8-1633102 
8.1683092 


.001848429 
.001845018 
•001841621 
.001838235 
-001834862 


546 
547 
548 
549 
550 


298116 
299209 
300304 
^301401 
302500 


23-3666429 
23.3880311 
23.4093998 
23.4307490 
23-4520788 


8-1733020 
8-1782888 
8.1832695 
8-1882441 
8-1932127 


.001831502 
•001828154 
.001824818 
.001821494 
-001818182 


551 
552 
553 
554 
555 


303601 
304704 
305809 
306916 
308025 


167284151 
168196608 
169112377 
170031464 
170953875 


23.4733892 
23-4946802 
23.5159520 
23-5372046 
23.5584380 


8-1981753 
8-2031319 
8-2080825 
8-2130271 
8-2179657 


.001814882 
.001811594 
•001808318 
•001805054 
-001801802 


556 
557 
558 
559 
560 


309136 
310249 
311364 
312481 
313600 


171879616 
172808693 
173741112 
174676879 

i75r.ifinno 


23-5796522 
23-6008474 
23.6220236 
23.6431808 
23-6643191 


8 2228985 
8-2278254 
8-2327463 
8.2376614 
8-2425706 


.001798561 
•001795332 
-001792115 
•001788909 
-001785714 


561 
562 
563 
564 
565 


314721 
315844 
316969 
318096 
319225 


176558481 
177504328 
178453547 
179406144 
180362125 


23-6854386 
23-7065392 
23-7276210 
23-7486842 
23.7697286 


8-2474740 
8-2523715 
8-2572633 
8-2621492 
8.2670294 


•001782531 
•001779359 
•001776199 
•001773050 
.001769912 


566 
567 
568 
569 
570 


320356 
321489 
322624 
323761 
324900 


181321496 
182284263 
183250432 
184220009 
185193000 


23-7907545 
23-S117618 
23-8327506 
23.8537209 
23.8746728 


8. 2719039 
8.2767726 
8-2816355 
8.2864928 
8.2913444 


.001766784 
.001763668 
•001760563 
.001757469 
•001754386 


571 
572 
573 
574 
575 


326041 
327184 
328329 
329476 

330625 


186169411 
187149248 
188132517 
189119224 
190109375 


23-8956063 
23-9165215 
23-9374184 
23-9582971 
23.9791576 


8-2961903 
8-3010304 
8-3058651 
8-3106£|41 
8 3155175 


•001751313 
.001748252 
•001745201 
.001742160 
.001739130 


576 
577 
578 
579 

580 


331776 
332929 
334084 
335241 
336400 


191102976 
192100033 
193100552 
194104539 
195112000 


24-0000000 
24-0208243 
24-0416306 
24-0624188 
24..P83.I89] 


8-3203353 
8-3251475 
8-3299542 
8.3347553 
R. 3395509 


.001736111 
.001733102 
•001730104 
•001727116 
.001724138 


581 
582 
583 
584 
585 


337561 
338724 
339889 
341056 
342225 


196122941 
197137368 
198155287 
199176704 
200201625 


24-1039416 
24-1246762 
24.1453929 
24.1660919 
24.1867732 


8-3443410 
8-3491256 
8-3539047 
8- 3586784 
8. 3634466 


•001721170 
•001718213 
•001715266 
.001712329 
.001709402 


586 
587 
588 
589 
590 


343396 
344569 
345744 
346921 
348100 


201230056 
202262003 
203297472 
204336469 
205379000 


24.2074369 
24.2280829 
24.2487113 
24-2693222 
24.2899158 


8-3682095 
8-3729668 
83777188 
8-3824653 
8-3872065 


.001706485 
.001703578 
.001700680 
.001697793 
•001694915 


591 
592 
593 
594 
595 


349281 
350464 
351649 
352836 
354025 


206425071 
207474688 
208527857 
209584584 
PI 0644875 


24.3104916 
24-3310501 
24 3515913 
24-3721152 
24-3926218 


8-3919423 
8-3966729 
8-4013981 
8-4061180 
8-4108326 


.001692047 
.001689189 
•001686341 
-001683502 
.001680672 

- 1 


596 
597 
598 
599 
600 


355216 
356409 
357604 
358801 
360000 


211708736 
212776173 
213847192 
214921799 
216000000 


24-4131112 
24-4335834 
24-4540385 
24-4744765 
24-4948974 


8-4155419 
8-4202460 
8-4249448 
8.4296383 
8-4343267 


.001677852 
.001675042 
.001672241 
.001669449 
.001666667 



813 





TABLE XVI. SQUARES, CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


601 
602 
603 
604 
605 


361201 
362404 
363609 
364816 
366025 


217081801 
218167208 
219256227 
220348864 
221445125 


24.5153013 
24.5356883 
24.5560583 
24.5764115 
24.5967478 


8.4390098 
8-4436877 
8-4483605 
8-4530281 
8-4576906 


.001663894 
.001661130 ! 
.001658375 
.001655629 
.001652893 ^ 


606 
607 
608 
609 
610 


367236 
368449 
369664 
370881 
372100 


222545016 
223648543 
224755712 
225866529 
226981000 


24.6170673 
24.6373700 
24.6576560 
24.6779254 
24.698i781 


8.4623479 
8-4670001 
8-4716471 
8-4762892 
8.4809261 


.001650165 
.001647446. , 
.001644737 
.001642036 
.001639344 


611 
612 
613 
614 
615 


373321 
374544 
375769 
376996 
378225 


228099131 
229220928 
230346397 
231475544 
232608375 


24-7184142 
24.7386338 
24.7588368 
24-7790234 
24-7991935 


8-4855579 
8-4901848 
8-4048065 
8-4994233 
8-5040350 


.001636661 
.001633987 
.001631321 
.001628664 
-001626016 i 


616 
617 
618 
619 
630 


379456 
380689 
381924 
383161 
384400 


233744896 
234885113 
236029032 
237176659 
23R328000 


24-8193473 
24-8394847 
24.85.96058 
24-8797106 
24-8997992 


8-5086417 
8.5132435 
8-5178403 
8.5224321 
8.5270189 


.001023377 
.001620746 
.001618123 
.001615509 
-00ir,l?>903 


621 
622 
623 
624 
625 


385641 
386884 
388129 
389376 
390625 


239483061 
240641848 
241804367 
242970624 
244140625 


24-9198716 
24-9399278 
24-9599679 
24-9799920 
25-0000000 


8-5316009 
8.5361780 
8-5407501 
8-5453173 
8.5498797 


.001610306 
.001607717 
.001605136 
.001602564 
.001600000 


626 
627 
628 
629 
630 


391876 
393129 
394384 
395641 
396900 


245314376 
246491883 
247673152 
248858189 
250047000 


25-0199920 
25"- 0399681 
25.0599282 
25.0798724 
25-0998008 


8-5544372 
8-5589899 
8-5635377 
8-5680807 
R. 5726189 


.001597444 
.001594896 
.001592357 
.001589825 
.001587302 


631 
632 
633 
634 
635 


398161 
399424 
400689 
401958 
403225 


251239591 
252435968 
253636137 
254840104 
256047875 


25.1197134 
25.1396102 
25-1594913 
25.1793566 
25.1992063 


8-5771523 
8-5816809 
8-5862047 
8-5907238 
8.5952380 


.001534786 
.001582278 
.001579779 
.001577287 
.001574803 


636 
637 
638 
639 
640 


404496 
405769 
407044 
408321 
409600 


257259456 
258474853 
259694072 
260917119 
262144000 


25.2190404 
25-2388589 
25.2586619 
25-2784493 


8-5997476 
8-6042525 
8-6087526 
8.6132480 
8.6177388 


.001572327 
.001569859 
.001567398 
.001564945 
.001562500 


641 
642 
64? 
644 
645 


410881 
412164 
413449 
414738 
416025 


263374721 
264609288 
265847707 
267089984 
268336125 


25-3179778 
25-3377189 
25.3574447 
25-3771551 
25.3968502 


8-6222248 
8.6267063 
8.6311830 
8.6356551 
8-6401226 


.001560062 
.001557632 
.001555210 
.001552795 
.001550388 


646 
647 
648 
649 
650 


417316 
418609 
419904 
421201 
422500 


269586136 
270840023 
272097792 
273359449 
274625000 


25-4165301 
25.4361947 
25.4558441 
25.4754784 
25-4950976 


8-6445855 
8-6490437 
8.6534974 
8.6579465 
8-6623911 


.001547988 
.001545595 
.001543210 
.001540832 
.001538462 


651 
652 
653 
654 
_655 


423801 
425104 
426409 
427716 
429025 


275894451 
277167808 
278445077 
279726264 
281011375 


25.5147016 
25.5342907 
25.5538647 
25.5734237 
25.5929678 


8.6668310 
8.6712665 
8.6756974 
8.6801237 
8-6845456 


.001536098 
.001533742 
.001531394 
.001529052 
.001526718 


656 
657 
658 
659 
660 


430336 
431649 
432964 
434281 
435600 


282300416 
283593393 
284890312 
286191179 
287496000 


25-6124969 
25-6320112 
25.6515107 
25.6709953 
25-6904652 


8.6889630 
8-6933759 
8.6977843 
8.7021882 
8.7065877 


.001524390 
.001522070 
.001519757 
.001517451 
.001515152 



814 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


661 
662 
663 
664 
665 


436921 
438244 
439569 
440896 
442225 


288804781 
290117528 
291434247 
292754944 
294079625 


25 
25 
25 
25 
25 


7099203 
7293607 
7487864 
7681975 
7875939 


8 
8 
8 
8 
8 


7109827 
7153734 
7197596 
7241414 
7285187 




001512859 
001510574 
001508296 
001506024 
001503759 


666 
667 
668 
669 
670 


443556 
444889 
446224 
447561 
448900 


295408296 
296740963 
298077632 
299418309 
300763000 


25' 
25 
25 
25 
25 


8069758 
8263431 
8456960 
8650343 
8843582 


8 
8 
8 
8 
8 


7328918 
7372604 
7416246 
7459846 
7503401 




001501502 , 

001499250 

001497006 

001494768 

001492537 


671 
672 
673 
674 
675 


450241 
451584 
452929 
454276 
455625 


302111711 
303464448 
304821217 
306182024 
307546875 


25 
25 
25 
25 
25 


9036677 
9229628 
9422435 
9615100 
9807621 


8 
8 
8 
8 
8 


7546913 
7590383 
7633809 
7677192 
7720532 . 




001490313 
001488095 
001485884 
001483680 
001481481 


676 
677 
678 
679 
680 


456976 
458329 
459684 
461041 
462400 


308915776 
310286733 
311665752 
313046839 
314432000 


26 
26 
26 
26 
26 


0000000 
Q192237 
0384331 
0576284 
0768096 


8 
8 
8 
8 
8 


7763830 
7807084 
7850296 
7893466 
7936593 




001479290 
001477105 
001474926 
001472754 
001470588 


681 
682 
683 
684 
685 


463761 
465124 
466489 
467856 
469225 


315821241 
317214568 . 
318611987 
320013504 
321419125 


26 

26 

26 

26. 

26. 


0959767 
1151297 
1342687 
1533937 
1725047 


8 
8 
8 
8 
8 


7979679 
8022721 
8065722 
8108681 
8151598 




001468429 
001466276 
001464129 
001461988 
001459854 


686 
687 
688 
689 
690 


470596 
471969 
473344 
474721 
476100 


322828856 
324242703 
325660672 
327082769 
328509t)00 


26 
26 
26 
26 
26 


1916017 
2106848 
2297541 
2488095 
2678511 


8 
8 
8 
8 
8 


8194474 
8237307 
8280099 
8322850 
8365559 




001457726 
001455604 
001453488 
001451379 
001449275 


691 
692 
693 
694 
695 


477481 
478864 
480249 
481636 
483025 


329939371 
331373888 
332812557 
334255384 
335702375 


26 
26 
26 
26 
26 


2868789 
3058929 
3248932 
•3438797 
3628527 


8 
8 
8 
8 
8 


8408227 
8450854 
8493440 
8535985 
8578489 




P01447178 
001445087 
001443001 
001440922 
001438849 


696 
697 
698 
699 
700 


484416 
485809 
487204 
488601 
490000 


337153536 
338608873 
340068392 
341532099 
343000000 


26 
26 
26 
26 
.26 


3818119 
4007576 
4196896 
4386081 
4575131 


8 
8 
8 
8 
8 


8620952 
8663375 
8705757 
8748099 
8790400 




001436782 
001434720 
00143266.5 
001430615 
001428571 


701 
702 
703 
704 
705 


491401 
492804 
494209 
495616 
497025 


344472101 
345948408 
347428927 
348913664 
350402625 


26 
26 
26 
26 
26 


4764046 
4952826 
5141472 
5329983 
5518361 


8 
8 
8 
8 
8 


8832661 
8874882 
8917063 
8959204 
9001304 




001426534 
001424501 
001422475 
001420455 
001418440 


706 
707 
708 
709 
710 


498436 
499849 
501264 
502681 
504100 


351895816 
353393243 
354894912 
356400829 
357911000 


26 
26 
26 
26 
26 


5706605 
5894716 
6082694 
6270539 
6458252 


8 
8 
8 
8 
8 


9043366 
9085387 
9127369 
9169311 
9211214 




001416431 
001414427 
001412429 
001410437 
001408451 


711 
712 
713 
714 
715 


505521 
506944 
508369 
509796 
511225 


359425431 
360944128 
362467097 
363994344 
365525875 


26 
26 
26 
26 
26 


6645833 
6833281 
7020598 
7207784 
7394839 


8 
8 
8 
8 
8 


9253078 
9294902 
9336687 
9378433 
9420140 




001406470 
001404494 
001402525 
001400560 
001399501 


716 
717 
718 
719 
730 


512656 
514089 
515524 
516961 
518400 


367061696 
368601813 
370146232 
371694959 
373248000 


26 
26 
26 
26 
26 


7581763 
7768557 
7955220 
8141754 
8328157 


8 
8 
8 
8 
8 


9461809 
9503438 
9545029 
9586581 
9628095 




001396648 
001394700 
001392758 
001390821 
001388889 '< 



815 





TABLE XVI. — SQUARES, 


CUBES, 


SQUARE ROOTS, 


No. • 


Squares. 

— : — i 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


721 


519841 


374805361 


26-8514432 


8-9669570 


.001386963 


722 


521284 


376367048 


26 


•8700577 


8 


-9711007 


.001385042 


723 


522729 


377933067 


26 


8886593 


8 


-9752406 


.001383126 


724 


524176 


379503424 


26 


.9072481 


8 


-9793766 


.001381215 


725 


525625 
527076 


381078125 


26 


.9258240 


8 


-9835089 


.0013793L0 ; 
.001377410 


726 


382657176 


26 


.9443872 


8 


-9876373 


727 


528529 


384240583 


26 


.9629375 


8 


-9917620 


.001375516 


728 


529984 


385828352 


26 


.9814751 


8 


-9958829 


.001373626 


729 


531441 


387420489 


27 


.0000000 


9 


-0000000 


.001371742 


730 


532900 


389017000 


27 


.0185122 


9 


-0041134 


-001369863 


731 


534361 


390617891 


27 


.0370117 


9 


.0082229 


.001367989 


732 


535824 


392223168 


27 


-0554985 


9 


-0123288 


.001366120 


733 


537289 


393832837 


27 


-0739727 


9 


-0164309 


.001364256 


734 


538756 


395446904 


27 


-0924344 


9 


-0205293 


.001362398 


735 


540225 


397065375 


27 


-1108834 


9 


-0246239 


.001360544 
•001358696 


736 


541696 


398688256 


27 


-1293199 


9 


-0287149 


737 


543169 


400315553 


27 


-1477439 


9 


.0328021 


.001356852 


738 


544644 


401947272 


27 


-1661554 


9 


.0368857 


.001355014 


739 


546121 


403583419 


27 


-1845544 


9 


.0409655 


.001353180 


740 


547600 


405224000 


27 


■2029410 


9 


.0450419 


-001351351 


741 


549081 


406869021 


27 


-2213152 


9 


.0491142 


.001349528 


742 


550584 


408518488 


27 


-2396769 


9 


.0531831 


.001347709 


743 


552049 


410172407 


27 


-2580263 


9 


.0572482 


.001345895 


744 


553536 


411830784 


27 


-2763634 


9 


.0613098 


.001344086 


745 


555025 


413493625 


27 


.2946881 


9 


.0653677 


.001342282 


746 


556516 


415160936 


27 


.3130006 


9 


.0694220 


.001340483 


747 


558009 


416832723 


27 


■3313007 


9 


-0734726 


.001338688 


748 


559504 


418508992 


27 


3495887 


9 


-0775197 


.001336898 


749 


561001 


420189749 


27 


.3678644 


9 


-0815631 


.001335113 


750 


562500 


421875000 


27 


.3861279 


9 


0856030 


-001333333 


751 


564001 


423564751 


27 


-4043792 


9 


-0896392 


•001331558 


752 


565504 


425259008 


27 


-4226184 


9 


0936719 


•001329787 


753 


567009 


426957777 


27 


4408455 


9 


0977010 


.001328021 


754 


568516 


428661064 


27 


4590604 


9 


1017265 


.001326260 


755 


570025 


430368875 


27 


4772633 


9 


1057485 


•001324503 


756 


571536 


432081216 


27 


4954542 


9 


1097669 


.001322751 


757 


573049 


433798093 


27 


5136330 


9 


1137818 


.001321004 


758 


574564 


435519512 


27 


5317998 


9 


1177931 


.001319261 


759 


576081 


437245479 


27 


5499546 


9 


1218010 


.001317523 


760 


577600 


438976000 


27 


5680975 


9 


1258053 


•001315789 


761 


579121 


440711081 


27 


5862284 


9 


1298061 


•001314060 


762 


580644 


442450728 


27 


6043475 


9 


1338034 


.001312336 


763 


582169 


444194947 


27 


6224546 


9 


1377971 


.001310616 


764 


583696 


445943744 


27 


6405499 


9 


1417874 


.001308901 


765 


585225 


447697125 


27 


6586334 


9 


1457742 


•001307190 


766 . 


586756 


449455096 


27 


6767050 


9 


1497576 


•001305483 


767 


588289 


451217663 


27 


6947648 


9 


1537375 


•001303781 


768 


589824 


452984832 


27 


7128129 


9 


1577139 


•001302083 


769 


591361 


454756609 


27 


7308492 


9 


1616869 


•001300390 


770 


592900 


456533000 


27 


7488739 


9 


1656565 


•001298701 


771 


594441 


458314011 


27 


7668868 


9 


1696225 


-001297017 


772 


595984 


460099648 


27 


7848880 


9 


1735852 


.001295337 


773 


597529 


461889917 


27 


8028775 


9- 


1775445 


.001293661 


774 


599076 


463684824 


27. 


8208555 


9. 


1815003 


.001291990 


775 


600625 


465484375 
467288576 


27 


8388218 


9. 


1854527 


-001290323 


776 


602176 


27. 


8567766 


9. 


1894018 


.001288660 


777 


603729 


4690O7433 


27. 


8747197 


9. 


1933474 


.001287001 


778 


605284 


470910952 


27. 


8926514 


9. 


1972897 


.001285347 


779 


606841 


472729139 


27. 


9105715 


9- 


2012286 


.001283697 


780 


608400 


474552000 


27-9284801 


9-2051641 


•001282051 



816 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots, ' 


Reciprocals. 


781 

782 
783 
784 
785 


609961 
611524 
613089 
614656 
616225 


476379541 
478211768 
480048687 
481890304 
488736625 


27 
27 
27 
28 
28 


9463772 
9642629 
9821372 
0000000 
0178515 


9 
9 
9 
9 
9 


2090962 
2130250 
2169505 
2208726 
2247914 


■001280410 
.001278772 
•001277139 
•001275510 
.001273885 


786 
787 
788 
789 
790 


617796 
619369 
620944 
622521 
624100 


485587656 
487443403 
489303872 
491169069 
493039000 


28 
28 
28 
28 
28 


0356915 
0535203 
0713377 
0891438 
1069386 


9 
9 
9 
9 
9 


2287068 
2326189 
2365277 
2404333 
2443355 


•001272265 
•C01270648 
•001269036 
•001267427 
.001265823 


791 
792 
793 

794 
795 


625681 
627264 
628849 
630436 
632025 


494913671 
496793088 
498677257 
500566184 
502459875 


28 
28 
28 
28 
28 


1247222 
1424946 
1602557 
1780056 
1957444 


9 
9 
9 
9 
9 


2482344 
2521300 
2560224 
2599114 
2637973 


•001264223 
.001262626 
•001261034 
•001259446 
.001257862 


796 
797 
798 
799 
800 


633616 
635209 
636804 
638401 
640000 


504358336 
506261573 
508169592 
510082399 
512000000 


28 
28 
28 
28 
28 


2134720 
2311884 
2488938 
2665881 
2842712 


9 
9 
9 
9 

9 


2676798 
2715592 
2754352 
2793081 
2831777 


.001256281 
.001254705 
•001253133 
•001251564 
.001250000 


801 
802 
803 
804 
805 


641601 
643204 
644809 
646416 
648025 


513922401 
515849608 
517781627 
519718464 
521660125 


28 
28 
28 
28 
28 


3019434 
3196045 
3372546 
3548938 
3725219 


9 
9 
9 
9 
9 


2870440 
2909072 
2947671 
2986239 
3024775 


.001248439 
.001246883 
•001245330 
•001243781 
.001242236 


806 
807 
808 
809 
810 


649636 
651249 
652864 
654481 
656100 


523606616 
525557943 
527514112 
529475129 
531441000 


28 
28 
28 
28 
28 


3901391 
4077454 . 
4253408 
4429253 
4604989 


9 
9 
9 
9 
9 


3063278 
3101750 
3140190 
3178599 
3216975 


•001240695 
.001239157 
•001237624 
•001236094 
•001234568 


811 
812 
813 
814 
815 


657721 
659344 
660969 
662596 
664225 


533411731 
535387328 
537367797 
539353144 
541343375 


28 
. 28 
28 
28 
28 


4780617 
4956137 
5131549 
5306852 
5482048 


9 
9 
9 
9 
9 


3255320 
3298634 
3331916 
3370167 
3408386 


•001233046 
•001231527 
•001230012 
.001228501 . 
•001226994 


816 
817 
818 

819 

, 820 


665856 
667489 
669124 
670761 
672400 


543338496 
545338513 
547343432 
549353259 
551368000 


28 
28 
28 
28 
98 


5657137 
5832119 
6006993 
6181760 
6856421 


9 
9 
9 
9 
9 


3446575 
3484731 
3522857 
3560952 
3599016 


.001225490 
•001223990 
•001222494 
•001221001 
.001219512 


821 
822 
823 
824 
825 


674041 
675684 
677329 
678976 
680625 


553387661 
555412248 
557441767 
559476224 
561515625 


28 
28 
28 
28 
28 


6530976 
6705424 
6879766 
7054002 
7228132 


9 
9 
9 
9 
9 


3637049 
3675051 
3713022 
3750963 
3788873 


•001218027 
.001216545 
•001215067 
•001213592 
•001212121 


826 
827 
828 
829 
830 


682276 
683929 
685584 
687241 
688900 


563559976 
565609283 
567663552 
569722789 
571787000 


28 
28 
28 
28 
28 


7402157 
7576077 
7749891 
7923601 
8097206 1 


9 
9 
9 
9 
9 


3826752 
3864600 
3902419 
3940206 
3977964 


•001210654 
•001209190 
•001207729 
•001206273 
.001204819 


831 
832 
833 
834 
835 


&90561 
692224 
693889 
695556 
697225 


573856191 
575930368 
578009537 
580093704 
582182875 


28 
28 
28 
28 
28 


8270706 
8444102 
8617394 
8790582 
8963666 


9 
9 
9 
9 
9 


4015691 
4053387 
4091054 
4128690 
4166297 


.001203369 
•001201923 
•001200480 
.001199041 
•001197605 


836 
837 
838 
839 
840 


698896 
700569 
702244 
708921 
705600 


584277056 
586376253 
588480472 
590589719 
592704000 


28 
28 
28 
28 
28 


9136646 
9309523 
9482297 
9654967 
9827535 


9 
9 
9 
9 
9 


4203873 
4241420 
4278936 
4316423 
4353880 


•001196172 
•001194743 
•001193317 
.001191895 
.001190476 



817 





TABLE XVI. SQUARES; CUBES, 


SQUARE ROOTS, 1 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


841 
842 
843 
844 
845 


707281 
708934 
710649 
712336 
714025 


594823321 
596947688 
599077107 
601211584 
603351125 


29.0000000 
29-0172363 
29-0344623 
29.0516781 
29-0688837 


9.4391307 
9.4428704 
9.4466072 
9.4503410 
9.4540719 


.001189061 
.001187648 
.001186240 1 
.001184834 
-001183432 


846 
847 
848 
849 
, 850 


715716 
717409 
719104 
720801 
722500 


605495736 
607645423 
609800192 
611960049 
614125000 


29-0860791 
29-1032644 
29-1204396 
29-1376046 
29-1547595 


9 4577999 
9.4615249 
9-4652470 
9-4689661 
9.4726824 


.001182033 
.001180638 
.001179245 
.001177856 
-001176471 


851 
852 
853 
854 
. 855 


724201 
725904 
727609 
729316 
731025 


616295051 
618470208 
620650477 
622835864 
625026375 


29.1719043 
29-1890390 
29-2061637 
29.2232784 
29.2403830 


9-4763957 
9-4801061 
9-4838136 
9-4875182 
9-4912200 


.001175088 
.001173709 
.001172333 
.001170960 
-001169591 


856 
857 
858 
859 
, 860 


732736 
734449 
736164 
737881 
739600 


627222016 
629422793 
631628712 
633839779 
636056000 


29-2574777 
29-2745623 
29-2916370 
29-3087018 
29-3257566 


9-4949188 
9-4986147 
9-5023078 
9-5059980 
9-5096854 


-001168224 
.001166861 
.001165501 
.001164144 
.001162791 


861 
862 
863 
864 
865 


741321 
743044 
744769 
746496 
748225 


638277381 
640503928 
642735647 
644972544 
647214625 


29-3428015 
29-3598365 
29-3768616 
29-3938769 
29.4108823 


9-5133699 
9-5170515 
9-5207303 
9 - 5244063 
9.5280794 


.001161440 
-001160093 
.001158749 
.001157407 
.001156069 


866 
867 
868 
869 
870 


749956 
751689 
753424 
755161 
756900 


649461896 
651714363 
653972032 
656234909 
658503000 


29-4278779 
9-4448637 
29-4618397 
29-4788059 
29-4957624 


9-5317497 
9.5354172 
9.5390818 
9.5427437 
9-5464027 


.001154734 
.001153403 
.001152074 
-001150748 
-001149425 


871 
872 
873 
874 
875 


758641 
760384 
762129 
763876 
765625 


660776311 
663054848 
665338617 
667627624 
669921875 


29.S127091 
29.5296461 
29.5465734 
29-5634910 
29-5803989 


9-5500589 
9-5537123 
9-5573630 
9.5610108 
9.5646559 


.001148106 
.001146789 
.001145475 
.001144165 
.001142857 


876 
877 
878 
879 
880 


767376 
769129 
770884 
772641 
774400 


672221376 
674526133 
676836152 
679151439 
681472000 


29.5972972 
29.6141858 
29-6310648 
29:- 6479342 
29-6647939 


9.5682982 
9-5719377 
9-5755745 
9-5792085 
9-5828397 


.001141553 
.001140251 
.001138952 
.001137656 
.001136364 


881 
882 
883 
884 
885 


776161 
777924 
779689 
781456 
783225 


683797841 
686128968 
688465387 
690807104 
693154125 


29.6816442 
29.6984848 
29.7153159 • 
29-7321375 
29.7489496 


9-5884682 
9-5900939 
9-5937169 
9-5973373 
9-6009548 


.001135074 
.001133787 
.001132503 
.001131222 
.001129944 


886 
887 
888 
889 
890 


784996 
786769 
788544 
790321 
792100 


695506456 
697864103 
700227072 
702595369 
704969000 


29-7657521 
29-7825452 
29-7993289 
29.8161030 
29. 8328678 


9-6045696 
9. 6081817 
9-6117911 
9-6153977 
9. 6190017 


.001128668 
.001127396 
.001126126 
.001124859 
.001123596 


891 
892 
893 
894 
895 


793881 
795664 
797449 
799236 
801025 


707347971 
709732288 
712121957 
714516984 
716917375 


29 8496231 
29.8663690 
29-8831056 
29-8998328 
29- 9165506 


9-6226030 
9.6262016 
9.6297975 
9. 6333907 
9.6369812 


.001122334 
-001121076 
.001119821 
.001118568 
-001117318 


896 
897 
898 
899 
900 


802816 
804609 
806404 
808201 
810000 


719323136 
721734273 
724150792 
726572699 
729000000 


29.9332591 
29-9499583 
29-9666481 
29.9833287 
30-0000000 


9.6405690 
9-6441542 
9-6477367 
9-6513166 
9-6548938 


.001116071 
.001114827 
.001113586 
.001112347 
.001111111 



818 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Scfuares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


901 
902 
903 
904 
905 


811801 
813604 
815409 
817216 
819025 


731432701 
733870808 
736314327 
738763264 
741217625 


30.0166620 
30-0333148 
30.0499584 
30.0665928 
30-0832179 


9 6584684 
9-6620403 
9 6656096 
9-6691762 
9-6727403 


.001109878 
.001108647 
-001107420 
.001106195 
.001104972 


906 
907 
908 
909 
910 


820836 
822649 
824464 
826281 
828100 


743677416 
746142643 
748613312 
751089429 
753571000 


30-0998339 
30-1164407 
30-1330383 
30-1496269 
30.1662063 


9-6763017 
9-6798604 
9.6834166 
9.6869701 
9.6905211 


.001103753 
.001102536 
.001101322 
.001100110 
.001098901 


911 
912 
913 
914 
915 


829921 
831744 
833569 
835396 
837225 


756058031 
758550528 
761048497 
763551944 
766060875 


30.1827765 
30.1993377 
30.2158899 
30-2324329 
30-2489669 


9.6940694 
9.6976151 
9-7011583 
9-7046989 
9-7082369 


.001097695 
.001096491 
.001095290 
.001094092 
.001092896 


916 
917 
918 
919 
930 


839056 
840889 
842724 
844561 
846400 


768575296 
771095213 
773620632 
776151559 
778688000 


?0- 2654919 
30-2820079 
30.2985148 
30-3150128 
30-3315018 


9-7117723 
9.7153051 
9-7188354 
9-7223631 
9-7258883 

9-7294109 
9-7329309 
9-7364484 
9-7399634 
9-7434758 


.001091703 
-001090513 
.001089325 
.001088139 
-001086957 


921 
922 
923 
924 
925 


848241 
850084 
851929 
853776 
855625 


781229961 
783777448 
786330467 
788889024 
791453125 


30-3479818 
30-3644529 
30-3809151 
30-3973683 
30-4138127 


.001085776 
.001084599 
.001083423 
-001082251 
.001081081 


926 
927 
928 
929 
930 


857476 
859329 
861184 
863041 
864900 


794022776 
796597983 
799178752 
801765089 
804357000 


30-4302481 
30-4466747 
30-4630924 
30-4795013 
30-4959014 


9-7469857 
9-7504930 
9-7539979 
9-7575002 
9-7610001 


.001079914 
.001078749 
.001077586 
.001076426 
.001075269 


931 
932 
933 
934 
935 


866761 
868624 
870489 
872358 
874225 


806954491 
809557568 
812166237 
814780504 
817400375 


30.5122926 
30.5286750 
30.5450487 
30.5614136 
30.5777697 


9-7644974 
9.7679922 
9.7714845 
9.7749743 
9.7784616 


-001074114 
.001072961 
.001071811 
.001070664 
-001069519 


936 
937 
938 
939 
940 


876096 
877969 
879844 
881721 
883600 


820025856 
822656953 
825293672 
827936019 
830584000 


30.5941171 
30-6104557 
30.6267857 
30.6431069 
30.6594194 


9.7819466 
9-7854288 
9-7889087 
9-7923861 
9-7958611 


.001068376 
.001067236 
.001066098 
.001064963 
.001063830 


941 
942 
943 
944 
945 


885481 
887364 
889249 
891136 
893025 


833237621 
835896888 
838561807 
841232384 
843908625 


30.6757233 
30.6920185 
30.7083051 
30.7245830 
30.7408523 


9-7993336 
9-8028036 
9-8062711 
9-8097362 
9-8131989 


.001062699 
.001061571 
.001060445 
.001059322 
.001058201 


946 
947 
948 
949 
950 


894916 
896809 
898704 
900601 
a02500 


846590536 
849278123 
851971392 
854670349 
857375000 


30.7571130 
30.7733651 
30.7896086 
30.8058436 
30-8220700 


9-8166591 
9-8201169 
9-8235723 
9-8270252 
9-8304757 


-001057082 
.001055966 
.001054852 
.001053741 
-001052632 


951 
952 
953 
954 
955 


904401 
906304 
908209 
910116 
912025 


860085351 
862801408 
865523177 
868250664 
870983875 


30.8382879 
30.8544972 
30.8706981 
30.8868904 
30.9030743 


9-8339238 
9-8373695 
9-8408127 
9.8442536 
9-8476920 


.001051525 
.001050420 
.001049318 
.001048218 
.001047120 


956 
957 
958 
959 
960 


913936 
915849 
817764 
919681 
921600 


873722816 
876467493 
879217912 
881974079 
884736000 


30.9192497 
30.9354166 
30.9515751 
30.9677251 
30. 9838668 


9.8511280 
98545617 
9-8579929 
9.8614218 
9.8648483 


.001046025 
.001044932 
.001043841 
.001042753 
.001041667 



819 



TABLE XVI. 


SQUARES, 


CUBES, SQUARE ROOTS, ETC. 

■- - 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


RecipBocals. 


961 
962 
963 
964 
965 


923521 
925444 
927369 
929296 
931225 


887503681 
890277128 
893056347 
895841344 
898632125 


31.0000000 
31.0161248 
31.0322413 
31.0483494 
31.0644491 


9-8682724 
9-8716941 
9.8751135 
9.8785305 
9.8819451 


.001040583 
.001039501 
.001038422 
.001037344 i 
.001036269 '. 


966 
967 
968 
969 
970 


933156 
935089 
937024 
938961 
940900 


901428696 
904231063 
907039232 
909853209 
912673000 


31.0805405 
31.0966236 
31.-1126984 
31.1287648 
31.1448230 


9.8853574 
9.8887673 
9.8921749 
9.8955801 
9-8989830 


.001035197 ; 

.001034126 

.001033058 

.001031992 

.001030928 


971 
972 
973 
974 
975 


942841 
944784 
946729 
948676 
950625 


915498611 
918330048 
921167317 
924010424 
926859375 


31.1608729 
31.1769145 
31.1929479 
31.2089731 
31-2249900 


9.9023835 
9.9057817 
9.9091776 
9.9125712 
9-915962^ 


.001029866 
.001028807 
.001027749 
.001026694 
.001025641 


976 
977 
978 
979 

980 


952576 
954529 
956484 
958441 
960400 


929714176 
932574833 
935441352 
938313739 
941192000 


31-2409987 
31.2569992 
31.2729915 
31.2889757 
31-3049517 


9. 
9. 
9. 
9. 
9- 


9193513 
9227379 
9261222 
9295042 
9328839 


.001024590 
.001023541 
.001022495 
.001021450 
.001020408 


981 
982 
983 
984 
, 985 


962361 
964324 
966289 
968256 
970225 


944076141 
946966168 
949862087 
952763904 
955671625 


31-3209195 
31-3368792 
31-3528308 
31.3687743 
31-3847097 


9- 

9- 

9- 

9 

9 


9362613 
9396363 
9430092 
9463797 
9497479 


.001019368 
.001018330 
.001017294 
.001016260 
.001015228 


986 
987 
988 
989 
990 


972196 
974169 
976144 
978121 
980100 


958585256 
961504803 
964430272 
967361669 
970299000 


31.4006369 
31.4165561 
31.4324673 
31-4483704 
31-4642654 


9 
9 
9 
9 
9 


9531138 
9564775 
9598389 
9631981 
9665549 


.001014199 
.001013171 
.001012146 
.001011122 
.001010101 


991 
992 
993 
994 
995 


982081 
984064 
986049 
.988036 
990025 


973242271 
976191488 
979146657 
982107784 
985074875 


31-4801525 
31.4960315 
31-5119025 
31-5277655 
31.5436206 


9 
9 
9 
9 
9 


9699095 
9732619 
9766120 
9799599 
9833055 


.001009082 
.001008065 
.001007049 
.001006036 
.001005025 


996 
997 
998 
999 
1000 


992016 
994009 
996004 
998001 
1000000 


988047936 
991026973 
994011992 
997002999 
1000000000 


31-5594677 
31-5753068 
31.5911380 
31-6069613 
31-6227766 


9 
9 
9 
9 
10 


9866488 
9899900 
9933289 
9966656 
0000000 


.001004016 
.001003009 
.001002004 
.001001001 
.001000000 


1001 
1002 
1003 
1004 
1005 


1002001 
1004004 
1006009 
1008016 
1010025 


1003003001 
1006012008 
1009027027 
1012048064 
1015075125 


31.6385840 
31-6543836 
31-6701752 
31-6859590 
31. 7017349 


10 
10 
10 
10 
10 


0033322 

0066622 

0099899 

-0133155 

.0166389 


.0009990010 
.0009980040 
.0009970090 
.0009960159 
.0009950249 


1006 
1007 
1008 
1009 
1010 


1012036 
1014G49 
1016064 
1018081 
1020100 


1018108218 
1021147343 
1024192512 
1027243729 
1030301000 


31-7175030 
31.7332633 
31-7490157 
31-7647603 
31-7804972 


10 
10 
10 
10 
10 


-0199601 
-0232791 
0265958 
-0299104 
•0332228 


.0009940358 
.0009930487 
.0009920635 
.0009910803 
.0009900990 


1011 
1012 
1013 
1014 
1015 


1022121 
1024144 
1026169 
1028196 
1030225 


1033364331 
1036433728 
1039509197 
1042590744 
1045678375 


31-7962262 
31-8119474 
31-8276609 
31-8433666 
31-8590646 


10 
10 
10 
10 
10 


.0365330 
.0398410 
.0431469 
-0464506 
-0497521 


.0009891197 
.0009881423 
o0009871668 
.0009861933 
.0009852217 


1016 
1017 
1018 
1019 
1030 


1032256 
1034289 
1036324 
1038361 
1040400 


1048772096 
1051871913 
1054977832 
1058089859 
1061208000 


31.8747549 
31-8904374 
31-9061123 
31.9217794 
31. 9374388 


10 
10 
10 
10 
10 


•0530514 
-0563485 
0596435 
-0629364 
-066227], 


.0009842520 
.0009832842 
.0009823183 
.0009813543 
.0009803922 



820 



TABLE XVII.— CUBIC YARDS PER 100 FEET OP LEVEL 
SECTIONS. SLOPE 1:1. 



Depth, 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


d 


12 feet. 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


48 


56 


63 


70 


78 


107 


115 


122 


2 


104 


119 


133 


148 


163 


222 


237 


252 


3 


167 


189 


211 


233 


256 


344 


367 


389 


4 


237 


267 


296 


326 


356 


474 


504 


533 


5 


315 


352 


389 


426 


463 


611 


648 


685 


6 


400 


444 


489 


533 


578 


756 


800 


844 


7 


493 


544 


596 


648 


700 


907 


959 


1011 


8 


593 


652 


711 


770 


830 


1067 


1126 


1185 


9 


700 


767 


833 


900 


967 


1233 


1300 


1367 


XO 


815 


889 


963 


1037 


1111 


1407 


1481 


1556 


11 


937 


1019 


1100 


1181 


1263 


1589 


1670 


1752 


12 


1067 


1156 


1244 


1333 


1422 


1778 


1867 


1958 


13 


1204 


1300 


1396 


1493 


1589 


1974 


2070 


2167 


14 


1348 


1452 


1556 


1659 


1763 


2178 


2281 


23?>5 


15 


1500 


1611 


1722 


1833 


1944 


2389 


2500 


2611 


16 


1659 


1778 


1896 


2015 


2133 


2607 


2726 


2844 


17 


1826 


1952 


2078 


2204 


2330 


2833 


2959 


3085 


18 


2000 


2133 


2267 


2400 


2533 


3067 


3200 


3333 


19 


2181 


2322 


2463 


2604 


2744 


3307 


3448 


3589 


20 


2370 


2519 


2667 


2815 


2963 


3556 


3704 


3852 


21 


2567 


2722 


2878 


3033 


3189 


3811 


3967 


4122 


22 


2770 


2933 


3096 


3259 


3422 


4074 


4237 


4400 


23 


2981 


3152 


3322 


3493 


3663 


4344 


4515 


4685 


24 


3200 


3378 


3556 


3733 


3911 


4622 


4800 


4978 


25 


3426 


3611 


3796 


3981 


4167 


4907 


5093 


5278 


26 


3659 


3852 


4044 


4237 


4430 


5200 


5393 


5585 


27 


3900 


4100 


4300 


4500 


4700 


5500 


5700 


5900 


28 


4148 


4356 


4563 


4770 


4978 


5807 


6015 


6222 


29 


4404 


4619 


4833 


5048 


5263 


6122 


6337 


6552 


30 


4667 


4889 


5111 


5333 


5556 


6444 


6667 


.6889 


81 


4937 


5167 


5396 


5626 


5856 


6774 


7004 


7233 


32 


5215 


5452 


5689 


5926 


6163 


7111 


7348 


7585 


33 


5500 


5744 


5989 


6233 


6478 


7456 


7700 


7944 


34 


5793 


6044 


6296 


6548 


, 6800 


7807 


8059 


8311 


35 


6093 


6352 


6611 


6870 


7130 


8167 


8426 


8685 


36 


6400 


6667 


6933 


7200 


7467 


8533 


8800 


9067 




6715 


6989 


7263 


7537 


7811 


8907 


9181 


9456 


38 


7037 


7319 


7600 


7881 


8163 


9289 


9570 


9852 


39 


7367 


7656 


7944 


8233 


8522 


9678 


9967 


10256 


40 


7704 


8000 


8296 


8593 


8889 


10074 


10370 


10667 


41 


8048 


8352 


8656 


8959 


9263 


10478 


10781 


11085 


42 


8400 


8711 


9022 


9333 


9644 


10889 


11200 


11511 


43 


8759 


9078 


9396 


9715 


10033 


11307 


11626 


11944 


44 


9126 


9452 


9778 


10104 


10430 


11733 


12059 


12385 


45 


9500 


9833 


10167 


10500 


10833 


12167 


12500 


12833 


46 


9881 


10222 


10563 


10904 


11244 


12607 


12948 


13289 


47 


10270 


10619 


10967 


11315 


11663 


13056 


13404 


13752 


48 


10667 


11022 


11378 


11733 


12089 


13511 


13867 


14222 


49 


11070 


11433 


11796 


12159 


12522 


13974 


14337 


.14700 


50 


11481 


11852 


12222 


12593 


12963 


14444 


14815 


15185 


51 


11900 


12278 


12656 


13033 


13411 


14922 


15300 


15678 


52 


12326 


12711 


13096 


13481 


13867 


15407 


15793 


16178 


53 


12759 


13152 


13544 


13937 


14330 


15900 


16293 


16685 


54 


13200 


13600 


14000 


14400 


14800 


16400 


16800 


17200 


55 


13648 


14056 


14463 


14870 


15278 


16907 


17315 


17722 


56 


14i04 


14519 


14933 


15348 


15763 


.17422 


17837 


18252 


57 


14567 


14989 


15411 


15833 


16256 


17944 


18367 


18789 


58 


15037 


15467 


15896 


16326 


16756 


18474 


18904 


19333 


59 


15515 


15952 
16444 


16339 


16826 


17263 


19011 


19448 


19885 


60 


1600O 


16889 


17333 


17778 


19556 


20000 


20444 



821 



TABLE XVII. CUBIC YARDS PER 100 FEET OF LEVEL 

SECTIONS. SLOPE 1.5 : 1. 



Depth 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


d 


12 feet. 
50 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


57 


65 


72 


80 


109 


117 


124 


2 


111 


126 


141 


156 


170 


230 


244 


259 


Z 


183 


206 


228 


250 


272 


361 


383 


406 


4 


267 


296 


326 


356 


385 


504 


533 


563 


5 


361 


398 


435 


472 


509 


657 


694 


731 


6 


467 


511 


556 


600 


644 


822 


867 


911 


7 


583 


635 


687 


739 


791 


998 


1050 


1102 


8 


711 


770 


830 


889 


948 


1185 


1244 


1304 


9 


850 


917 


983 


1050 


1117 


1383 


1450 


1517 


10 


1000 


1074 


1148 


1222 


1296 


1593 


1667 


1741 


11 


1161 


1243 


1324 


1406 


1487 


1813 


1894 


1976 


12 


1333 


1422 


1511 


1600 


1689 


2044 


2133 


2222 


13 


1517 


1613 


1709 


1806 


1902 


2287 


2383 


2480 


14 


1711 


1815 


1919 


2022 


2126 


2541 


2644 


2748 


15 


1917 


2028 


2139 


2250 


2361 


2806 


2917 


3028 


16 


2133 


2252 


2370 


2489 


2607 


3081 


3200 


3319 


17 


2361 


2487 


2613 


2739 


2865 


3369 


3494 


3620 


18 


2600 


2733 


2867 


3000 


3133 


3667 


3800 


3933 


19 


2850 


2991 


3131 


3272 


3413 


3976 


4117 


4257 


20 


3111 


3259 


3407 


3556 


3704 


4296 


4444 


4593 


21 


3383 


3539 


3694 


3850 


4006 


4628 


4783 


4939 


22 


3667 


3830 


3993 


4156 


4319 


4970 


5133 


5296 


23 


3961 


4131 


4302 


4472 


4642 


5324 


5494 


5665 


24 


4267 


4444 


4622 


4800 


4978 


5689 


5867 


6044 


25 


4583 


4769 


4954 


5139 


5324 


6065 


6250 


6435 


26 


4911 


5104 


5296 


5489 


5681 


6452 


6644 


6837 


2f 


5250 


5450 


5650 


5850 


6050 


6850 


7050 


7250 


28 


5600 


5807 


6015 


6222 


6430 


7259 


7467 


7674 


2@ 


5961 


6176 


6391 


6606 


6820 


7680 


7894 


8109 


80 


6333 


6556 


6778 


7000 


7222 


8111 


8333 


8556 


SI 


6717 


6946 


7176 


7406 


7635 


8554 


8783 


9013 


32 


7111 


7348 


7585 


7822 


8059 


9007 


9244 


9481 


83 


7517 


7761 


8006 


8250 


8494 


9472 


9717 


9961 


84 


7933 


8185 


8437 


8689 


8941 


9948 


10200 


10462 


83 


8361 


8620 


8880 


9139 


9398 


10435 


10694 


10954 


36 


8800 


9067 


9333 


9600 


9867 


10933 


11200 


11467 


87 


9250 


9524 


9798 


10072 


10346 


11443 


11717 


11991 


se 


9711 


9993 


10274 


10556 


10837 


11963 


12244 


12526 


39 


10183 


10472 


10761 


11050 


11339 


12494 


12783 


13072 


40 


10667 


10963 


11259 


11556 


11852 


13037 


13333 


13630 


41 


11161 


11465 


11769 


12072 


12376 


13591 


13894 


14198 


42 


11667 


11978 


12289 


12600 


12911 


14156 


14467 


14778 


43 


12183 


12502 


12820 


13139 


13457 


14731 


15050 


15369 


44 


12711 


13037 


13363 


13689 


14015 


15319 


15644 


15970 


45 


13250 


1358 


13917 


14250 


14583 


15917 


16250 


16583 


46 


13800 


14141 


14481 


14822 


15163 


16526 


16867 


17207 


47 


14361 


14709 


15057 


15406 


15754 


17146 


17494 


17843 


48 


14933 


15289 


15644 


16000 


16356 


17778 


18133 


18489 


49 . 


15517 


15880 


16243 


16606 


16969 


18420 


18783 


19146 


50 


16111 


16481 


16852 


17222 


17593 


19074 


19444 


1981S 


61 


16717 


17094 


17472 


17850 


18228 


19739 


20117 


20494 


52 


17333 


17719 


18104 


18489 


18874 


20415 


20800 


21185 


53 


17961 


18354 


18746 


19139 


19531 


21102 


21494 


21887 


54 


18600 


19000 


19400 


19800 


20200 


21800 


22200 


22600 


55 


19250 


19657 


20065 


20472 


20880 


22509 


22917 


23324 


56 


19911 


20326 


20741 


21156 


21570 


23230 


23644 


24059 


57 


20583 


21006 


21428 


21850 


22272 


23961 


24383 


24805 


58 


2126? 


21696 


22126 


22556 


22985 


24704 


25133 


25563 


59 


21961 


22398 


22835 


23272 


23709 


25457 


25894 


26331 


60 


22667 


23111 


23556 


24000 


24444 


26222 


26667 


27111 



822 



TABLE XVII.— CUBIC YARDS PER 100 FEET OF LEVEL SECTIONS. 
CORRECTIVE PERCENTAGE FACTORS. 

To be applied when cross-sections are not level. See § 95. 

Side slope =1.5:1 or ^=33°41'. 



Trans- 
verse 

surface 
slope. 



a" 



5 
10 
15 
20 
SO 



Per- 
cent 



9 
18 
27 
36 
57 



6=12 feet 
and d= 



10 

feet. 



% 
1.9 
8.2 
21 

45 
327 



20 
feet. 



% 

1.8 

7.7 

20 

44 
324 



50 
feet. 



% 

1.8 

7.5 

19 

43 
317 



6=20 feet 
and d— 



10 


20 


feet. 


feet. 


% 


% 


2.1 


1.8 


9.0 


8.0 


23 


21 


51 


45 


358 


336 



50 
feet. 



% 

1.8 

7.6 

20 

44 
321 



6=30 feet 
and d= 



10 


20 


feet. 


feet. 


% 


% 


2.3 


2.0 


10.0 


8.4 


26 


22 


57 


48 


400 


354 



50 
feet. 



% 

1.8 

7.7 

20 

44 
326 



Side slope = 1 : 1 or ^ = 45°. 



Trans- 
verse 

surface 
slope. 


6 =12 feet 
and d= 


6=20 feet 
andd= 


6=30 feet 
and d— 


a° 


Per- 
cent 


10 

feet. 


20 

feet. 


50 

feet. 


10 

feet. 


20 

feet. 


50 

feet. 


10 
feet. 


20 
feet. 


50 
feet. 


5 
10 
15 
20 
30 


9 

18 
27 
36 
57 


% 
0.9 
3.7 
9.0 

18 

58 


% 
0.8 
3.4 
8.2 

16 

53 


% 
0.8 
3.2 
7.8 

15 

50 


% 

1.0 

4.3 

10.3 

20 

67 


% 
0.9 
3.6 
8.7 

17 

56 


% 
0.8 
3.3 
8.0 

16 

51 


% 

1.2 

5.0 

12.1 

24 

78 


% 
0.9 
4.0 
9.5 

19 

61 


% 
0.8 
3.4 
8.2 

16 

53 



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824 



INDEX. 

Numbers refer to sections except where specifically marked pages (p.). 

Abandonment of existing track 534, c 

Abutments for trestles 176 

Accelerated motion, application of laws to movement of trains 514 

Acceleration-speed curves. 462 

Accidents, danger of, due to curvature 507 

Accuracy of earthwork computations 125 

numerical example 117 

tunnel surveying 197 

Additional business; methods of securing (or losing) it. . ... 532 

Adhesion of wheels and rails 421, 422 

Adjustments of dumpy level — Appendix pp. 619, 620 

instruments, general principles — Appendix p. 612 

transit — -Appendix pp. 613-617 

wye level — Appendix pp. 617-619 

Advance signals, in block signaling , 391 

Advantages of re-location of old lines 533 

tie-plates 286 

Air-brakes 424, 425 

Air resistance — see Atmospheric resistance. 

Allowance for shrinkage of earthwork 128 

Alternating current, used in signaling 394 

American locomotives, frame 401 

equalizing levers 412 • 

wheel, base 400 

Rwy. Eng. Assoc, formula for train resistance 439 

system of tunnel excavation 205 

Aneroid barometer, use in reconnoissance leveling 7 

Angle-bars, angles and dimensions of standard designs — Table XXIV . . 284 

cost 447, d 

efficiency of 280 

for various weights of rail — ^Table XXXII 447, d 

number per mile of track — Table XXXV 447, e 

standard 284 

Angle of slope in earthwork. 90 

Annual charge against a tie, at 5% interest — Table XVIII p. 824 

Antiseptics 38 

Appliances, medical, surgical 37 

Apprehension of danger, effect on travel 508, c 

ARCH CULVERTS 226, 227 

design 226 

example 227 

Area of culverts, computation 212, 217 

A. S. C. E. standard rail sections. 267 

825 



826 INDEX. 

Ash pits , 363 

Asphyxiation, treatment 42 

Assistant engines — see Pushei* engines and Pusher grades. 

Atmospheric resistance, train 430 

Atlantic locomotives, wheel base 400 

Austrian system of tunnel excavation 205 

Automatic air-brakes 425 

signaling, track circuit 394 

stokers 407 

Averaging end areas, volume of prismoid computed by 101 

Axle, effect of parallelism 396 

effect of rigid wheels on 395 

radial, possibihties of 397 

size of standard M.C.B 420 

Azimuth, determination pp. 620-626 

" Bad order " tracks 378 

Balance of grades for unequal traffic 529-531 

determination of relative traffic. .. 531 

general principle 529 

theoretical balance 530 

BALLAST— Chap. VIL 

cost 238, 447, a 

cross-sections , 233 

laying 237 

materials 232 

pressure on, by ties, under traffic 543 

proper depth 236 

Banjo signals, in block signaling 392 

Barometer, reduction of readings to 32° F. — Table XI p. 799 

use of aneroid in reconnoissance leveling 7 

Barometric elevations — Table XII p. 800 

coefficients for corrections for temperatures and 

humidity — Table XIII p. 800 

Beams, strength of stringers considered as 190 

Bearings, compass, use as check on deflections 20, 21 

in preliminary surveys 11 

Beds, camping 33 

Belgian system of tunnel excavation 205 

Belpaire fire-box 403 

Blasting 149-155 

use in loosening earth 138 

Bleeding, treatment 40 

BLOCK SIGNALING— Chap. XIV. 

" Body tracks "..... 378 

Boiler compounds 323 

for locomotive 402-404 

Boiler-power of locomotives, relation to tractive and cylinder power. . . 414 
Bolts — see Track bolts. 

Bonds of railroads, security and profits 469 

Borrow-pits, earthwork 120 

Bowls (or pots) as rail supports 239, 263 

Box-cars, size and capacity 416 



INDEX. 827 

Box culverts 222-224 

old-rail 224 

stone 223 

wooden 222 

Bracing for trestles 174, 175 

design 193 

Brakes— see Train-brakes. 

Brake resistances 434 

Bridge joints (rail) 282 

spirals 5 

warning 375 

Bridges and culverts, cost of repairs and renewals — Table XLI 485 

Bridges of standard dimensions for small spans 230 

in block signaling 392 

Bridges, trestles, and culverts on railroads, cost • 446 

Broken-stone ballast 232 

Burnettizing (chloride-of-zinc process) for preserving timber 251 

Burnt clay ballast 232 

Capital, railroad, classification of 469 

returns on 469, 470 

Caps (trestle), design 192 

Car mileage, nature and cost — Table XLI, § 485, and 495 

Cars , 416-420 

brake-beams 418 

capacity and size 416 

cost of renewals and repairs — Table XLI 485 

draft gear 419 

gage of wheel and form of wheel -tread 420 

stresses in car frames 417 

resistance, track, freight 438 

passenger 439a 

truck frames 418 

use of metal 418 

wheels, kinetic energy of 435 

Cars and horses, use in earthwork 140, e 

and locomotives, use in earthwork 140, / 

Carts and horses, use in earthwork 140, a 

Cattle guards 228 

passes 229 

Center of gravity of side-hill sections, earthwork 123 

Central angle of a curve 51 

Centrifugal force, counteracted by superelevation of outer rail 71, 72 

of connecting-rod, etc., of locomotive 413 

Chairs' as supports for double-headed rails 267 

Chats for ballast • 232 

Chemical composition of rails 273, 274 

purification of water , 321 

Chert for ballast 232 

Cinders ior ballast 232 

Circular lead rails for switches o 304 

Classification of excavated material 156 

railroads 234 



828 INDEX. 

Cleaning, mechanical, locomotive boilers 320 

Clearance card in permissive block signaling ; . . 388 

spaces in locomotives 410 

Clearing and grubbing for railroads, cost 444 

Clothing, surveying parties 35 

Coal consumption in locomotives 407, 452 et seq. 

effect of increasing rate 457 

varying quality 458 

per car- mile. ; . . . . 407 

Coaling stations — see LoComotive coaling stations. 

Columbia locomotives, wheel base 400 

Compass, use of, in preliminary surveys 11 

Competitive traflSc 498 et seq. 

Competitive rates, equality, regardless of distance. . 499 

Compensation for curvature 510, 511 

rate . 511 

reasons ' 510 

rules for 511 

Compensators in block signaling 393 

Compound curves 67-70 

modifications of location 69 

nature and use 67 

mutual relation ^ of the parts 68 

Compound sections, earthwork 91 

Computation of earthwork 101-128 

approximate, from profiles 126 

using a slide rule 106 

Concrete pipe culverts 221 

Conducting transportation, cost of 489-495 

Coning wheels, effect . 397 

Connecting curve from a curved track to the inside 310 

from a curved track to the outside 309 

from a straight track 308 

Consolidation locomotives, equalizing levers 412 

frame 401 

wheel-base 400 

Constants, numerical, in common use — Table XV p. 803 

Construction of tunnels 203-208 

Contours, obtained by cross-sectioning 12 

Contractor's profit, earthwork 147 

Control points, in general route for a railroad 2 

Cooking utensils, camping 29 

Corbels for trestles 178 

Cost of ballast 238 

of blasting , 155 

of chemical treatment of timber 256 

of earthwork 137 et seq. 

of framed-timber trestles 184 

of metal ties 262 

of pile trestles 168 

COST OF RAILROADS.— Chap. XVII. 

detailed estimate 451 



INDEX. 829 

Cost of rails 278 

of station buildings 329 

of ties 248 

of treating wooden ties 256 

of tunneling 209 

Counterbalancing for locomotives 413 

Creosoting for preserving timber 250 

Cross-country routes — reconnoissance 4 

Cros&ings, one straight, one curved track 316 

two curved tracks * 317 

numerical example „ . . 317 

two straight tracks 315 

Cross-over between two parallel curved tracks, straight connecting 

curve 312 

straight tracks 311 

Cross-sectioning, for earthwork computations 98 

for preliminary surveys . 12 

irregular sections for earthwork computations 118 

Cross-sections of ballast 233, 235 

of tunnels 198 

Cross-ties — see Ties. 

Crown-bars in locomotive fire-box 403 

Cubic yards per 100 feet of level sections — Table XVII pp. 821-823 

CULVERTS AND MINOR BRIDGES.— Chap. VI. 

Culverts, arch 226, 227 

area of waterway 212-217 

iron-pipe ^ 220 

old-rail 224 

reinforced-concrete « . . 225 

stone box 223 

tile-pipe 221 

wooden box 222 

CURVATURE.— Chap. XXII. 

compensation for. 510, 511 

correction for, in earthwork computations 121-124 

danger of accident due to -. 507 

effect on travel 508 

extremes of sharp 512 

general objections 506 

of existing track, determination 65 

proper rate of compensation 511 

Curve, elements of a 1 ° 53 

location by deflections 55 

by middle ordinates '. 59 

by offsets from long chord 60 

by tangential offsets 58 

by two transits , 57 

notation, alinement curves 50 

resistance of trains 395, 396, 433 

Curves, compound, — see Compound curves. 

elements of 61 

instrumental work in location 55 



830 INDEX. 

Curves, limitations in location , 64 

method of computing length 49 

metric 47 

modifications of location 63 

mutual relations of elements 52 

obstacles to location 62 

simple, method of designation 46 

transition — see Transition curves. 

use and value of other methods of location (not using a transit) 61 
vertical — see Vertical curves. 

Cylinder power of locomotives, relation to boiler and tractive power. . 414 

Dating nails, for marking ties 247 

Deflecting rods for operating block signals 393 

Deflections for a transition curve 78 

Degree of a curve 46 

Design of culverts ; 211 d seq. 

framed trestles 185-193 

bracing 193 

caps and sills 192 

floor stringers 190 

posts 191 

nutlocks 295 

pile trestles 165 

tie-plates : 287 

track bolts 294 

tunnels 202 

distinctive systems 205 

Development, definition 5 

example, with map 5 

methods of reducing grade 5 

Disadvantages of re-location of old lines , 534 

Diseases, medicines 41 

DISTANCE.— Chap. XXI. 

effect of change on business done 505 

on division of through rates 500 

justification of decrease to save time 504 

relation to rates and expenses 496 

Distant signals in block signaling 390 

Ditches to drain roadbed 94 

Dividends actually paid on railroad stock. 469 

Double-ender locomotives, wheel-base 400 

Double-track, distance between centers 92 

Draft gear 419 

"continuous" 419 

Drainage of roadbed, value of 94, 95 

Drains in tunnels 202 

Draw-bar pull, locomotives 456 

Draw-bars 419 

Drifting, locomotives, relation to speed curves. 464 

Drilling holes for blasting 150, 151 

Drinking water, camping parties 39 

Driving-wheels of locomotives , , 40f 



INDEX. 831 

Driving-wheels of locomotives, section of , 413 

Drop tests for train resistance 437 

Drowning, treatment 42 

Durability of metal ties 259 

rails 275, 273 

wooden ties 242 

Dynamometer tests of train resistance 436 

Earnings of railroads, estimation of 473 

per mile of road 473 

EARTHWORK.— Chap. III. 

Earthwork computations, accuracy 125 

approximate computations from profiles 126 

level sections, approximate volume 102 

numerical example 103 

probable error 116 

relation of actual volume to numerical results 96 

simple approximations 101 

Earthwork, cost 137 et seg., 445 

limit of free haul 136 

method of computing haul 130 et seq. 

shrinkage 127 

surveys 96-100 

Eccentricity of center of gravity of earthwork cross-section 122 

Economics, railroad, nature and limitations 478 

of ties 240 

of treated ties 257 

Efficiency, loss, in steam pressure 410 

Electric shock, treatment 42 

Elements of a 1° curve 53 

simple curve 51 

transition curves — Table IV pp. 637-639 

Embankments, method of formation 129 

usual form of cross-section 88 

Empirical formulae for culvert area 214 

accuracy required 217 

value 215 

Engine-houses for locomotives 341''355 

doors 342 

drop pits 348 

electric lighting 351 

engine pits 345 

floors 347 

form 341 

heating .' 349 

hoists 354 

length 343 

materials of construction 344 

piping 352 

smokejacks 346 

tools 353 

turntables 355 

window lighting. ...,.,.,,,,, 350 



832 INDEX. 

Engineering, proportionate and actual cost, in railroad construction. . , 442 

Engineering News formula for pile-driving 163 

Engineer's duties in locating a railroad. 479 

Engine-houses for locomotives 341-355 

Enginemen, basis of wages 490 

English system of tunnel excavation 205 

Enlargement of tunnel headings 204 

Entrained water in steam 410 

Equalizing-levers on locomotives . 412 

Equivalent sections in earthwork, determination of area. 104 

Estimation of probable volume of traffic and of probable growth 473 

Evaporation per pound of fuel — Table XXXVI, and 452 

Excavation, usual form of cross-section 88 

Exhaust-steam, effect of back-pressure 410 

Expansion of rails , 271 

Explosives, amount used 152 

firing 154 

tamping 153 

use in blasting ,. 149 

Expenditure of money for railroad purposes, general principles 477 

External distance, simple curve 51 

table of, for a 1° curve — Table II pp. 632-634 

Facilities, traffic, effect of increase 475 

Factors of safety, design of timber trestles 189 

Failures of rail joints 283 

Fastenings for metal cross-ties. 261 

Fences 366-371 

braces 369 

concrete posts 370 

construction details 371 

posts. . 368 

types 367 

wire fences 366 

Field work for locating a simple curve 56 

a spiral 80 

Fire-box of locomotive . 403 

area of grate 404 

Fire-brick arches in locomotive fire-box 403 

Fire protection on trestles 182 

Fixed charges, nature and ratio to total disbursements 480 

Flanges of wheels, form 420 

Flanging locomotive driving-wheels, effect 398 

Floor systems for trestling 177-184 

Foaming and priming, in locomotive boilers 322 

Formation of embankments, earthwork 127-129 

railroad corporations, method 468 

Formulde for pile-driving 163 

required area of culverts 214 

train resistance 438 

trigonometrical — Table XIV pp. 801, 802 

useful, and constants-^Table XV p. 803 



INDEX. 833 

Fouling point of a siding. 394 

Foundations for framed trestles. . , 173 

Fractures, bone, treatment 43 

FRAMED TRESTLES .* 169-193 

abutments 176 

bracing 174, 175 

cost , 184 

design 169, 185-193 

foundations . 173 

Joints. . . . . . 170^ 

multiple story construction 171 

span 172 

Frame of locomotive, construction 401 

Free haul of earthwork, limit of 136 

Freight houses „, i^.;. 330-339 

dimensions. 332 

doors ' 335 

fire risk.. 331 

floors 334 

lighting 337 

platforms 333 

ramps..... 339 

roofs 336 

scales 338 

two types, in-bound, out-bound. 330 

Freight yards 378-383 

general principles 379 

minor yards 381 

relation of yard to main track 380 

track scales 382 

. transfer cranes 382 

French system of tunnel excavation 205 

Friction, laws of, as applied to braking trains 422 

Frogs, diagrammatic design 297 

for switches 297, 298 

to find frog number 298 

trigonometrical functions — Table III pp. 635, 636 

Fuel for locomotives, cost of 485, 491 

pumps and engines, cost — Table XXVIII 325 

Gauge of wheels, form of wheel-tread 420 

German system of tunnel excavation ,■ •. i! ^,lii<i ... 205 

GRADE.— Chap. XXIII. 

(see Pusher grades, Ruling grades.) 

-accelerated motion of trains on 514 

distinction between ruling and minor grades 513 

effect on tractive power of locomotives. 461 

in tunnels 199 

line, change in, based on mass diagram 135 

resistance of ^ 432 

starting resistance at stations, reduction 637 

undulatory, advantages, disadvantages, and safe limits. ..... 518 



834 INDEX. 

Grade, virtual , 515 

use, value, and misuse 517 

Grade resistance of trains '. 432 

Gravel ballast .'. 232 

Gravity tests of train resistance 437 

Grate area of locomotives .... — 404-409 

ratio to total heating surface 409 

Gravity, effect on trains on grades 432 

tests of train resistance 437 

Ground levers for switches , 301 

Growth of railroad trafiGlc 473 

affected by increase of facilities 475 

Guard rails for switches . 303 

for trestles 179 

Guides around curves and angles (signaling mechanism) 393 

Gumbo, used for ballast. 232 

Hand brakes 423 

Haul of earthwork, computation of length 130 et stq. 

cost 140, 148 

Umit of profitable 148 

method, depending on distance hauled 141 

Headings in tunnels 203 

Heating surface in locomotives 409 

Hoosac Tunnel, surveys for 194, 197 

Hump yards 379 

I-beam bridges, standard 230 

IMPROVEMENT OF OLD LINES.— Chap. XXIV. 

classification 532 

Inertia resistances 435 

Insect bites, treatment 44 

Instrumental work in locating simple curves 56 

spirals 80 

Interest on cost of railroads during construction , 449 

Iron pipe culverts ; 220 

Irregular prismoids, volume 108 

numerical example 109 

sections in earthwork, computation of area 107 

Joints, framed trestles 170 

rail 279-285 

Journal friction of axles 431, 6 

Kinetic energy of trains 514 

Kyanizing (bichloride-of-mercury or corrosive sublimate process) for 

preserving timber 252 

Ladder tracks , 381 

Land and Land damages, cost , 443 

Lateral bracing for trestles , 175 

Length of rails 270 

a simple curve , 49 

a spiral 81, 83 

Level, dumpy, adjustments of — Appendix p. 619 

wye, adjustments of — Appendix , , . . p. 6x7 



INDEX, 835 

Leveling, location surveys 20 

Level sections, volume of prismoids surveyed as 102 

numerical example 103 

Life of locomotives 415 

Limitations in location of track 64 

of maximum curvature 512 

Lining of tunnels 200 

Loading earthwork, cost 139 

of trestles 188 

Local traffic, definition and distinction from through 498 

Location of stations at distance from business centers, effect 476 

Location Surveys — paper location 18 

surveying methods 20 

Locomotive coaling stations 356-359 

coal conveyors 359 

coaling trestles 358 

hand shoveling 356 

locomotive crane 357 

rating 467 

resistances 429 

Locomotives, cost of renewals and repairs 488 

general structure 401-414 

life of 415 

resistance — Table XXIX 429 

types permissible on sharp curvature 509, & 

Logarithmic sines and tangents of small angles — Table VI ... . pp. 660-662 
sines, cosines, tangents, and cotangents — Table VII. pp. 663-707 
versed sines and external secants — Table VIII ... pp. 708-752 

Logarithms of numbers — Table V pp. 640-659 

Long chords for a 1° curve — Table II pp. 632-634 

of a simple curve 51 

Longitudinal bracing of a trestle 1.74 

Longitudinals (rails) 239, 264 

Loop — see Spiral. 

Loosening earthwork, cost 138 

Loss in traffic due to lack of facilities 476 

Maintenance of equipment, as affected by pusher engines 528 

cost of 488 

Maintenance of way as affected by pusher engines 528 

cost of 485-487 

Mallet locomotives, wheel-base 400 

Map chest, for field parties 31 

Maps, use of, in reconnoissance 1, 6 

Mass curve, area 133 

properties 132 

diagram, effect of change of grade line 135 

haul of earthwork 131 

value 134 

Mathematical design of switches 304-312 

Measurements, location surveys . v •. s •.•... •. 21 

Mechanism of brakes. ....,....•..,<.-... , , 423-425 



836 INDEX. 

I^edical and surgical treatment 36-45 

METAL TIES — see Ties, metal 258-263 

Metric curves , . 47 

Middle ordinate oi a simple curve 51 

Mileage, car , . , 495 

locomotives, average annual » 415 

Mikado locomotive, power of one typical engine under various con- 
ditions 465 

wheel-base 400 

Minor openings in roadbed 228-230 

Minor stations, rooms required, construction 329 

MISCELLANEOUS STRUCTURES AND BUILDINGS.— Chap. XII. 

Modifications in location, compound curves 69 

simple curves 63 

Mogul locomotives, wheel-base , 400 

Monopoly, extent to which a railroad may be such 471 

Mountain routes — reconnoissanoe. 5 

♦• Mud " ballast 232 

sills, trestle foundations 173, b 

Multiple story construction for trestles 171 

Myer's formula for culvert area 214 

Natural sines, cosines, tangents, and cotangents — Table IX . . . pp. 753-775 

versed sines and external secants — Table X pp. 776-798 

Non-competitive traffic, definition 498 

effect of variations in distance 502, 503 

extent of monopoly 471 

Notes— form for cross-sectioning 12 

location surveys 21 

reconnoissanoe 7 

Number of a frog, to find 298 

of trains pex. day, probable 474 

Nut-locks, design 295 

Obstacles to location of trackwork ..,.,. 62 

Obstructed curve, in curve location. . . , , , . 62, c 

Odometer, use in reconnoissanoe 8 

Oil-burning locomotives 408 

houses 360 

Old-rail culverts 224 

Open cuts vs. tunnels 208 

OPERATING EXPENSES.— Chap. XX. 

detailed classification — Table XLI 485 

per train mile 481 

reasons for uniformity per train mile 482 

Operation of trains, effect of curvature on , 509 

Oscillatory and concussive velocity resistances, train 430 

Ordinates of a spiral 78 

Paper location in location surveys 18 

preparation of notes for field-work 19 

Physical tests of steel splice bars 285 

steel rails 274-14 

Picks, use in loosening earth 138, b 



INDEX. 837 

Pile bents 161, 165 

driving 162, 167 

driving formulae 163 

points and shoes 164 

trestles, cost 168 

design 165 

PILE TRESTLES 161-168 

Piles, timber, specifications 166 

Pilot truck of locomotive, action 399 

PIPE CULVERTS 218-221 

advantages. ■. 218 

construction 219 

iron 220 

tile .' 221 

Pipe compensator, '.y.-.-.m 393 

Pipes, use in block signaling-. 393 

Pit cattle guards 228 

Platforms, station ■ 328 

Ploughs, use in loosening earth 138, a 

Point of curve. 51 

inaccessible, in curve location . 62, b 

Point of tangency 51 

inaccessible, in curve location 62, b 

Point-rails of switches, construction 300 

Point-switches 300 

Pony truck of locomotive, action 399 

Portals, tunnels, methods of excavation 207 

Posts, trestle, design of 191 

Pounds of steam per I.H.P. hour at various cUt-offs 455 

per pound of coal 452 

POWER OF A LOCOMOTIVE— Chap. XVIII. . 

Preliminary financiering of railroads, Chap. XIX, and 441 

Preliminary surveys — cross-section method 11 

"first" and "second" 17 

general character 10 

value of re-surveys at critical points 17 

Preservative processes for timber, cost 256 

general principle 249 

methods 249-255 

Prismatic compass, use in reconnoissance 8 

Prismoidal correction for irregular prismoids, approximate value 115 

in earthwork computations, comparison of exact 

and approximate 
methods.. . . . 116, 117 
for equivalent sections 113 
for irregular sections. 115 

for level sections 112 

for three-level sec- 
tions 114 

for triangular pris- 

moid Ill 

formula, proof 110 



838 INDEX. 

Prismoids, in earthwork computations 97 

Profit and loss, dependence on business done 472 

small margin between them for railroad promoters 470 

Profits (and security) in the two general classes of railroad obligations . . 469 
Profit, in earthwork operations 147 

PROMOTION OF RAILROAD PROJECTS.— Chap. XIX. 

Provisions, for camping 32 

Pumping, for locomotive water-tanks 325, 326 

Pusher grades 523-528 

comparative cost 528 

general principles 523 

required balance between through and pusher grades. . . 524 

required length 527 

Pusher engines, cost per mile — Table XLIV. .'. 528 

operation 526, 

service 528 

Radial stays, in locomotive boilers 403 

Radiation from locomotives 410 

into the exhaust-steam 410 

Radii of curves — Table I pp. 628-63 1 

Rail bending and depression due to traflfic 541 

braces , 286 

expansion, resistance at joints and ties to free expansion 293 

RAIL FASTENINGS.— Chap. X. 

Rail gap, elTect of, at joints 281 

joints 279-285 

effect of rail gap 281 

efBciency of any type 280 

failures 283 

standard designs and dimensions, Table XXIV 284 

specifications 285 

" supported " 280, 282 

" suspended " 280, 282 

theoretical requirements for perfect 279 

sections 266, 267 

A.S.C.E. 267 

" bridge " 266 

" bull-headed " 266, 267 

compound 281 

" pear " section 266 

radius of upper corner, effect 267 

reversible 267 

" Stevens " 266 

" Vignoles " 266 

stresses due to counterbalancing of locomotives 545 

wear, experimental determination 277 

RAILS.— Chap. IX. 

angles and dimensions of standard designs. Table XXIII 267 

branding 273, d 

cast-iron 266 



INDEX. 839 

Rails, chemical composition 273, a 

classification 273, e 

cost .- . , 278, 447, c 

■ cost of renewals of 278, 485 

dimensions and drilling > 273, e 

■ effect of stiffness on traction 269 

expansion 271 

stresses caused by prevention of expansion 271 

rules for allowing for 272 

finishing, .v. * 273,/ 

flow of metal i 275, a 

inspection ........... i 273, a 

intensity of pressure on .... » i 275 

length 270 

allowable variation 273, e 

45- and 60-foot rails 270 

life 274 

No, 2. . 273, c 

physical requirements 273, 6 

relation of weight, strength, and stiffness 269 

specificationa. 273 

temperature when exposed to &un . 272 

testing ..., ^ .,.,.. 273, 6 

tons per mile— Table XXXI 447, c 

wear on curves .......*, 276a, 277 

tangents 276, 277 

weight, for various kinds of traffic 268, 447 

Rates based on distance, reasons 497 

through, method of division of 499 

Rating of locomotives 467 

Receipts (railroad), effect of distance on 498-505 

Reconnoissance over a cross-country route 4 

surveying, leveling methods 7 

surveys 1-9 

character of 1 

cross-country route ■ 4 

distance measurements 8 

mountain route 3 

selection of general route 2 

value of high grade work 9 

through a river valley 3 

Reduction of barometer reading to 32° F. — Table XI p. 799 

Reheaters, in locomotives .......... i 406 

Reinf orced-concrete culverts ;.....'... 225 

ties 265 

Renewal of rails, cost of 485 

of ties, cost of 246, et seq. 485 

regvilations governing it 246 

Repairs and renewals of locomotives^ cost. 488 

Repairs of roadway, cost of 486 

Repairs, wear, depreciation, and interest on cost of plant; cost for earth- 
work operations / ...>•.•...-.-.-.•.•.•.%-.-..,,. 145 



840 INDEX. 

Replacement of a compound curve by a curve with spirals ..,.»...,,» 83 

simple curve by a curve with spirals 81 

Requirements, nut-locks 295 

perfect rail-joint 279 

spikes , 289 

track-bolts 293 

Resistances internal to the locomotive •r»o:«j$c4i^Ki • . • 429 

(see Train Resistance.) 

Retardation-speed curves 463 

Revenue, gross, distribution of 480 

Roadbed, form of subgrade , , 93 

width for single and double track 92 

Roadway, cost of repairs of .... 486 

as affected by pusher engines 528 

Roadways, earthwork operations, cost of keeping in order 143 

Rock ballast 232-236 

Rock cuts, compound sections , 91 

Rolling friction of wheels 431, a 

ROLLING STOCK.— Chap. XIV. 

Rotative kinetic energy of wheels of train 435, 514 

Route, selection as affected by locomotive power 466 

Rules for switch-laying 313 

Ruling grades 519-522 

choice of 520 

definition 3, 519 

proportion of traffic affected by 522 

Run-off for elevated outer rail 73 

Sand houses 362 

used for ballast 232 

Scales, track 383 

Scrapers, use in earthwork 140, d 

Screw-spikes, as rail-fastenings 291 

Section houses, value, construction 340 

tool houses. . 361 

Selection of a general route for a railroad 2 

Semaphore boards, in block signaling 392 

Setting tie-plates, methods. 288 

Shafts, tunnel, design 201 

surveying 195 

Shifting centers for locomotive pilot trucks, action 399 

Shoveling (hand) of earthwork, cost 139, a 

(steam) of earthwork, cost 139, b 

Shrinkage of earthwork , 127 

allowance 128 

Side-hill work, in earthwork computations 119 

correction for curvature 122 

Signaling, block, " absolute " blocking 388 

automatic 389 

manual systems 386-388 

permissive < 388 

Signals, mechanical details 392 



I 



INDEX. 841 

Signs 372-374 

division posts 374 

highway signs 372 

marker posts 374 

mile posts , ....,..;.,,,.., 374 

trespass signs ,...,.•>>•«•> 373 

whistle signs 374 

Sills for trestles, design , 192 

Simple curves ...,,,..,.. 46-66 

Skidding of wheels on rails 421-422 

Slag, used for ballast ^ .. ^ ....... . 232 

Slide-rule, in earthwork computations. .i««ei:.a. ^ . . . 106 

Slipping of wheels on rails, lateral 396 

longitudinal 395 

Slips, for switchwork 314 

Slopes in earthwork, for cut and fill 90, 92 

effect and value of sodding 95 

Slope-stake rod, automatic 100 

Slope-stakes, determination of position 99 

Snake bites, treatment 44 

Snow fences .*....... 364 

sheds .,,,,.,,,..... 365 

Sodding slopes, effect and value • * • •, ' 95 

Spacing of ties 244 

Span of trestles ...,.,,.. ....,.,.. 172 

Specifications for earthwork , 157 

steam shoveling, earthwork 139, 6 

steel rails 274 

steel splice-bars . , .,..,,.. 285 

timber piles , . *»..•* • • ..... 166 

wooden ties ,.,.,..,.... ii^,^,Vfc .^kj^ « ),, , ,'. , . . 245 

Speed of trains, reduction due to curvature 508, a 

relation to superelevation of outer rail ., 71, 72 

relation to tractive adhesion 422, e 

Spikes 289, 291 

cost . . . . ,.,..., 447. 4 

driving. 290 

number per mile of track — Table XXXIII 447, e 

screw.. 291 

requirements in design 289 

" wooden " for plugging spike-holes 292 

Spirals, bridge and tunnel 5 

(see Transition Curves.) 
Splice-bafs— gee Angle-bars. 

Split stringers, caps, and sills 161, 177 

Spreading earthwork, cost 142 

Stadia method, form of notes 14 

for preliminary surveys 13 

for reconnoissance 8 

methods of work 13 

organization of party 13 

reduction of observations 15 



842 INDEX. 

Stadia method vs. cross-section method 16 

Stand pipes for locomotive water supply , 327 

Starting grade at stations, reduction of 537 

Station buildings, cost 329 

platforms 328 

Staybolts for locomotive fire-boxes . . . . i ; i * 403 

Stays, in locomotive fire-box 403 

Steam pile-drivers 162 

Steam-shoveling of earthwork *,....*....;... 139, 6 

weight per foot of stroke . 454, 455 

Stiffness of rails, effect on traction . ; ; ; i 269 

Stocks of railroads, security and profits 369 

Stone ballast 232-233 

box culverts 223 

foundations for framed trestles 173, c 

Straight connecting curve between two parallel curved tracks 312 

from a curved main track 310 

Strength of timber 187 

factors of safety 189 

required elements for trestles 186 

STRESSES IN TRACK.— Chap. XXV. 

Action of track as an elastic structure ^ . . . < 539 

Bending moment and depression of rail 541 

Counterbalancing — effect on track stresses. .....•.■.-.. .-.•.■. 545 

Depression of track for static loads 540 

Instruments, special, used for tests 542 

Nature of the subject 538 

Pressure by ballast on ties ............; 543 

Rail stress 545 

Transverse stresses in ties 544 

Stringer bridges, standard, steel ; 230 

Stringers, design 190 

for trestle floors 177 

Stub-switches ; 299 

Subchord, length . . 48 

Subgrade, of roadbed, form ;;;;;... ^ . ^ ;....;...... 93 

Superelevation of the outer rail on curves, L. V. R. R. run-off 73 

on trestles 181 

practical rules 72 

standard on N. Y. N. H. 

&H. R. R 72 

Table XIX 71 

theory 71 

Super-heaters, in locomotives 405 

Superintendence, cost in earth operations 146 

Supported rail-joints ..........;.;... 282 

Surface cattle guards ^ . ; . , 228, 6 

surveys for tunneling 194 

Surveying parties, maintenance 23-35 

number of men required 22 

Surveys and engineering expenses for railroads, cost 442 

accuracy .........,,,. 197 



INDEX. 843 

Surveys for tunneling 194-197 

with compass 11 

Suspended rail-joints 282 

Swinging pilot truck on locomotive 399 

Switchbacks 5 

Switch construction 296-303 

essential elements 296 

frogs 297, 298 

guard rails 303 

point 300 

stands 301 

stub 299 

tie rods 302 

SWITCHES AND CROSSINGS.— Chap. XI. 

Switches, curved lead rail, rectangular coordinates — Table XXV . . 313 

mathematical design 304-307 

using circular lead rails 304 

using straight frog rails and straight 

point rails 305 

resistance of cars through 438 

Switching engines, wheel-bases 400 

used in pusher-engine service 526 

Switch leads and distances — Table III pp. 635, 636 

laying, practical rules 313 

slips 314 

stands 301 

Tables, dining, camp 28 

drawing, camp 30 

Talbot's formula for culvert area 214 

Tamping for blasting 153 

Tangents for a 1° curve — Table II pp. 632-634 

Tangent distance, simple curve 51 

Tanks, water, for locomotives 324 

track 326 

Temperature allowances, while laying rails 272 

Ten-wheel locomotives, wheel-base 400 

Telegraph lines for railroads, cost 450 

Tent floors 26 

stoves 27 

Tents 25 

TERMINALS.— Chap. XIII, 

inconvenient, resulting loss 476 

justification for great expenditures 476 

Terminal -pyramids and wedges, in earthwork 89 

Tests for splice bars 285 

for rails 274 

to measure the efficiency of brakes 426 

Three-level sections in earthwork, determination of area 105 

numerical example 105 

Throw of a switch 304 

Through traffic, definition 498 

division of receipts between roads 499 



844 INDEX. 

Througli traffic, effect of changes in distances on receipts 500 

Tie-plates , 286-288 

advantages ....,,..,.,.... 286 

elements of design 287 

method of setting 288 

Tie rods, for switches ; 303 

TIES.— Chap. VIII. 

cost of renewal of 246 et seq. 

metal 258-263 

durability 260 

economics . 261 

extent of use 258 

form and dimensions 259 

number per mile of track — Table XXX 447 

number and value, used in U. S. in 1915 — Table XXII 241 

on trestles 180 

reinforced concrete 265 

wooden, preservative processes . . . 249-256 

regulations for relaying 246 

Ties, wooden 241-257 

choice of wood 241 

construction 245 

cost 249, 447, b 

dimensions 243 

durability 242 

economics 240 

quality of timber 245 

> spacing 244 

specifications 245 

transverse stresses under traffic 544 

Tile drains, to drain roadbed 94 

pipe culverts 221 

Timber, choice for trestles 183 

piles 161 

ties 241 

moduli of rupture — Table XX 187 

strength of 187 

working unit stresses — Table XXI 187 

Topographical maps, use of, in reconnoissance 6 

Track, action as an elastic structure 539 

bolts, average number in a keg of 200 pounds 447, d 

cost 447, d 

design 294 

essential requirements 293 

for various weights of rail — Table XXXII 447, d 

number required per mile — Table XXXV 447, d 

circuit for automatic signaling 394 

depression for static loads 540 

laying on railroads, cost 447, ? 

scales 383 

stresses in — Chap. XXV 



INDEX, &4S 

Tractive power of locomotives — Tables XXXIX and XLIII and 411 

effect of grade 461 

relation to boiler and cylinder power ... 414 

variation with velocity 457, 460 

Tlraffic, classification of 498 

estimation of probable volume 473 

TRAIN-BRAKES 421-427 

automatic 425 

brake-shoes 427 

general principles 42 1 , 422 

hand-brakes 423 

straight air-brakes , 424 

tests for efficiency 426 

'IVain length limited by curvature 509, a 

maximum on any grade 521 

loads, methods of increasing 532, 6, 535 et seq, 

TRAIN RESISTANCE.— Chap. XVI. 

formulae for freight cars 439 

passenger cars 439, a 

through switches 438 

Train service, cost of> 492, and Table XLI 541 

supplies and expenses, cost of, in conducting transportation 493 

and Table XLI 541 

wages — see Train service. 

Transfer cranes in freight yards 382 

Transit, adjustment of— Appendix pp. 6i3-<5i7 

Transition-curves 71-83 

Table IV pp. 637-639 

application to compound curves 82 

field-work 80 

fundamental principle 74 

replacing a compound curve by curves with spirals . . 83 
simple curve by a curve with spirals. ... 81 

required length . 76 

symbols 77 

their relation to tangents and simple curves 79 

to find the deflections from any point 78 

ordinates Y8 

use of Table IV 78 

varieties 75 

from level to inclined track 73 

Transportation, surveying parties 34 

TRESTLES.— Chap. IV. 

cost , 184 

extent of use 158 

framed 169-184 

Trestle, pile 161-168 

posts, design - 191 

required elements of strength 186 

sills, design 192 

stringers, design 190 

timber 183, 187 



846 INDEX. 

Trestles vs. embankments 159 

Trimming cuts to proper cross-section 144 

Trucks, car 418 

four-wheeled, action on curves 396 

locomotive pilot 399 

with shifting center 399 

TUNNELS.— Chap. V. 

cost 209 

vs. open cuts 208 

Tunnel cross-sections 198 

design 198-202 

drains 202 

enlargement . 204 

grade 199 

headings 203 

lining 200 

portals 207 

shafts 195, 201 

spirals 5 

Turnout, connecting curve from a straight track 308 

from a curved tract to the outside 309 

to the inside 310 

dimensions, development of approximate rule 306 

from inner side of curved track . 307 

from outer side of curved track 306 

Turnouts with straight point rails and straight frog rails, dimensions 

of — Table lit .' pp. 63S. 636 

Turntables for locomotives 292 

Underground surveys in tunnels 196 

Undulatory grades, advantages, disadvantages, and safe limits 518 

Unit chord, simple curves 46 

Useful formulee and constants — Table XV p. 803 

trigonometrical formulae — Table XIV pp. 801, 802 

Valley route — reconnoissance 3 

Velocity head applied to theory of motion of trains 514 

as applied to determination of train resistance 437 

of trains — Table XLII 515 

Velocity of trains, method of obtaining 535 

resistances, train 430 

Ventilation of a tunnel during construction 206 

Vertex inaccessible, curve location 62, a 

of a curve 51 

Vertical curves, mathematical form 86 

necessity for use 84 

numerical example 87 

required length 85 

Virtual grade, reduction of 535-537 

profile, construction of 515 

use, value, and possible misuse-. 517 

Wages of engine-men , 490 

trackmen 486 

trainmen 492 



INDEX. 847 

Wagons, use in hauling earthwork 140, b 

Water for locomotives, chemical qualities 319 

consumption and cost 318, 325 

methods of purification 319-321 

reagents for removing corrosive or incrusting matter — Table 

XXVI 321 

stations and water supply 318-327 

location 318 

pumping 325 

required qualities of water 319 

stand-pipes 327 

tanks 324 

track tanks 326 

table in locomotive fire-box 403 

tanks for locomotives 324 

protection from freezing 324 

way for culverts 212-217 

Watering stock 469 

Wear of rails on curves 276 

on tangents 275 

Weight of rails 267, 268 

Westinghouse air-brakes 425 

Wheelbarrows, use in hauling earthwork 140, c 

Wheel-bases of locomotives, types 400 

Wheel resistances, train 431 

Wheels and rails, mutual action and reaction 395-399 

effect of rigidly attaching them to axles 395 

White oak, use for trestles 161, 183, 187 

ties 242 

Wire-drawn steam 410 

Wires and pipes, used in block signaling 393 

Wooden box culverts 222 

spikes for filling spike holes 246, 292 

Wounds, treatment 45 

Yard-engine expenses 489 

YARDS AND TERMINALS.— Chap. XIII. 

Yards, definitions 377 

engine 384 

general principles 378 

hump - 380 

ladder tracks 381 

minor freight 379 

transfer cranes 383 

,track scales 382 

value of proper design 376 

Zinc-creosote, emulsion process 254 

two-injection process 255 

-tannin process for preserving timber 253 



